Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1765
3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo
Thomas Kerler Volodymyr V. Lyubashenko
Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners
123
Authors Thomas Kerler Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, Ohio 43210, USA
Volodymyr V. Lyubashenko Institute of Mathematics National Academy of Sciences of Ukraine 3, Tereshchenkivska st. Kyiv-4, 01601 MSP, Ukraine
E-mail:
[email protected] E-mail:
[email protected] Cataloging-in-Publication Data applied for
Mathematics Subject Classification (2000): 16W30, 18D05, 18D10, 18E10, 57N10, 57N13, 57N70 Physics and Astronomy Classification (1999): 11.10.Cd, 11.10.Kk, 11.25.Hf, 11.25.Sq ISSN 0075-8434 ISBN 3-540-42416-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10847519
41/3142-543210 - Printed on acid-free paper
Contents
0.
0. 1
0.2 0.3 0.4
1.
.........................
..........................................
..........................................
.......
The Double
Category of Framed, Relative 3-Cobordisms Category of Surfaces with Boundaries
..........
4.
1.3
Consequences of the Double Category Picture Mapping Class Groups, Framed Braid Groups, and Balancing Some Facts about Handle Decompositions b -4 Obb The Central Extension 04 -+ Basic
.............
35 51
.....................
68
...
4.2 4.3
97 99
.....................
104
..........
109
......................................
116
...................
143
...................
153
Isomorphism between Tangle and Cobordism. Categories 3.1 Trading and Eliminating Handles Stratified Function Spaces and External Strands on W 3.2 3.3 From Tangle Classes to Cobordism Classes 3.4 Verification of Compositions
..........
166 173
.............................
175
............
187
.....................
199
.................................
207
categories and monoidal 2-categories categories Hopf algebras in braided categories Abelian categories form a monoidal 2-category Ribbon monoidal.
27
......................
...............
Monoidal
15
.....
Tangle-Categories and Presentation of Cobordisms 2.1 Local Ingredients of Tangle-Diagrams and Horizontal 1-Arrows 2.2 Admissible Tangles and Vertical 1 -Arrows 2.3 Equivalence Moves of Tangles, and the 2-Arrows in Tg1 2.4 Tangles in Three-Space 2.5 Alternative Calculi and Further Equivalences 2.6 Compositions and 7'gl as a Double Category 2.7 Special Cases and Applications
4.1
8
18
2-Arrows from Cobordisms with Comers
...............................
3.
6
23
1.2
1.6
1 3
......................
The 0- 1 -Arrow
1.5
1
............
1.1
1.4
2.
Summary of Results ........................... Atiyah's TQFT Axioms via Categories Double Categories Extended TQFT's Statement of Main Result on the Class of Extended TQFT's
Introduction and
..................
217
..................................
217
...........................
226
..................
242
V1
5.
6.
Contents
Coends and construction of Hopf algebras 5.1
The coend
5.2
Braided function
........................
6.2
Colorations, Natural Transformations, and Liftings
6.3
Topological Invariance Compositions over Colored Surfaces Lifting V (M) to Color-Independent Natural Transformation Horizontal Compositions Topological moves imply the modularity
6.7
................................................
..............
294
....................................
299
.......................
304
............................
7.3
Sketch of the construction of enhanced TQFT
7.4
Examples
categories as a
...............................
double functor in the extended
case
315
.................................................
Quantum Field Theory to Axiomatics Theory and Conformal Field Theory A.2 Developing the Axiomatics for Extended TQFT's A.3 Generalized TQFT's in Gauge Theory
.....................
Categories and Double Functors Categories Double pseudofunctors
C.2 Index
tangles
335
.......
335
................
338
.........................
341
........................
343
..........................................
343
......................................
345
...............................................
bicategory of thick tangles Representation of thick tangles by abelian categories Monoidal
313
315 320
Witten-Chern-Simons
Thick
.....
313
...................
From
C. 1
283 284 292
Formulation of TQFT
B.2
283
.......
7.2
Double
270
290
Enhanced cobordism.
B. 1
261
..........................
7.1
Double
261
......................................
Generalization of a modular functor
A. 1
C.
....................................
Main result
6.6
B.
algebra
6.1
6.5
A.
.................................................
Construction of TQFT-Double Functors
6.4
7.
.......................
353
..........................
353
.............
365
..........................................................
377
0. Introduction
and
Summary of Results
decade quantum field theory and string theory have strongly impacted of low dimenthe geometry and topology especially many areas of mathematics, mathematical were structures of intriguing In particular, a wealth sional manifolds. theories called inherent (TQFT's) be to so discovered to quantumfield topological In the last
these notions refer to a class of conformal field theories (CFT's). Originally, Chernthree dimensional which field theories, among physical quantum conformal field theory are some of the rational Simons theory and two dimensional that the abstract most prominent ones. It was soon realized setting of category theof these and structures of data the zoo to it makes organize efficiently possible ory demathematical into evolved notions, field theories. purely TQFT's Eventually, Axiomatic and functors. TQFT's in the language of categories fined axiomatically and
concrete
and similar
theories
gebraic topology,
homology.
Atiyah
into
an
axiomatic
framework
pendently
and at about the
definition
of CFT's,
time G.
was
the first
to other
in his seminal
Segal [Seg88]
functors
mathematician
[Ati88].
work
formulates
a
in al-
to cast
the
Inde-
mathematical
The noand functors. based on categories which very similarly constructions which and here introduce our will on that we TQFT's
of extended
tion
same
similar
rather
in nature
as
TQFT's
of
notion
therefore,
are,
such
higher category theory, namely double categories and double notion of a TQFT in dimension three and both Atiyah's of CFT notion categorias special cases, though they appear on different Segal's of unification and conceptual will not only be a natural cal levels. The definition of allow to construct will new classes abstractions us but further theories, previous other from different that are manifestly TQFT's, TQFT's, namely non-semisimple defined ones and in some cases describe TQFT's based on classical combinatorially be based involves
will
functors.
It thus
gauge theories. In order to
TQFT.
Let
boundaries. scribe
0.1
explain
start
This
manifolds
Atiyah's
Following to a
us
contains
with will with
of an extended we give next our definition Atiyah's axioms for manifolds with smooth be subsequently using double categories to degeneralized, to define an extended TQFT. and double functors corners the main results a
of
TQFT Axioms via Categories
to the axioms
d-dimensional
recollection
of
oriented
Atiyah manifold
[Ati88], Ed
a a
TQFT V
vector
T. Kerler, V.V. Lyubashenko: LNM 1765, pp. 1 - 14, 2001 © Springer-Verlag Berlin Heidelberg 2001
space
in dimension
V(Ed)
,
d + 1
assigns
and to an oriented
2
Introduction
0.
and
manifold,
d + 1-dimensional manifolds -
if
-Zod Zod with
z0g,
is
we
glue
Summary of Results
and
the
v
d,
a
whose
opposite
two such d + 1-manifolds
of in their
tion
of the linear
maps of the individual the language of categories
Using Atiyah's axioms Definition
0.1.1 between
functor
concisely
very
([Ati88]). symmetric
Here k-vect
a
of characteristic closed
objects
Zod
mensional
manifold,
union
of d-dimensional
map for d + 1-manifolds.
and functors,
as
in
[Mac88],
The manifold
field
that
requires
d-submanithe
composistate
we can
follows:
quantumfield topological theory in dimension categories [Mac881 asfollows:
Cobd+1
:
d is
a
--+
k-vect.
and Zd is
veca
field
morphisms, between composition.
of
maps with the usual d-dimensional oriented
d-manifolds
disjoint
monoidal
0. The set
the set of linear
a
the category, whose objects dimensional are finite for instance, k, which we assume to be perfect,
denotes
over
spaces
as
A
V
tor
is
along a common(closed) the composite has to be
together
the linear
boundaries,
fold
boundary
-+ V(Zld). : V(Zod) map V(Md+l) orientation. The gluing axiom in [Ati88]
linear
manifolds.
two vector spaces is simply The category Cobd+1 has as A morphism. between two such
meaning an oriented d + 1-di_ZOd Lj Zd is the disboundary gMd+l speaking we consider as morphisms (Strictly joint union of the two d-manifolds. cobordisms modulo relative Given another homeomorphisms or diffeomorphisms). cobordism Nd+1 between r 1d and Z2d in the above sense, we define the composite Md+1 U_rdN d+1 The union U_,,t stands for the quotient space by Md+1 o Nd+1 of the disjoint union, in which we have glued the two d + 1-manifolds along the Zd The identity surface Zd in their boundaries. on a d-manifold common (closed) d in Cobd+1 is easily identified as the (class of) the cylinder E X [0, 1] with canonical boundary identifications. Atiyah's gluing axiom is now implied by functoriality: a
Md+l,
d + 1-cobordism,
whose
=
,
=
V(M o N)
=
.
V(M)
-
The term monoidal structures
on
the two
V(N). in Definition
categories.
0. 1. 1
The tensor
means
product
V respects
that
the natural
tensor
Cobd+1 is given by disjoint product Ok. These conditions
on
by the usual tensor product on k -vect the remaining set of axioms from [Ati88]. form Atiyah's axioms associate to any d + 1 -manifold Note, that in their original Mwithboundary a vector V(M) in V(OM). The assignment of linear maps and from the additional and axioms for tensor products rule follow the composition union,
and the
allow
us
duals
via
-
to infer
V(OM)
the identifications
=
V(-Zo
U
Zi)
-
V(Zo)*
0
V(Zl)
Hom(V(Zo), V(Zi)). inspired by Witten's investigation particular, theory, giving rise to a TQFT with 3-dimenused by sional cobordisms, (d 2). Although the functional integral formulation the implied Witten is a priori it lends itself not rigorous, nicely to illustrate propin Appendix A. Witten also writes down formulae for erties of a TQFT as outlined for some closed defined partition the heuristically sums of the Chem-Simons theory The axioms
[Wit89]
in
[Ati88]
were,
of the Chern-Simons =
field
in
manifolds ical
invariants
them as topologbut the arguments used to identify to simultaneous Almost from and far rigorous. physical and Turaev, in their ground breaking paper [RT9 11, gave
via surgery,
obtained
purely
are
work Reshetikhin
Witten's
Categories
Double
0.2
invariants of the 3-manifold using quantum groups as well as rigorous definition invariants These can be considered for their a systematic computation. procedure The Chem-Simons of Witten's realization mathematical generalization theory. as a and TQFT's is developed in detail in Turaev's to cobordisms of their constructions a
book [Tur94]. At around
cal structures therein. tum
the
time
Segal,
field
[Wit89] also realized that the restriction theories)
field
boundary OMyields
in CFT is that
be "sewn
one
along
together"
on
(again of
precisely surfaces
considers
Moore and
theories
Witten
Mto its
can
same
conformal
for
on a
a
in their
the circles
of
theory important
CFT. The
boundaries
[MS89] and references level
the heuristic
so
categori-
similar
found
see
Chern-Simons
such with
Seiberg
surfaces,
that
boundaries.
physical
on a
quan3-manifold
new
surfaces In the
ingredient
themselves
physical
in-
the locations
or punctures give terpretation observables in Cheminto the theory. The corresponding where charges are inserted that run along Wilson lines, thus creatin 3-dimensions Simons theory are currents the their end bounding surface. See Appendix A for a more points on ing a charge at from which we excise consider This leads to detailed 3-cobordisms, us exposition. at an end point of a line on of embedded lines. The excision tubular neighborhoods
in the 2-dimensional
the holes
thus results
the surface
in the removal
of
a
surface
disc from the surface
at this
location.
sewing operations of theory, we need to excisions tubular obtained with 3-manifolds the them comers extend to along by formulated be of notion can the no a As lines. by a consequence, TQFT longer but we have to pass to higher category theory. ordinary categories and functors, is the of double categories of an extended TQFT as a double functor The definition In order
the related
subject
generalize
to
CFT's besides
of the next
0.2 Double
Atiyah's the gluing
axioms to include
also the
axioms of Chem-Simons
paragraphs.
Categories
Atiyah's TQFT axioms and the axioms of CFT rehigher category theory was realized by many people quires which allows us is that of a 2-category, The simplest generalization independently. let us give the basic In this section to talk about morphisms between morphisms. will be definitions The outlined and double categories. of 2-categories definitions of the details For book. on in this more for the constructions sufficient theory given or the the reader refer [KV94]. to [KS74], [B6n67], we original papers 2-categories See also Appendix B. 1. of an ordinary consists it firstly A 2-category category (ti, with objects and between a as as them, operation between 1-morcomposition I-morphisms ,well to In we associate and if the addition, are source matching. objects target phisms the with source and same : -+ Ot : two 0. Ot Ai Q, Af I-morphisms any The fact
the
that
use
a
unification
of
some sort
of
of
Introduction
0.
4
and
HOM2(Ai,
Summary of Results
Af )
2-morphisms, denoted B : Ai =: Af Wehave a (vertical) 2-morphisms if the target 1 -morphism composition operation of of the one coincides with the source 1-morphism of the other. The composition extends second of of to a (horizontal) composition 2-morphisms 1-morphisms type over an intermediate object. Finally, the two compositions are required to be mutually distributive. The objects Cat of categories. One standard example is given by the 2-category small categories that of Cat are essentially to are (such categories whose equivalent form of functors from the class consists a Horn, (C, Ct) set), 1-morphism objects for functors between the of and is the to same pair HOM2(Fi Tf), Ct, C, categories, from Fi to Ff. The vertical and horizontal transformations set of natural composiand target
a
set
of
.
between two
I
of natural
tions
be interested
transformations in its
k-linear,
abelian
left
functors,
exact
Another
are
2-subcategory categories with and natural
in standard
additional
a
ways,
field
given
finiteness
[Mac8g].
see
Weshall
small, essentially (see Chapter 4),
k of
conditions
transformations.
of interest
2-category
given
AbCat for
Cob"'d+1
cobordisms,
is that of relative
-
The
1-morphisms objects category e i cobordisms Md and Nd beCobd from above. For two d-dimensional Xd-1 and yd-1, we can consider the closed d-manifold 1-manifolds tween d MdU (Zd-I x [0, 1]) UNd, where Zd-1 is the disjoint union Xd-1 Lj yd-1. In S 9Md _- Zd-1 x 0 and o9N d _- Zd-1 X 1. the definition of S we make identifications underlying
of
is identical
and
the cobordism
to
category
-
=
manifolds Wwith the 2-morphisms M- N are given by d + 1-dimensional 5--aW S. boundary The (vertical) composition of 2-morphisms over I -morphisms is given by gluing For the (horizontal) the d + 1-manifolds together over the bounding d-manifolds. the d 1-manifolds we over + together along the cylinders composition objects glue d 1-manifolds. over the respective source and target of 2-categories and In this book we define TQFT's using certain generalizations Double and double were 2-functors, doublefunctors. namely categories categories introduced by Ehresmann in [Ehr63a]. Let us give an equivalent version of his defiThe
-
nition.
Definition
0.2.1.
A double
Z
category
consistsfirstly
X and Y, there
For any pair ofobjects, vertical and horizontal
are
sets
of
a
class
Hom'(X, Y)
Zo of objects. and
HoMh(X, y)
The objects and the vertical 1-morphisms or 1-arrows. 1-morphisms by themselvesform an ordinary category Z1, and an analogous cate1-morphisms. gory Zh for the horizontal We call a square S a set of four objects 0, 1, two vertiXij, with i, j cal 1-morphisms gj E Horn, (Xoj, X1j) and two horizontal 1-morphisms fj E
of
=
Homh(X jo,
Xj1 )so
that
they
X00 S
=
got X10
can
be
arranged
in
a
square
diagram
as
follows:
X01
lol_1_4 zx,6 g1t 0,
X1 I
7
a
E
HOM2(S)
-
(0.2.1)
S,
For any square,
has
one
HOM2(S)
HOM2(S) of 2-morphisms.
set
a
Categories
Double
0.2
include
We often
an
above.
diagrammatic 1-morphisms, S', we have gi g6 for the vertical S' then we define the horizontal Oh S to be the one with vertical composite square I-morphisms fo'Oh fo and fo'Oh fo. A double I-morphisms go andgi, andhorizontal with is horizontal a composition equipped category element
E
a
Iffor
in the
S and
two squares,
Oh
HOM2(S')
:
notation
x
as
=
HOM2(S)
HOM2(SI
-+
S)
Oh
:
(a, 0)
i-
a
j6-
Oh
there is a vertical composition -y o, a declared if the target horizontal Analogously, 1 -morphism of y. with the source horizontal I -morphism of a coincides rise to to give We require both compositions (Dh,2 Oh) and (0,2 o,), categories In particu1 and whose objects are the vertical horizontal -morphisms respectively. '
Oh and o,
lar,
are
associative.
that two composition states law for double categories the interchange Finally, I -morphisms arhave twelve distributive. More precisely, are mutually suppose we 6 and and are 2-morphisms a, 0, -Y, ranged in a square ofsquares as depicted below, in the HOM2-sets of thefour squares of I-morphisms:
t
t can
be
the operations interchanged.,
(6 In
does
Appendix not require
From
a
B.2
only squares, Conversely, if
Oh
'Y)
Ov
we recall
(P the
for we
category which have
t
t
a
we
Oh
a)
the horizontal
(6
0
we can
have
Xoj (t
2-category
Ov
)
extract
Xij
we can
and gj
a
is
construct
of (t in (Ehr63b], the double category of quintets and vertical Wechoose both the horizontal categories
Hom 2D (S) The horizontal I
91
0
fo,
0
fo
=*
composition fl, 0 go 0 fo
of =
=
(ti
composi-
a).
ov
(0.2.2)
of Ehresmann
called
(t, that is, (Q(t)v 1 = egory underlying 1 -morphisms as in (0.2. 1) the associated
and vertical
which
[Ehr63a],
1-, and 2-morphisms.
readily =
(,y
Oh
definition
original
between 0-,
the distinction
double
t
of performing
that
We require tions
t
a
and
Hom"-'2 (g,
o
fo, f,
we
for
to be identical =
sets
o
(ti.
For
a
consider
j 0
0, 1. Q(t,
to the cat-
square
S of
are
go).
Q(t is 2-morphisms in 0 I 0 0 0 0 = fl fo f, fl, go. gI =
if
a double category as follows.
(Q(t)h 1
2-morphism
2-category the identity
the obvious
composite
0.
and
Introduction
0.3 Extended
Summary of Results
TQFT's
of a This quintet construction yields the first example relevant to our definition left Abelian categories, TQFT, namely the double category QAbCat, of k-linear and natural transformations. The precise definition of the topological exact functors
double category it is as follows. The set
numbered, Hoin
h
(a, b)
dibn
used in
our
definition
of
is given as IS" of objects : circles. union of a oriented disjoint
between two such 1 -manifolds
a
TQFF is E
a
Z>o},
more
involved.
where Sua is
The set of horizontal of connected
consists
In outline
a
fixed,
1-morphisms
oriented
surfaces,
by (and is homeomorphic to) _SUa U SUb All of the same genus g are homeomorphic, such surfaces so we may leave only one in each homeomorphism class, parametrized by g E Z,>o. Weshall, representative to keep several more convenient however, find it technically isomorphic copies of 29 horizontal morphismsl namely g+1 g of them for the class with genus g. To be with the the horizontal isms are in one to one correspondence mo more concrete, set of combinatorial plane graphs that consist of an interval at the boundary of the in the inand of g non-intersecting arcs in the half plane with endpoints half-plane The standard surface and illustrations. terval. See Sect. 7. 1. 1 for precise definitions of a thickening to the graph G is then obtained as the boundary ZG corresponding whose
of G in
boundary
is
parametrized
.
R3.
1-morphism set Homi (a, b) of aibn is empty if a 54 b. Weidentify Sa with the symmetric group of an a-element endomorphism set Homi (a, a) Hence, in a square S the two horizontal 1-morphisms ZO and ZH always lie in h same set Hom (a, b) as shown in the following diagram. The vertical
the set.
the
=
EG
S
=
0,1
b
t
4 ZH
-+
b
bounding circles of each of the standard surfaces homeomorphism. We now sew the surfaces parametrization circle source together by connecting the j-th source circle of ZG to the a(j)-th S' x [0, 1]. Here a E Sa is the left vertical in the boundary of ZH by a cylinder 1-morphism. of the diagram. Doing the same for the target circles, we obtain a closed surface Es. The cylinders lines in Chem-Simons theory. as Wilson are interpreted for a given square S is now a homeomorphism class of triples A 2-morphism. M [(M, 0, a)], where each triple (M, 0, a) consists of a compact, oriented 3-manifold with corners, M, a homeomorphism, 0 : OM=-+ Es, and a 2-framof a structure ing of its tangent bundle, a : TME) TM= R' x M. The additional 2-framing is motivated by the Chem-Simons gauge theory. In this book we choose as an extensions an equivalent by signatures of bounding 4-manifolds. description definition of this extension is given using so In [BHMV95] yet another equivalent The
are
a
source
and b target
numbered due to
=
7
Extended TQFT's
0.3
[Wal] and later Turaev [Tur94] define extensions by enlarging the set of objects to pairs (Z, L), where Z is a surface and L C H, (Z) a Lagrangian subspace to indicate a bounding handlebody. The stanMoreover, dard surfaces in our approach are thus equipped with fixed Lagrangians. called
Walker
Moreover,
p, -structures.
or
cobor-
formula.
signature
3-manifolds,
of the
extension
of linear
definitions
that is put into their via Wall's definition
cocycle involving Lagrangians in our is implicit dism categories the signature If we disregard the
we
and
have natural
two 3-mani-
and horizontal vertical compositions obtained by gluing pieces ZG in their boundary or the cylindrical together along the horizontal define a double catthat these compositions verifies One readily pieces respectively. with 2-framings so to 3-manifolds can be extended egory Cobn. The compositions can In we that the axioms of a double category are fulfilled. analogy to group theory well
defined
folds
dib
thus view
n
as a
1
extension
central
94
-4
bn
'
cobordism
S?4 is the smooth 4-dimensional
where
Cobn
cyclic group generated by [Cp2 ] and the signature sign(W) is an isomorphism (see, for example [GS99] There is
a
natural
of
notion
a
double
strict
it is defined
Analogous to ordinary functors 1-morphisms and 2-morphisms
(0.3.1)
1
_+
S?4
Note that
group.
map 124 =4 Z Sect. 9. 1).
between double
functor
:
is
a
free
[W]
categories.
map between classes of objects, with which is a functor categories,
as a
of two double
Its weak version (ordinary) category structure. is defined in Appendix B.2. pseudofunctor there is a natural that we consider, For objects of the main two double categories and for unions of circles, For dibn it is given by the disjoint tensor multiplication. We now are of abelian 0 1]. tensor [Del9 categories QAbCat by Deligne's product field of an extended topological in a position theory to give a definition quantum
-
and vertical
to both horizontal
respect
double
-
Definition
0.3.1.
TQFTover afield
An extended
V between double the level
categories of objects.
bn
:
k is
a
QAbCat,
--+
above, which is compatible
as
dolublepseudofunctor
with
tensor
structures
on
Hence,
V(Sua) associated
where C is the category interest
as a
Definition circle
generator,
0.3.2.
category
let
us
Let V be
of V is
then
give
an
a
_
...
ZC
to one
circle.
formal
definition
as
CV
Since this
TQFTin the
extended
defined
CM
=
V(S').
as
category
is of obvious
follows.
sense
of Definition
0. 3. 1. The
Introduction
0.
Statement
0.4
To state
our
monoidal
main result
let
us
will
adjectives 0.4.1.
monoidal,
braided,
be
the Class of Extended
on
that
note
A few additional
Definition
categories
circle
assumptions explained after
briefly
TQFT's
always carry following
lead to the
a
braided
definition.
the definition.
k is a bounded abelian, over a field rigid, that (ribbon) category C with a special Hopfpairing The endomorphism ring of the unit object is supposed to be k.
A modular
category
balanced
non-degenerate.
C is k-linear.
particular,
In
Summary of Results
of Main Result
structure.
The various
is
and
definitions
precise
The
and formulations
of these
given in Chapter 4. The notion of an of abelian category allows us to consider subobjects, quotients and decompositions of tensor products monoidal and rigid imply the existence objects. The properties X (9 Y and duals X' of objects. These notions are part of classical category theory The word braided implies in [Mac88]. as described a natural isomorphism. cx,y : X 0 Y =4 Y 0 X, which does not necessarily Balanced (or square to the identity. ribbon) refers to a natural isomorphisms X =4 X" compatible with the braiding. for example, by Joyal and Braiding and balancing in categories were introduced, Street [JS911, see also [RT90]. Wecall a category bounded if it is equivalent to a category of finite dimensional modules over a finite dimensional algebra. This turns conditions,
which
be
equivalent
out to
details
on
coends
bounded
precisely
The
feature
new
be found in
will
be
of the coend F
to the existence can
Chapter isomorphism.
of
approach is Hopf pairing
our
semisimple. degenerate, is defined in which is a Hopf algebra. The
Now we state
our
Sect.
w :
an
f
classes
abelian
F (9 F
XEC a
of
semisimple category simple objects is finite.
modular
category
1, which
-+
for the coend F
=
XZXI inCMC. The
we
(OF
does not have
require
-XEC
is
to be non-
X 0 Xv E C,
main result.
For every
which has C as circle
5.2.2
that
=
example,
5. For
when the set of
to be
Theorem 0.4.2.
natural,
all rather
are
modular
C there
category
exists
an
extended
TQFTVC,
category.
CV from the class Q_1' of by Theorem 0.4.2 a the that 9A is 9A such ( C -+ on F-+ : VC identity. composite map Replacing a modular category C with an equivalent one, and replacing the strucmonoidal 2-category of AbCat with an equivalent ture of symmetric one, we can In summary, we have an extended TQFT's to the class
achieve
0.4.1
that
VC is
Specializations
strict
0E
-+
9A of modular
9X
:
V
categories,
-+
and
double functor.
and Generalizations
of &ibn is that all One assumption in our definition Surfaces. can be, 1-morphisms should be connected. This constraint representing surfaces from of a TQFT for disconnected overcome, by constructing principle,
1. Disconnected
surfaces in
a
assignment
0.4
of Main Result
Statement
on
the Class of Extended
9
TQFr's
in [Ker9 8b] In the the procedure described TQFT's for connected ones following modification this of the TQFT axof non-semisimple a slight TQFT's requires have in been introduced which or as [Ker98b] non-semisimple ioms, half-projective surfaces in this to disconnected TQFT's. We will not carry out the generalization thedescribe formal to considerable since it as a general a book, requires apparatus .
case
ory
Field
Conformal
2.
Theory.
might
of CFT one
spirit
In the
consider
a
double
cate-
a7bn,
has the same objects and 1-morphisms as The latter gory Surf of surfaces. namely circles and surfaces. The 2-morphisms, however, are homeomorphisms bethe surfaces
tween
class
group of
a
of cobordism.s.
instead
Chapter
In
the invertible
with
surface
1
cobordism
aibn
we
identify
classes
the
from this
mapping surface
to
subcategory. TQFT double functor, Vc, turns out to be a See again Apversion of what Segal, Moore and Seiberg call a modularAnctor. Vmod The also double functor A details. for and context more implies propendix C Hence,
itself.
Vgtod
tion
jective
we can
Surf
:
of Surf
think
C
QAbCat of
-+
The restric-
double
a
mapping
of the
representations
as a
class
groups that
are
with respect
compatible
of surfaces.
to concatenations
TQFT. In the double category a4bn we can consider the subcategory, in which all objects are empty I -manifolds. This means we are dealing with closed surfaces and the only relevant composition is the gluing over these surfaces in vertical 3.
Atiyah's
direction.
Wethus
Atiyah's
in
obtain
a
central
disms between empty 1-manifolds, a
result,
To
we
obtain
a
projective manifold,
associated are
naturally
of
a
identified
TQFT given by
a
version
of
surfaces,
closed
to
with
functor
VC0
:
Cob3 seen
as
used
as
cobor-
vector
spaces.
Cob3
-+
As
k-vect
the between empty surfaces, associated normalization which is the thus to TQFT assigns a number, up 3-manifold. of the underlying topological
closed
2-framed
a
6-ob3
extension
The functors
definition.
projective invariant
as a
seen
cobordism
Theory. Reshetikhin and Turaev gave in [RT91] a construc0. 1. 1. The deTQFT in the sense of Atiyah as in Definition book [Tur94]. tails are worked out in Turaev's They use a semisimple modular and in our in [Tur94] is defined differently category C as input data. Modularity for sernisimple as we are equivalent book, however, both definitions categories 4. Reshetikhin-Turaev tion
of
a
projective
show in Section
7.4.1.
When C is
of the Reshetikhin-Turaev
phic out
to the
the theories. trace
VC1. Besides,
above
problems quotients
to
a
sernisimple
construction
disconnected
in
to
sernisimple
case
our
the restriction
category,
surfaces
is
construction
a
TQFT, isomorextends
with-
giving a complete agreement of for instance, can be produced, as semisimple categories of quantum groups representation
surfaces
Semisimple categories from non-semisimple
modular
connected
as
well,
[RT91, And92, Ker92, TW931. of closed 3-manifolds In [Hen96] Hennings defines an invariant Hennings 77teory, from a possibly ribbon Hopf algebra A. quasi-triangular non-semisimple, This invariant of extends to a TQFT as shown in [Ker97]. The invariants naturally closed 3-manifolds and TQFT's are again special cases of Vc, if we insert the rep5.
directly
resentation
category
C
=
A-mod and restrict
ourselves
to
closed
surfaces.
10
Introduction
0.
to
vertical different
ism, is
Vertical
extend
of the
one
of
corners
ing
product
the Cartesian
Aut'11
(a) k,
is identical
Obbn. Further
of
become
The
means
with
is identical
category we can
to other
vertical
surfaces
are
with
arbitrary
have
in
with
obtained
are
the
objects
I -manifold
a
the group of invertible with S'. Clearly, we have used Sa, which is the restriction
of Construction
of
discovery
new
by
a
tak-
1-cobordisms in the definition
possible
quickly
but
Studying
closed
3-manifolds
partition
x
S1 andL
via
a
3-manifold
a
link
to the
transverse
was
example
remarkable
down in [Wit89]
wrote
of functions partition surgery performed along
tween the
S1
of new preceded by constructions in [Jon87] the Jones polynomial and Turaev quantum groups by Reshetikhin of the Chern-Simons quantum field theory on
for
functions
Witten
Summary of Content
invariants
invariants, defined
invariants
[RT90].
and
3-manifold
of knot and link
and the ribbon
M=
Cob, and
impractical.
families
L with
the vertical
that
cobordisms
generalizations
Strategy
0.4.2
to more
square. Thus, a morphism in Homi (a, b) is The vertical surfaces and b target endpoints.
a
endpoints
source
This
surface.
same
An obvious
Surfaces.
construction
our
of 1-dimensional
category the
Summary of Results
question is whether it is possible and senthe general classes of surfaces representing 1-morphisms instead of mere cylinders connecting the boundary pieces of that we can easily deal with in our formalA small modification, surfaces. to another to allow a cylinder to connect a boundary component of a surface
6 General sible
and
and the
S'-fibres
Mwith ones
of the
Witten's
linear
included
relations
Wilson
3-manifold.
original
treatment
suggests
be-
along
lines
For
to com-
braid group element in the functions over the associated as traces pute the partition field theory. conformal obtained from the corresponding Although representations functions level of rigor, his program for computing partition written at the physical indicated
the existence
In their
time to construct consistent
of their
dients. a
The first
[RT91]
of links,
Reshetikhin
quantum 3-manifold
is the
on
invariants
invariants.
and Turaev in
a
succeeded
rigorous
data of quantum groups that of framed links in S3 The other .
which establishes
and
for
the first
mathematically
inspired by Witten's heuristic ideas, the and use of two new crucial the discovery
algebraic
of invariants
large family
type of 3-manifold
a new
way. Although partially construction was based
calculus
when two framed links
in
allows is the
them use
S3 describe
of
success
ingreproduce Kirby's
the
same
via surgery. They discover and prove that a comunder Kirby's is invariant moves and thus constitutes
(with empty boundary)
3-manifold bination
of
famous article
of their
link
invariants
to assign they generalize their constructions represented by ribbon graphs and links rather S1, so that they obtain a TQFT in the sense of Atiyah. and Turaev [RT91] using embedded ribbon graphs The approach of Reshetikhin caterealized was fully by Turaev in his book [Tur941 for the case of sernisimple gories. The related question of extending Kirby's calculus of links to manifolds with boundary to determine which tangles describe the same cobordisms is addressed in moves [MP94] and [Ker99]. [Tur94] and in some versions requires additional an
invariant
linear
of 3-manifolds.
maps to cobordisms, than only links in
Moreover,
which
are
now
0.4
of Main Result
Statement
on
the Class of Extended
TQFT's
11
of the extended TQFT functor, we as given in Theorem 0.4.2, combinatorial of first a presentation producing analogous strategy surgery data of the algebraic 3-cobordisms of the relative -with comers and then assigning In
follow
an
construction
an
abelian
modular
a
double
tangles
7'gll
category
of certain
assignment
of the
invertible
an
-
The methods
:
2-morphisms
double
functor
the TQFFfunctor
g1n
n
are
the two
respects
6urg.
data to combinatorial
Obb
employed
work of the authors.
data
replacing
Hence, it will
structures.
whose
presentation
algebraic
Vc*. In summary, we construct functors as follows: Ve
The combinatorial
composition
itself,
The
type. and, hence, constitutes the
to it.
category
needs to encode the two
also as
our
VC as V'-
framed link
equivalence compositions
classes
Likewise,
formulate
tangles as a the composite
)
a
be formulated
QAbCat.
we
of
structures
double
functor
of two double
(0.4.1)
on the techniques developed in previous techniques are further developed and refined. for orthe one from [Ker99] given by 6uto generalizes
In this
here
are
based
book the
The surgery presentation dinary cobordisms between closed surfaces. In addition to this, we have to include of &7bn and to and presentation the cylindrical boundary pieces in the definition & bn and 7gln, so that we obtain doudefine the horizontal composition both in ble categories and double functors compatible with the 2-framing extension. The the methods C, generalizes algebraic assignment Vc*, for possibly non-semisimple -manifold and of of 3 the mapping used in the construction invariants representations class groups in [Lyu95c, Lyu961. In particular, we extend here the coend techniques and natural transformations, instead of just objects like F functors to also construct and F. The appearance of the symmetric group necessitates more careful investigations of its action on Z-products of abelian categories. We also expand and refine which allow very the theory of braided Hopf algebras in braided tensor categories, and of concise invariance dictionary proofs conceptual type. Wehave organized this book by devoting one or two chapters to the construction of (0.4. 1). of each of the five ingredients and investigation with the comThe first three chapters of this book, therefore, concern themselves binatprial 6urg : T'gl =-+ a7b. The double categories Irgi and dib representation differ from 7'gl' and d4bn appearing in (0.4.1) only in that they have one horizontal I -morphism instead of several isomorphic 1 -morphisms. In Chapters 1 and 2 definitions of discuss and characteristics double the the categories d7b and 7-gl we The double in Chapter 3. functor constructed is e5urg isomorphism. respectively. of the double catin Chapter 1 with the discussion More specifically, we start
3-cobordisms. that Gob contains We discover a canonical egory Obb of relative balanced braided tensor category. The mapping class group of a surface is identified with the group of invertible As a specobordisms of Obb on that surface object. cial subgroup we also discuss the image of the framed braid groups on'a surface
corresponding mapping class groups. In the last part of Chapter I we define eWb of Gob using bounding 4-manifolds. the 2-framing extension Gluings are extended to 3-dimensional cylinders over the respective surfaces. Weshow that these
in the
12
Introduction
0.
and
Summary of Results
operations factor into homeomorphism and cobordism classes, and verify that the the axioms of a double category. composition structure on the classes fulfills The 2-morphisms double 2 the tangle In Chapter category 7-gil is introduced. admissible of Oasses ;,_n- generdc projections are given as equivalence tangles with form of a list of the in The equivalences several types of strands. are expressed of the devoted this is to of moves. A large part finding equivalent description chapter BL classes show that where 7T; 113L, we 7-gl g IS2 are category Tgl. In particular, gS2 in the thickened in the sense of [Ker98a], of bridged link diagrams, sphere S2 X IBL of the will be closer The version to cobordisms, ,r gS2 surgery presentations [0, 1]. data. Finally, we while 7-gl is more adequate for the assignment of the algebraic fact in do for and that and horizontal defined vertical 7-gl they compositions prove of the usual give rise to a double category. The compositions are mild modifications and operations. juxtaposition stacking 1 b is constructed by doing 113L := In Chapter 3 the functor E5urg : r gS2 obtained from a respective tangle and surgery along a link in a sum of handlebodies the stanWerecall the presentation thus generalizes [Ker99] for closed surfaces. and Morse and Cerf of handle attachments dard tools such as surgery manipulations theory, and review the resulting surgery calculi on non-simply connected manifolds. factors into an isomorphism e5urg on the equivWeprove that the surgery operation n-,
alence
classes
of
-rglBLS2
and
5 b.
Wealso show that
6urg respects
the vertical
and
for the latter requires a more detailed analyFunctoriality of the bounding 4-manifolds. sis of the handle structure Chapter 4 through 7 are concerned with the second composite V* : 7-gl QAbCat of the TQFT double functor as it is given in (0.4. 1). of the functors, In Chapter 4 the algebraic building blocks for the construction of ordinary of the properties discussion we give a thorough Vc are laid. In particular, braided braided tensor categories balancings, (BTC's) such as braided, reflexive Graphical calculi for both BTC's and Hopf algebras in BTC's and their integrals. of their and find criteria We study Hopf pairings Hopf algebras are introduced. of this In the last section in terms of integrals. non-degeneracy (side-invertibility) Z of Deligne's and properties tensor product chapter we recall the basic definitions of modof categories Wefirst consider only the 2-category for abelian categories. dimensional ules over finite algebras inside a strict version of the category of vector induces a strict accategory we ensure that the 2-braiding spaces. For this strictified tensor products ZCN tion of the symmetric group SN on the multifold C, 0 C2 0 horizontal
compositions.
...
of
categories
of modules.
As
a
AbCat inherits
result,
the structure
of
a
weak sym-
monoidal 2-category. Chapter 5 we begin with a discussion of a large class of coends, in abelian such as (&, 0, that are determined by an expression with operations tensor categories 1 -morphisms are obtained associated to horizontal the functors and -1. In particular, of the braided Hopf algebra form. We review the construction as coends of this
metric
In
structure
for the
Weconstruct
Modularity
a
of
a
special coend F special Hopf pairing
=
bounded, ribbon
f
XEC
w :
category
X 0 X1 in
F 0 F
C
-+
means,
a
1 for
bounded, abelian such
by definition,
a
BTC C.
Hopf algebra F. non-degeneracy
0.4
of the form
factor
w.
through
and that
on
the Class of Extended
non-degenerate
Weprove that w is w. In the modular
the natural
Homc(F, 1)
of Main Result
Statement
if and
only
if
TQFT's
13
integral-functionals
of F are two-sided case we prove that integrals in functor induced by the integral of the identity
transformation
object of C. pseudofunctor V* : 7-gl -+ QAbCat on Chapter The cobordisms. which invariance, meaning proof of topological tangles, represent is obtained by a classes of tangles, the fact that V* is well defined on equivalence axioms. The proof, that of elementary moves to algebraic dictionary style translation is straightforward. with the vertical the double functor is compatible composition, to The horizontal isomorphism. composition is, however, respected only up V : &7b -4 In the first part of Chapter 7 we lift the double pseudofunctor V : ebibn _+ QAbCat using an analogous QAbCat to a double pseudofunctor via a tangle double category rgl n. It can be made strict after replacing presentation of AbCat with an equivalent one. the structure of symmetric monoidal 2-category In the remainder of Chapter 7 we consider two special cases for the input cateabelian category C, for which our gory C. The first is the example of a semisimple double functor extends the Reshetikhin-Turaev theory. In the second case we conC A-mod for a general quantum group A, which sider the Tannakian situation the form of the Wediscuss in detaill yields an extension of the Hennings invariant. The relations for both types of categories. braided Hopf algebras and their integrals factors 6
In
through
1 ED
...
ED I for
every
the double
we construct
=
to
cellular
quantum invariants
are
also outlined.
background that leads us Appendix A we discuss the physical and historical Westart with an field theory in terms of double functors. to defining an axiomatic exposition of the topological aspects of Chern-Simons theory, that were investigated of conformal field theories. Other various formulations by Witten, and the functorial In
axiomatic
frameworks,
presented
and their
that
relation
attempt to
TQFT constructions gauge theoretic Lee, and Fukaya are outlined. Werecall
these two theories,
and axiomatize
to unite
the double
functor
explained.
picture
by Frohman, Nicas, Donaldson, of double
the Ehresmann definition
pendix B.l. In Appendix B.2 we discuss weak horizontal notions The related pseudofunctors. -
categories versions
and vertical
natural
to
Hutchingson,
from [Ehr63a]
of double
are
Relations
functors
Ap-
in -
the
transforma-
in those is explained Our interest are also described. by the fact that we first study a version of the TQFT functor V, which is a double functor in the weak sense. of the category of multiple in Appendix C. 1, we give a description coFinally, ends, which are associated to higher genus surfaces, together with natural isomorphisms between them. Wedo it in terms of the monoidal bicategory of thick tanobject. gles, which can be thought of as a free bicategory generated by a self-dual and in terms of generators of this bicategory Weobtain a combinatorial. presentation Coherence of the above mentioned which have graphical relations, presentations. from the comis asserted in the form of a functor functors and their isomorphisms binatorial bicategory to AbCat.
tions
-
Acknowledgments. L. Crane, P. Deligne,
Weare
grateful
G. Felder,
to J.
Baez, Yu. Bespalov, A. Casson, F. Cohen, J. Fr6hlich, V Jones, M. Karowski,
Z. Fiedorowicz,
0.
14
D.
Introduction
and
Summary of
Results
Kazhdan, T. Le, R Quinn, N. Reshetikhin, E. Witten, D. Yetter for attention
V. Turaev,
invaluable
R.
Schrader, J. Stasheff, A. Sudbery, discussions and work, fruitful
to our
advices.
Commutative
diagrams
package diagrams. The work of T.K.
in this
book
are
drawn with
the
help
of the Paul
Taylor
tex.
partially supported by NSF grant DMS-9305715. Early for Advanced completed while T.K. was at the Institute parts of Study, Princeton, supported by NSF grant DMS9304580, and at the University California at Berkeley, Ke 624/1. supported by DFGForscherstipendium. V.L. began to work on this project of York, U.K., partially at the University EPSRC research 42976. GR/G of Part this work had been comsupported by grant while de VL. Institut Recherche at was pleted Math6matique Avanc6e, Strasbourg, France. Further work of VL. was partially supported by NSF grant 530666 while he Kansas State University, was visiting Manhattan, U.S.A. Final touches were added of this
while
was
work had been
VL. visited
Max-Planck-Institut
fiir
Mathematik
in Bonn.
1. The Double
of Framed, Relative
Category
3-Cobordisms
whose objects cobordisms in dimension d + 1 is a category, The category of relative d classes of see [ES52]. whose 1-manifolds, + and d-manifolds are are morphisms
The definition The
of this
picture
we consider
a
category
was
of cobordisms
recalled
in Sect.
emerges very f : M-+ [0,
Morse function
0.2.
1]
topology
in differential
naturally f
with
-
1
f 0, 11 M, see [Mil69]. =
OMwith
when N dis-
It yields a decomposiexpressed categorically decomposition is an elementary o MN, where each Mj tion into a product M M, o M2 o of this A classical critical has which one on application cobordism, point. f only considered be which as a Theorem h-Cobordism of is the view [Mil65], may point dimensions. in the ecture Poincar6 of con higher generalization inherent to cobordism the structures discuss in detail In this chapter we will cobordisms relative 3-dimensional of the for d 2, meaning category categories
critical
tinct
values
d + 1-dimensional
on a
manifold
of Mwhich is thus
handle
=
as a
...
=
surfaces
between 2-dimensional
ferential
or
piecewise be mostly
will
Our focus
in the
of [ES52].
sense
on
In three
dimensions
the dif-
[Moi52].
theory topological equivalent theory of this following two important generalizations to the
is
linear
the
cat-
dibn
double category of the topological egory, which will enter the construction extended topological that we use to define and construct quantum fields theories. The first
generalization
arises
from the fact that
we want
The usual
with comers. and 3-manifolds boundaries, boundaries d-manifolds Vt and V, with diffeomorphic with
there
is
a
d + 1-dimensional
manifold
OVt
to
consider
definitions ---
OV,
2-manifolds
says that two if are cobordant
Mwith
am = Vt
U -V,. ev
aV into the product aV x out the 1-manifold modify it by thickening contains three pieces, namely the boundary of 3-manifold M, therefore, surfaces Vt, V, and aV x [0, 1]. Given another pair of d-manifolds Vt' and V,,' that Vt 24 V " are cobordant by a d+ 1-dimensional manifold M', and a diffeomorphism cobordism by gluing the boundary piece Vt construct a composite one can naturally of Monto the boundary piece V,' of M' as follows:
Weslightly
[0, 1].
The
mom,
=
M Vt
U -
MI.
V"
T. Kerler, V.V. Lyubashenko: LNM 1765, pp. 15 - 95, 2001 © Springer-Verlag Berlin Heidelberg 2001
16
The Double
1.
3-Cobordisms
of Framed, Relative
Category
gluing operation by identifying components of (9V x W' of x [0, 1] corresponding [0, 11 C M. The second components law relative second between a 3-cobordisms, gluing operation yields composition defined in (1.0. 1). which is distributive with respect to the composition over surfaces the that generalize the notion of a category by incorporating Algebraic structures for For combination of two operations our are, example, 2-categories. purposes we of double relative coborthe notion In more fact, categories. slightly prefer general form double categories and thus provide much richer disms with corners naturally and algebraic than ordinary cobordism categories. stru 'ctures topological that will consume the larger part of our exposition The second generalization that the data of a topological manifold arises from the circumstance by itself does field theories. Geometrical not suffice to construct interesting topological quantum and physical models suggest that we have to consider isotopy classes of framings data. The latter of the cobordisms as additional can be encoded in or 2-framings this the signature that is bounding the 3-cobordism. of a 4-manifold Algebraically, the that of cobordism extension our so an morphisms are implies integer category Mbut Z with manifolds a E no longer topological pairs (M, a) of these integers under composiThe cocycle that expresses the non-additivity results V' from of theWall [Wal69] be tion over the surfaces _can Vt computed for It obtained by related is also a as in [Ker99], see [Ati90] computation. closely in obvious the 4-manifolds the to an bounding gluing operation extending way. the extension For the second composition to the over the OV x [0, I]-cylinders, 4-dimensional setting is, however, not quite as naYve, and involves a series of addiThis complication is justified tional handle attachments. mainly by properties inherafter which the ent to the physical topological examples, quantum field theories are We may define
another
C Mwith
-
modeled. The relative no
independence
longer interchange
distributiveness
expressed by cocycles
cobordisms, trivial
or
the
of the two
interchange
have to match
since the
compositions
law of double as
well.
Wegive
for the extended
categories, a geometric
is
now
proof
law.
of the
Summary of Content
chapter we will assume Appendix B. 1. The main goal In this
the definition
of
a
double category
the double
is to construct
category
in Section
eWbn
0.2 and
and prove
Theorem 1.6.8. In Section
1. 1
we
the
introduce
egories that appear in this chapter. ing 1 -manifolds and 2-manifolds, we
associate
morphic a
to
a
to each
non-negative
copies
of the circle,
objects
In both one
and 1 -arrows cases
for the cobordism
sets
the sets will
be choices
homeomorphism class. object, given as a 1 -manifold triple of non-negative integers
for each
integer
a an
and to each
of genus g and a + b boundary components. between Among the holes on the surface we distinguish
cat-
of representIn particular, homeo-
[g, a/b]
surface
and b target
boundaries
so
that
the surface
itself
may be
a source
regarded
as a
boundaries
cobordism
in
The Double
1.
dimension
d
1. The fact
=
omorphism order to identify
that
confine
we
the
ourselves
3-Cobordisms
only
one
However, in
[92, b1c]
and
17
per home-
surface
to handle.
better
[gi, a/b]
of two surfaces
composition
to
representations
makes the technical
class
of Framed, Relative
Category
with
cho-
a
homeomorphism that is not canonical. In Chapter 7 where we set of 1-arrows, a larger we will remedy this situation by introducing 1 class. choose a finite surfaces for each number of homeomorphism 2g) g+1 (g 1 is that all surfaces are connected. Another constraint we work with in Chapter and TQFT functors It is not very difficult of presentations to find generalizations for disconnected surfaces using results in [Ker99] and [Ker98b] as outlined again in Chapter 7. The vertical 1-arrows are given as permutations among the holes induced by the cylindrical boundary pieces. Wewant to allow non-trivial permutations in order the full braid group instead of just the pure braid group on surfaces to incorporate in our description. instead of just Weare thus forced to consider double categories 2-categories. definition of what constitutes In Section 1.2 we give the detailed a homeomorand vertical phism class of relative cobordisms associated to a square of horizontal 1-arrows. and vertical Wedefine the horizontal compositions as gluings. A routine verification shows that this gives rise to a double category Gob. sen
surface
Section
we
1.3 deals with
Particularly,
pretation. and relative
category. surfaces sense
of
The
need to define
important with exactly
some
we
cobordisms An
standard
elaborate
functors
on
transformations
and natural
on
the
same
vertical of the
[JS91].
(Diff
group 7ro 1.4 with the group of
whose horizontal
1-arrows
gate the
of the
structure as
the fundamental
sets
framed
are
of small
+
of
a
surface
groups
as
well
Z*
==
as
_r
-
are
over
a
is identi-
surface
in
we
00b,
investi-
EP, which
are
spaces of discs on Z (instead the central extension over the
its kernel D2 Un 1
also
boundary cobordisms
Z. Furthermore,
) (Z)
The disc
Z with
relative
groups of the configuration braid groups). In particular,
number of punctures. genus of the framed braid group extension or
(Z))
invertible vertically homeomorphic to
braid
of points for the ordinary ordinary braid groups is determined + phism. into iro (Diff (Z*)), where tral
consequences of the double category interthat assigns to surfaces the correspondence
the vertical is that we can identify application category one boundary component as a braided tensor category in
mapping class
fied in Section
defined
a
under the natural .
They
rotations
topologically
are
that
homomor-
non-trivial
generate
related
in
the
to the
cases cen-
ribbon
underlying braided tensor category. 1.5 we provide several standard tools of handlebody theory that allow us to describe and manipulate handle decompositions of cobordisms in arbitrary dimensions. Werelate this in more explicit terms to surgery on the coborded manifolds. of 3-manifolds and handle Special attention is given to surgery presentations in particular, and cancellation the sliding decompositions of 4-manifolds, operations between the 1-handles and 2-handles in dimension 4. Finally, a couple we include of technical lemmas that explicitly describe the effect of a 2-surgery which passes through 1-handles that have been added to a 3-manifold. element of the In Section
plicitly
1.6
in Section
construct
Mwe consider
cobordism
preliminaries the framing
the technical
Equipped with
a
Relative
of Frarried,
Categojy
The Double.
1.
18
(M,)
closure.
3-Cobordisms
previous
from the
b
extension and
identify
a
section,
we ex-
of Gob. For
a
relative
of Mwith
2-framing
the
(M") A vertical comsignature of a of 4-manifold Wwith corners such that'DW position is, as before, na1vely constructed by gluing the 4-manifolds along cylinders The horizontal however, inZ x [0, 1] over the bounding surfaces. composition, =
volves
a
The 2-arrow
of Ware retained
composition of double
as
In this
we
bounding
the
given
are
as
Mand the
3-manifold
of
instead
equivalence
classes
signature
and vertical Weprove that the induced horizontal law these sets are well defined and obey the interchange
operations on Hence, dib categories.
section
Colb
information.
The 0-1-Arrow
1.1
only
in which
of these 4-manifolds,
of
sets
and 2-handles
1-handles
of 4-dimensional
of attachments
series
boundary identifications.
-
define
is
a
double category
Category
of Surfaces
category
of 1+1-dimensional
a
as
stated
with
in Theorem 1.6.8.
Boundaries which consists
cobordisms,
cobordisms.
Wepresent
underlying the double category of 2+ 1 -dimensional We in it terms of natural generators and algebraic relations.
also discuss
the choices
between standard
and 1 -morphisms
objects
of the
An object
an
6, we admit
Chapter
in
tions
is
phisms
ordered
This
class.
and we denote it
as
set of oriented
only
depends only
of course,
object,
Here,
as
well
as
on
in the
generaliza-
in each homeomor-
manifold
one-dimensional
one
composites.
and their
surfaces circles.
the number of components,
follows:
sl
sUa
Li
...
Li
S1
a
objects of the cobordism category Cbb2. The morphisms in To be more cobordisms. Gbb2 homeomorphism classes of 1+1-dimensional cobordisms. all are nonof the first define COB2 Objects we bicategory precise, 1-morA Sul circles. of union the a with identified disjoint negative integers a, with a oriented Sub is and Sul Z, surface, a between together compact, phism Sub. Its U SUa are aZ::::-4 homeomorphisms : 2-morphisms homeomorphism. k : (Z, ) =4 (Z', . Two 1-morphisms, (Z, ) and (Z', '), ') such that ' o k k : Z =4 Z' connecting if is there in a 2-morphism. Gbb2 are considered equal
They
are
also
the
are
_
=
them.
book
In this
we
want to confine
ourselves
generated by connected surfaces. It is clear can be enumerated using the genus, g, of by
[g, a/b] If
b
>
[gi, a/b]
1 then =
[g,
the
composition
+ 92 + b
-
1, a/c].
law
:
__+
subcategory
morphisms
in
Gob'21,
which
HOMGbm (Sua, 2
Wedenote the
the surface.
Sul is
to the
that the
is
SLJb)
g-th morphism
sub.
given by
the
simple
formula
[92, b1c]
o
Category of Surfaces with Boundaries
The 0-1-Arrow
1.1
The category Gbb2 is product to be the ordered, a
embedding
natural
of
clearly a symmetric disjoint union, i.e, the symmetric group
in
=
a
Uordered. letters
we define
if
category
tensor
U
19
the tensor
we obtain particular, automorphisms of
In
the
into
Sua, i.e., S,,
Aut02b2 (Sua)
c
7r*.
Ir
:
the union of 7r*, associated to a permutation cobordism, 7r, is as a two-fold 5--S11 S' US' identifications The x x x are cylinders boundary [0, 1] [0, 1]. [0, 1]U such that the j-the cylinder from the source one-dimensional connects the j-th circle
The
...
manifold
7r(j)-th
the
to
circle
in the target
manifold.
It is easy to see that any morphism in and the tensor product of morphisms from
Cbb2
is the
composite
of
permutation
a
Gobl2n.
algebraic way of describing Obb2 as a freely generated tensor category A detailed is provided in the following proof can be found in [Abr96], proposition. and [DJ94] for expositions of this fact. see also [Bae97] A useful
1.1.1. Cbb2 is the free, Proposition commutative freely by an associative, Let
us
every
objects
the
assume an
in
more
strict
detail
what the contents
impose
A, with
the
which satisfies
of the
...
way. The fact that we speak of a symmetric the morphism. sets contain as generators isomorphisms,
that
(1. 1.3), compatible group as which is a unit with respect to U. that we have A is an algebra-object, we imply
symmetric object 1,
and which
in
respective the relation
relations that
of on
(D. Also
characterizes
a
this
commutativity we assume a morphism with respect
unit
to
side-inverse
object
The
of yet another generator, to the pairing 0* = tr
A is identified
with
a
namely q5 o
single
e
:
mean
circle,
I
that
A,
(i).
a
generating morphism. to Implicit
A U A, which is the
1
AUA
:
which U. We
morphism
e :
same
is the existence
two-sided
a
and
associativity
tensor
with
are
to Frobenius algebras implies further way the restriction relations first and them. The is a trace, morphisms meaning among this has also to be non-degenerate. tr : A --- 1. For a Frobenius algebra
In the
unit,
is
tensor
When we say that : A U A --+ A, and the conditions we
generated
category,
in the obvious
means
implement also
a
object is given
category
symmetric tensor algebra-object,
freely generated category freely generated by an object means that category of U on the level is of the form A U U A, and the tensor product
explain
To say that
is.
strict,
Frobenius
1.
and 1 with
the empty set. As The morphisms
Gob2 by product disjoint given & is mapped to the threecorrespond to discs, Zo,,, the multiplication holed sphere ZO,3, viewed as a cobordism S' U S' --+ S1, and the symmetric group generators are mapped onto each other. Thus almost all of the generators of Gob2, which we single out by this descripThe cobordisms of sense. tion, are elementary cobordisms in the Morse theoretical index 0 and 2 are e and tr, and the fusing index 1 cobordism is (D. The fissing index I elementary cobordism is given by the combination before, e
the tensor
U is
the
in
union.
and tr
(D
*
=
(
& U
1A) (11A 0
U
10),
and, conversely,
0
Similarly, generators
left
e
combination
a
an
the
same
elementary
into
The elemen-
cobordism.
map from the x-axis. the horizontal
1. 1. Here the cobordisms
Figure
in
index 2 cobordism.
an
above translate
describe is the
the Morse function
i.e.,
1 and
index
discussed
that
depicted
also
are
right,
to the
of
relations
between Morse functions
moves
tary
is
algebraic
the
3-Cobordisms
Category of Framed, Relative
The Double
1.
20
projection
on
G
:
0*
0
:
Fig.
1.1.
:
tr
There is
to
3-manifolds
we
lence
classes.
problem
fundamental
a
cobordisms
when
need to be able to talk
The natural
we
wish
candidate
about
to
extend
In order
1+1+1 cobordisms.
of
category
a
of 0ob2
Cobordisms
Generating
surfaces
specific
to construct
a
to
double
COB2 category of all compact oriented surfaces. Thus, we should select a subset of surfaces
a
category
and not
category
of
1+1
cobording just equiva-
describe
from would be
-
in
each
morphism
set,
which
,
is
and which
finite
is
composition
under
closed
CoBfinite 2
C COB2. In Chapter 7 we describe Roughly speaking, first, we choose specific and then we describe tr, and the permutations,
category.
0% e,
and thus
in these
generators.
Although, classes
even
are
positions
finite
sets,
cobordisms
morphism. formally
in
Gbb2 If .
described
only we
by
CoBf2 inite
we
and select
anymore,
mensional. be
of
the definition
for
such a subway of picking surfaces for the generators (D every other surface as a word
a
,
be chosen such that the
can
not insist
also confine
a
subcategory
a
isomorphism having a subcategory under comof the 3-dithe purpose of finding presentations surface in each isomorphism class, i.e., for each
shall
one
defines
on
to connected
ourselves
this
cobordisms,
can
2-pseudofunctor
(X, a)
:
Gobc2'
C
)
COB2-
surface It associates a standard objects this functor is the identity. formal definition of In we some give Zg,alb place Zg,alb to the morphism [g, a/b]. 1.2. in the form of a picture, as, e.g., the left surface in Figure of the source holes is from the left to the right, The labeling 1...... a, for the reversed choose b holes first a holes. For the we 1t. In bebt, ordering, target of which and horizontal in we that direction, are aligned tween we have g handles On the level
of
.
depicted
the first
Besides contains
an
.
.
,
and the last.
being a map on the sets isomorphism between
of the
objects and morphisms, a pseudofunctor also composite of the images of two morphisms
1.1
Category
The 0-1-Arrow
t
0'
_
.........
of Surfaces
0
image
and the
composite category,
for the functor
in
a[g1,a/b:92,b/c]
:
S
Zg,alb
Q92, b1c])
-
of
X
a
of
a
pseudofunctor
of 1+1-dimensional
case
the Specifically, homeomolphisms,
also
system of
LJSUb
requires
Egi,a/b
that
a
:::-4
-
holds
with
in the strict
g > 0
or
sense.
a, b > 1 these
For
our
additional as
be data
follows:
satisfies
Zgi+g2+b-1,a/can
can
purposes surfaces.
indeed
(1.1.5) condition
associativity
of the form a[1:2,3] (a[2:3] USUb 111) = a[1,2:3] (113 Usuc a[1:2]), where we compose with a third surface, situation, Z93,c/d- If to surfaces
will
cobordisms,
Qgj, a/b])
E g2,blc
The notion
isomorphism.
The
category.
in the
between surfaces.
(1. 1.4) consists E
original
in the
which
homeomorphisms
of the
consists a
of the
in another
21
..
Fig. 1.2. Standard Cobordism,
contained
with Boundaries
if we
we
consider
confine
be found such that
it suffices
to
verify
the
ourselves
the
equality
associativity
up to
isotopy, which can be done for all the isomorphism in (1. 1.5) as follows: instead of the we construct Specifically, gluing over circles we can also consider the surface, where we have inserted small as indicated cylinders between the boundary components that are to be identified, in Figure 1.2. Hence the composite is homeomorphic to the surface, which in the middle has the form as depicted in Figure 1.3. Moreover, the obvious on the left homeornorphism. is unique up to isotopies.
------------
4t
2t
3t
2
3,
4s
Fig. 1.3. Inductive
The slide
side of
of the tube
Figure
Z92,b/,
1.3, defines
LJSub
4t
Definition
connecting 2, to 2t, which a homeomorphism:
Egi,a/b
3----
3t
Z92+1,b-1/c
of
%
2
a
is indicated
LJsu(b-1)
on
the
Zgj,a/b-1-
right
hand
1.
The Double
we
have identified
22
Here, the
cylinder 1.4, we explicitly
pushed
Figure
resulting
the
out
as an
Fig.
of
The definition
ter
if
be irrelevant
will
is
a
cobordisms.
Also,
associativity
condition
as a
gluing
b
over
-
circles,
1
and
in the second surface.
handle
In
identification.
Cylinder-Handle
Defonnation
chosen in The convention completed by induction. The lathomeomorphism uniquely up to isotopies. consider homeomorphism classes of 2+1-dimensional
now
the
we
the holes
does not affect
a
part
92 + 1-st
the latter
1.4.
determines
1.3, thus,
Figure
middle
additional,
illustrate
3-Cobordisms
Relative
of FraTned,
Category
easily slightly
of the middle
outside
part
so
that
the
realized.
is
of a, which defines the same construction different Let us also give a of cobordisms of the compositions the description homeomorphism. It will facilitate In essence, it is given by the following direction. or tangle diagrams in horizontal which produce the standard manifold of operations, between composites identity and from starting zg2,blcZgi,a/b Zgl+92+b-l,a/c a o
f Glue
j,
to
jt
jGlue
=
for
bj
j , 0.
an exAlexander's argument [Ale23]: part is basically + -1 E 'Diff (e-sD 2, 0) given by defining T1, (f ) by R(s) ofoR(s) in D2, where R(s) is as in the proof of Lemma1.4.7. disc of radius e on the smaller whose distance from the origin is between e-' and 1, we On the annulus of points, since f leaves S' function This yields a continuous set T1, (f ) to be the identity.
Proof. The proof of plicit contraction!P,
the first
is
-I
This result
=,
idD2. SO(3)
that
recall
For the second part
lence.
(f )
and TV,,
fixed,
pointwise
is due to Kneser,
see
Diff
2.4
in
[Kne26],
+
(S2) for
is a homotopy equivahomeomorphisms and to fibrations over naturally
[Sma59] for the smooth case. Now, both spaces are is a map of fibrations S2 and it is easy to see that the inclusion
Smale ,
S1
SO(3)
C-
Diff+(S2,P)
C
C,
as
follows:
Diff+(S2) (1.4.4)
id
s2 From this
map between
spective In
[Bir74]
fibrations 1
the assertion
Private
long
exact
it is discussed
of the
automorphism
communication
by
follows
s2 an
easy
application
of the 5-Lemma to the
re-
sequences of the two fibrations.
in detail
groups can be obtained from In view of the previous surfaces.
how the braid
groups of
punctured
40
1.
The Double
Category of Framed, Relative
3-Cobordisms
lemma we expect to find groups that are slightly from the ordinary different braid boundaries. group if consider instead surfaces with parametrized when we consider the following fibration of spaces of maps, They arise naturally, in analogy to the fibration in Theorem 4.1 in [Bir74]: +
Diff
(-T9,
a
/
b
7SLja+b)
+
Aff
(Zg,
Sua)
a/0,
IT ,6mhd+(UbD2'
(1.4.5)
Zg,alo)lSb
+
Diff (Z7 SUN) is the group of automorphisms of Z SUN has to coincide with the canonical 9Z which restricted to the boundary action of some element of the symmetric group SN on SUN. The base space is the space of embeddings of b discs into the surface of genus action of Sb. The projection and modulo the canonical a holes, -r is defined by g b discs Of the in the to an surface, which have to automorphism restricting Zg,alo that
Recall
in
From this as
convention
- g,alb
in order to obtain
be deleted ends
our
fibration
we
obtain
-
long
a
exact
d.
Bb(Zg,a)
70
(Aff
+
(Zg,alb, i*
Here
we
notation
sequence of
homotopy
which
groups,
follows:
used that for
the base space of theframed braid group
) 70
Sua+b
(Viff
+
embeddings
(Zg,alO, Sua))
is connected,
__
1.
(1.4.6)
and introduced
the
( eM-bd+(UbD Zg,alo)lSb) 2
Bb(Zg,a)
-=
Irl
,
braid group Bb(-37g, a), we use here a symmetrized conAs opposed to the ordinary For the sake of simplicity, we shall figuration space of discs instead of just points. often confine ourselves of the fibration in (1.4.5), where we assume to the covering that the homeomorphisms leave the boundary components pointwise fixed (without and where we omit the division permutation), by Sb of the base space. To this end let us also introduce for the pure braid group and the mapping class group notations with trivial permutation,
'Pb
As for exact
the
ordinary
(Zg,a)
(E"d+ (Ljb D2, + 70 (Diff (Zg,alb,
ir,
M(g,
a/b)
braid
groups,
we
have for
their
9,a/0)) la LJ 1b)) framed
7
(1.4.7) counterparts
the short
sequence, 1-4
b
(Z)
(Z)
Sb
---+
1
(1.4.8)
Mapping
1.4
and
Groups,
Framed Braid
Groups,
Class
41
Balancing
to the pure case d. : or its restriction b (Eq, a) map in (1.4.6) 1.4.2 Mvia Proposition if we identify be described very explicitly 2 Ljb the in D closed t __+ -+ ft : path Eg,alo Aut0ob(IaJb) ( - g,alb). For a
connecting
The
a/ b)
M(g, with
can
embeddings by
space of
-"g,a/0
UbD
F:
three
2
a
complement:
[0, 1]
X
manifold
As the
define
we can
1g,a/b
c-
- 'q,a/o
fo and f,
[0, 1]
x
(d, t)
:
cylinder
tubes into
of solid
over
(ft (d), t).
--+
for the associated
cobordism
given by
is
image(F).
-
embeddings of b discs
the
are
[Oil]
X
candidate
the obvious
Mf Since
embedding
an
the standard
into
positions,
target
the
and we choose for them canonical upper and lower boundary piece of Mf is lg,alb, the of The identifications. pieces of 09Mf will cylindrical parametrization boundary be
given by
Lemma1.4.4.
Wepick
Proof
a
Morse-function
the upper and lower to the cylindrical pieces, are
be directed
will
has
no
h
and
from
is
an
>
of
pieces
S'
and x
inside
points
critical
(Vh, Vp)
have
f
The
assignment the connecting map d,,.
realizes
LjbSl
of F to
the restriction
pull
[0, 1]
X
Mf,
-+
and
h-'(0)
be
parallel
S'
x
.
along these pieces
back of the flow
S'
Mf,
and
h
1. The function
x
that
so
for
projection x
to
be modified
so
[0, 1] that
it
Zg,a/0 X [0, 1] we Zg,alo x [0, 1] -+ [0, 1]
extension
an
Zg,q0
cylinder
can
above,
as
h : Mf [0, 1], such that h-1 (1) oMf Moreover, the gradient Vh will
0 to
of h to the
[0, 1].
X
boundary identifications
with
0, where p is the canonical
extension
Ub D2
C
[0, 1].
to
It is easy to
see
that
the
slice to another Vh maps a slice p-1 (ti) of the vector field (vh,vp) of U1 (t) to t), while preserving Mf Let us denote by U(t) the restriction Zq'./0Zq'./0 X 10} + It is clear that p U(t) E Diff (Zg,,,10) is a lifting of ft in the fibration (1.4.5) the image of the connecting so that U(1). Moremap is given by the class of p the =4 X over zg,alb the : cylinder Mf, [0, 1] over, given isomorphism U(t) Eg,alb the bottom, identifications at canonical if in class use the we same Cob(*), yields the cobordism. it is assothe at but and cylindrical Thus, top-piece. U(l) pieces, p 1.4.2 to mapping class of p o U(1) ciated as in Proposition -1
U1 (t)
flow
p-' (t,
+
.
=
o
o
o
-
It
follows
also
immediately
from Lemma 1.4.4
d* is
that
a
group
hornomor-
phism. of Braid
Groups,
1.4.2
Framing Extension
Next,
let
groups. where
Clearly, EMU+(LjbD 2, Zg'alo) cj is the center point of the j-th
us
discuss
the
connection
and the Ribbon Element
between
the
maps onto disc
so
that
framed
and the
&zbd+(Jc1.... for
ir,
of this
ordinary )
braid
Cb}i Zg,alo),
map we obtain
a
42
Category
The Double
1.
C.
projection
of Framed, Relative
Bb- SurJectivity
-4
3-Cobordisms
of C. is
easily
but the kernel
seen,
C,, is
of
not trivial.
Specifically, with
j
=
1,
.
the standard
keeps
for every center ,
.
.
disc
b,
define
we can
define
we can
the center
E
Zg,,,10 6j 0(0)
the rotation
-9b
E
D'
:
D2
:
of the standard
element
an
If V
fixed.
point
point'jj-
Z'q,
,
by
Zg,.10,
in
following path. For D2 by an angle 0, which the standard embedding
-+
is
./o
discs
the
) Zgalo parametrized embedding io : UbD2 target get ii' ii o 0(0) if j 0 j' and disc. This yields a closed path in on the j-th by using Embd+(Lj b D2, Zg' a/0) and, thus, an element 6j E Bb, which maps to the constant path in 97nbd+ (I cl, Cb }, Zg, a/0) so that Jj E ker (C.).
disc,
j-th
of the
we
.
.
The additional as
Lemma1.4.5.
6j Proof.
Let
Zg,alo f
k
free
we
,
.
disc,
i.e.,
we
[0, 1]
x
path
as
well
b
be
represented
k
is
(t, _)
Jj
d,,
an
[0, 27r]
boundary
the
f
by functions embedding of
Embd+(I_jb
-+
(.A4 (g, a/b)).
E center
embedding of the k-th replace fi by fi o (id
on
group
group:
and
the standard
H to the
restrict
.
Ljbk =1 f
H: If
.
b
E
f (t,
j-th
of the
1,
=
element
an
class
(-j3'b(Zg,a/0))
E center
ik, i.e.,
(0,
j
both in the pure framed braid
central
are
mapping
For all
such that
rotation
7
.
generators
permutation
the
r-
a
disc. x
x
D2
k
f (1, we compose f with
If
we
and
obtain
a
a
function
Ig,alo)going from (0, 0)
rectangle
of the
[0, 1]
:
b discs,
0(0)),
D2 ,
k
to
- 'b
(t, 0) (1, 2,ir) (1, 0), composite path 6j 7 E Going from (0, 0) to (1, 27r) over (0, 27r) instead we obtain the path representing f * Jj E Pb. From H we easily construct a homotopy between these two paths, and then to
=
Jj
which shows that The rotation
o,
in
will
identity
In the collar
with
&
component of
by that
an
-
of
define
in
0 < t
Zg,a/b,
on
Ia)
outside we
If the disc a
disc
polar
-
Clearly,
in
for
an
Z9,a/0)
one
can
some
be lifted
o (1
+
is, thus,
et, 0)
represented
collar
0 (0)
identity inside & so that,
(I
a/b)
+
can
in
Et, 0 +
be
of the e-collar.
with
a
particular,
0 D2
(I
-
Dehn twist
boundary represented
It is obvious
d*6j
braid
group
we
is
the
(Diff+ (Zg,alb,
projection
Sua+b)).
from
=
get instead
jc(g)(j)'
(1.4.8),
centrality
of
Vg and
an
E
is central
the relation
Rb,
analogous
(1.4.9)
equation
for
-
path,
inside
by the j-th target
of the
in M(g,
element
=
to
coordinates,
local
and the rotation
6
e-thick
in
*
a/b).
9Jj9-1 (
d*Jj
any other which is
automorphism Of Zg,alb, commutes an automorphism
For the full
where
1 +
coordinates
such
in M(g,
7ro
of radius
is localized
the
is of radius
1. The element
1. Then there
7
S'
nor
is
a
S1,
x
short
exact
sequence
Proof.
First,
we
following
=
Ho
b(Z)
=
A, are the standard Embd+(Ub D2, _p) :(t, of
[0, 1]2
from
represented
[A]
=
[A]
For ZE
proof
same
isotopy
of the
discs
by a path embeddings,
Zg,a
that
analogous
s
in Z. Let
As
-+
then
consider
we can
Thus K yields
in
Ht
K the
a
:
E
element
[0, 1]
[0, 1]
x
in
A0
such that
--+
paths along the edges
_ ,b (Z)
in
[Bir741
ker(d,,) + 'Diff (Z) with
element in
an
suppose another
us
in the
lies
to Theorem 4.1
o
to
of d,,
-E"d+ (Ub D', Z),
E
1.
--
the kernel
s) -+ Ht As. If we restrict K to (1, 1), we get the composite paths
0 S2 D2'S1 for
isotopy
leaves
the
[0, 1], S'
x
surfaces
punctured
arguments
ambient
=
a/0)
M(g,
--o*
of the elements
[-F,, H]
homotopy, proving
[7-*H].
the group Z (61, braid t -+ Ht E an
Z
H and A in both orders.
For closed
trivial. The
(0, 0)
by *
proof
the
id to the b standard
represented
be
surfaces
a/b)
is exact we know that groups. Since (1.4.6) of a closed path t -+ by the restriction
ordinary braid can be represented for
H,
all
show for
b(Z),
of
center
M(g,
(Eg,a)
I
with
x
S'
Eg,,,
surfaces,
.
.
,
Jb)
of Lemma1.4.7.
S"d+ (1pi, we can
.
.
.
A },
+
It follows
Z)
a
Ht It is, therefore,
E
-+
an
isotopy
constant
Diff
+
sufficient
(Z),
braid.
in
'Pb (Z) is [Bir74].
by exactly
>, 3, follows
a
book.
Thus, there is into
,
also deform t
points IT, fixed.
2g
same
of
the center
in Lemma4.2.2
is proven
with
from Theorem 1.4 of the
know that
we
g > 2 this
that
d*H lies
that
deforms
Extending
such that the to consider
the in the
this
resulting
the fibration
to
The Double
1.
48
+(Z' UbD2)
Diff
and show that
c-
+
Diff
.
(T')
7r,
of Framed, Relative
Category
--,.
-
+
n,*ff
(Recal!
((Ub (D 2,
gnthd+
7,
(Z' jp73.1)
is trivial.
3-Cobordisms
(Z, X)
means
{jT 3
Pj
point-wise
X is
that
fixed). a path -y between a given point, say YFI E Z, point onOZ, all of which are fixed by a closed path + Viff (Z, JIT.73 1. To the cut defined by 7 we consider the covering space Z generated by JjT3 }, where the covering group --- Z is cyclically
Now, if and either t F-+
Ht
P(Y
Z7
c
:
E
(,E^I)
Ht'Y
o
=
Ht
Ht
p y,
o
be lifted
can
Hj'Y
and
is
a
to
an
Ht'Y
isotopy
covering
on
transformation,
Ho"
Z'Y, such that i.e., H11
=
id,
=
c"
with
E Z.
wi
that
Weclaim
end
To this
Z'Y.
+ b
I
-
covering
w,
0. In order to
=
consider
we
along
punctured
surface
of which,
one
space is
now
see
this
Z'
the surface
holes,
the
cutting a
or a
-
Now, the isotopy
p'Y
choose
we can
point 1T.
-
+
Diff
E
+ b > 2
a
another
let
us
Z
=
given by gluing
two
copies
an
U
has the
7. The result
S1, contains
give
(7
-
of
explicit
lp37}), same
copies of Z' together
of
construction
which genus
we
obtain
Z but
as
by only
boundary. The along the -y-pieces in
-Y in its
SC: z'Y
=
...
Uly EC Uly EC U^Y
Let -yo be one of the paths with p'Y o 7o = -y. Then Ht' deforms 7o into of Ht' o -yo remain in two components, copy 6" o -yo, where the endpoints
C2, of OZ'Y, which
cover
components C,
C2
gives
bj
rise
to a
denotes
a
a-
the
are
of the closed
contraction
small
respective
R in o9Z'y
c---
holes in Z
segment in
-
fp371
sc
=
bi
*
(--yo)
b2
*
shifted
C,
(CW1 -Yo).
*
and
-Y. Since the
Ht' actually
the deformation
contractible,
path
by
connected
a
0
Here
Cj.
suppose that x(O is the path along the boundary component S' of the of Zt, then sc is easily seen to be hok-th copy of Z' in the above presentation (ko+l) X(ko). X(ko+wi) Since 7r, (Z-Y) is the corresponding, * * * X MotopiC to Let
us
...
infinite,
free
is contractible
product
of the 7r,
(Z')
it follows
If w, > 0, this
in Z'.
compatible with the assumption is trivial. and the image of ir, (-r') the For remaining case, with b is not
=
is
that
s' is contractible
only possible that
1 and
Z is neither
a
=
0,
we
if Z'
D'
-
in Z'Y
D2, which,
nor
have to consider
a
if
x
however,
S2 Hence,. .
only w,
=
0
refinement
let us simplicity coverings. of the automorphisms compactification fpl 1) by (again) replace Diff (Zq,O Zq, 1, which preserve the boundary, although not pointwise. Instead for 1r, (-F') we + ' of 7r, (-r"), where -r" : Diff (Zq, 1) --- Diff (Sl) is the may show triviality We are going to consider the universal restriction to the boundary. covering pl : F the free is whose Z 97 1 (a,, b,.... covering group group Zg, 1, ag, b.) in 2g of the above argument,
that
involves
+
-
non-abelian
For
the
,
As before we lift the isotopy Ht to an isotopy t 1-4 Ht' E Diff+ (Zg7j). generators. --- IR, on each of which o S' has components Now, the restriction pl aZ9'71 the covering rl[aj, bj]. Since Ht preserves the group is c--- Z, generated by c and, hence, boundary it also must preserve each component of this sub-covering, =
Hl'
must be a power c".
Mapping
1.4
Groups, Frarned Braid Groups, and Balancing
Class
49
point q E OZ, , and choose a closed path -y in Zgj, that starts and ends in the respective point in OEg,,. In Z,.'71 this lifts to a path -yo from q to The deforms this path into the path 76 isotopy [-y] (q). Ht' Hl' o 7o that joins c" (q) to c" o [7] (q). But in the projection id is to -y since H, again mapped -y6 also joins c" (q) to [-y] o c-1 (q). By uniqueness of liftings and on Zg, 1. It, therefore, the fact that the covering from this [[-Y], c"] we infer 1, i.e., group acts freely, Select
0
any
now
=
=
=
c"
lies
The
right
that
in the center
of
a
Hence,
group.
w,
=
0 and -7r,
(-r")
1.
=
side of the sequence from Lemma 1.4.8 can, in fact, be expressed in of Gob using Proposition 1.4.2, as well as the fill functor from
of elements
terms
(1.3.5)
from Sectionl.3.
.A4
(g, a/ b)
Hence, for the above
-Fyil
on
the
Shil
of
we can
sphere
with
Lemma1.4.9. I
two
For g
--+
71
(Tiff
or
0
=
2.
we
describe
+
1
group less holes.
we
have
1
--*
the framed braid
(S2)) C
)
b
the maps from
Pb has non-trivial
short
C
j3 b (D 2)
--+
j3 b(ZO,2)
Z/ 2
_ --
group
as
the kernel
of
cobordisms.
thefollowing
-
3.'
define
group of invertible
for which the braid
cases, a
AutCob(a,o) (Zg,alo)
l.
cases
respective
diagram:
commutative
M(g,alO)
op
In the next two lemmas
special
following
Wehave the
AutCob(a,b) (Eg,alb)
case
free
b 2
)
to
Mfor the remaining
center.
exact
Webegin with
the
sequences:
(S2)
)
M(0,
01b)
--+
1.
M(0, 1 /
b) ---+ 1. M(O, 2/0)
M(O, 2/b)
c---
Z
1.
-+
Proof 1.
+(ZO,b,
For the first
by
follows
09ZO,b)) sequence observe that 7r,(Viff induction from the long sequence for the fibration
,Diff+(S2,ub+lD2) using that
+(S2, LjbD2)
1.This
Embd+(D2, ZO,2),
1. Moreover, we have that 7ro (-Diff + (S2)) (S"d+ (D2, ZO,b) + Z 2. (S2)) 7r, (Diff / The latter group is obtained from the for which 7r,(is) is ) : Diff+(S2) of fibrations, SO(3)
7r2
1 and that
=
=
is
clusion
Diff
c--
=Oofb,>
isomorphism.
in-
c
The claim
Diff+(ZO,b,99ZO,b)
r
follows )
Diff
now
+
(S2)
long E"d+
from the )
an
sequence for the fibration b (Lj D2, S2). Note, that for
b > 2 the kernel
for the map from the braid to the mapping class group is exfor the corresponding braid group actly sequence for the ordinary and punctured surfaces, where it coincides with the center of the braid group the same exact sequence We, thus, have precisely (see Lemma4.2.3 in [Bir74]). the
as
same as
in Lemma 1.4.7,
spective
images
if
in the
replace the mapping class
we
groups groups.
b
(S2)
The
and Pb (S2
cases
b
=
1,
2
) by are
their
re-
consistent
50
1.
2.
(0, 0/1)
since.A4
with Lemma1.4.7 Dehn-twist
around
The next
sequence
1/1)
I and M(0,
=
=
Z, generated
by
the
hole.
one
immediately
follows
Diff+(ZO,11b)OZO,11b)
from
that
fact,
the
(D 2'S1)
fibration D2)
I of Lemma1.4.3.
by part
is contractible
the
in
Srnbd+ (LjbD 2,
Tiff+ (ZO,1/0, aZO,110)
-+
3-Cobordisms
of Framed, Relative
Category
The Double
space Diff The map on the framed braid group should not be confused with the respective 2 plane E with holes as in map into the mapping class group of the Euclidean of the b points or discs in the braid group 21r-rotation The collective [Bir74]. the entire
is realized
E2
ther
around
class
mapping
in the
D2. In -7ro
or
+
(Diff
group
freely
the puncture
points + for,7ro (Diff (D 2 Ljb D2'S1)) points on the boundary can not _
(,ETnbd+ (D2
72
D2))
Finally, class
I
-+
7r,
long
+
(S'
X
S1))
Snthd+
extensions
exact
J b (SI
injected
are
(4)
==
into
Z (Ji....
)
Jb)
X
to the
SI)
respective
-
map-
-4
5---
SL (2,
Z)
the mapping class
group,
in
M(1,
)
lar, im
(Ub D27 ZO,2)
sequence
M(1, 01b)
, ,
thefiraming
and
then fol-
Z. The claim
group of the torus
Z ED Z
5--
d.
where
the
==
+
Viff+(ZO,2/0)'9)
C
thefollowing
Wehave
(Diff
the
Z, using E7nbd+ (D 2D2 )
(D 2,9) (,Embd+ (D 2 D2))
the map from the framed braid group is given as follows
Lemma1.4.10.
in ei-
rotate
we can
sequence for the fibration
'Diff+ (ZO,2/b 719) ping
'7rO(Viff+(ZO,2))
1 and
Diff
1 and 7r,
=
since
be moved.
C.
lows from the
is trivial
But the map from 'Pb has a kernel. is a non-trivial the 27r-twist element, since the
(Tiff+ (ZO,27 09)) Viff+(XO,209)
fibration
around the hole
Dehn twist
this
that
so
We have Irl
3.
as a
(E2- Ipj}))
ED
Pb (S1
X
0)
-+
1,
particu-
Sl)
center +
( -5'1,0/b)) is trivial for b > 0 Bir69a] that 71 Miff and Z (91) 92) for b 0, where the generators g, and 92 are represented by uniform from the The exact sequence then follows rotations along one of the S' -directions. Pro
I.t is implied
of.
by [Bir69b,
=
same
Diff
fibration
+
(S'
SI)
x
Now, the flow of gi defines X
which
:
S"d+ (1pi,
gives
a
covariant
with
Z (gi,
C
g2)
splitting
Diff
respect +
(S1
of X. Thus, also the the Pb(S' x Sl)-part center
see
-
.
,
pb I
before.
vector
field
S'
S')
x
on
S'
x
S1, and, thus,
a
(UbD 2'S1
X
-c'mbd+
--
section
Sl),
of the sequence in Lemma1.4.7. By construction X is also of the the that the action of the so to subgroup image gj in the image in EMU+(Ub D2'S1 x SI) lies entirely x Sl)
image
again [Bir69b,
.
as
a
of 7r,
of the
(Diff
splitting.
Bir69a].
+
(S'
x
S'))
For the fact
in
_ b (S'
that
there
x
S')
lies
it coincides
entirely
in
with
the
SomeFacts about Handle
1.5
Some Facts about Handle
1.5
Decompositions
51
Decompositions
The purpose of this section is to provide some basic technical tools that are needed for Section 1.6 as well as for the proof of the main presentation theorem in Chap-
handlebody decompositions and surgery are introduced in Secthe method of simplifying handle decomposition'by canparticular, is described. cellation This is applied 1.5.2 to describe 4-dimensional in Section handle decompositions and 3-dimensional in terms of links and, surgery explicitly links. Cancellations handle and slides more generally bridged are, thus, given by diin Section we 1.5.3, agrammatic equivalences. Finally, explain the effect of a surgery which on a 3-manifold a curve along through an attached 3-handle. passes parallelly ter
of
3. The notions
1.5.1.
tion
1.5.1
In
Surgery,
Handles, section
In this
let
us
Isotopies
review
and Cancellation
some
elementary
facts
about handle
attachments
and
dimensions as well as the surgery in arbitrary and 4. The two notions are closely intertwined
features of dimensions 3 particular and yield basic procedures to build manifolds. The existence and equivalences of handle decompositions or arbitrary is based on the theory of Morse functions. This will be dissurgery presentations cussed in greater detail in Chapter 3. Here let us only use this as a motivation and continue our discussion using only the notion of handle attachments. k + 1-dimensional a compact, connected, differentiable Vo U (W x [0, 1]) U V1, where the pieces are fit along 19V
Assume for
that ON=
-
Weconsider
f
:
(N,
VO,
which restrict
boundary.
functions
differentiable
Vi)
([0, 1], 10}, I 1})
--+
to the
In this
from N to the unit
projection
Va
the second factor
on
following
the
situation
with
is
a
basic
interval
=
on
fact
f
manifold x
Jj 1
N
09Vj.
5---
of the form
-'(a)
for
the OV x
a
=
0,
[0, 1]-part
1
,
of the
theory.
of Morse
(e.g., [Mil691). Let f be a Morse function on N as above, which has exactly one non-degenerate singularity of Morse index j. Then there is a homeomorphism between N and the cylinder over VO with an attachment of a handle of index j: Lemma1.5.1
N
Vo
x
[0, 1]
Uejk+l
.
9
Here the
boundary
is
j-handle given as
ejk+l
is
a
k + 1-ball
written
as
o9ejk+l
--
Si-1
x
U
Dk+l-j
Di
si-IXS' -i
Moreover,
G denotes
Di
x
Dk+l-j
the union
an
embedding !9
:
Si-1
X
Dk+1-i
V0. f
X
Sk-j.
So
that
its
52
The Double
1.
The
Si-I
Category
3-Cobordisms
of Framed, Relative
quotient space in Lemma1.5.1 is then obtained by identifying with the corresponding x Dk+l-j C ejk+l point (!9(x), 1)
(0, 1].
is glued In other words, the k + I-ball cylinder along a tubular neighborhood of an basic lemma shows that isotopic The following considered as equivalent.
Vo
x
of the
point
every E
Vo
X
x
11}
to the upper k-dimensional embedded j 1-sphere.
E
C
side
-
ought
attachments
handle
to be
into the boundary If two embeddings go and!9j of Sj-1 x Dk+l-j then there is a homeomorphism between N are isotopic, L9N of a k + 1-manifold two manifolds NG,, and. Ng, obtained by a j -handle attachment as in Lemma1. 5. 1, k+1 which is supported in a vicinity of o9N and the handle ej
Lemma1.5.2.
-' x 9t : [0, 1] -+ 9mbd+ (Sj Dk+l-j, ON) can be extended to an ambient isotopy. See Chapter 8 (Theorem 1.3) + t -+ Pt : [0, 1] -+ Diff This means that there is a diffeotopy in [Hir76]. (19N), a us to construct such that gt on ON. This allows Pt o go and 4 0 is the identity
Proof.
It
is
a
standard
that
fact
isotopy
an
t
i-+
=
diffeomorphism.
on
TV
on
V maps the N-part
N UONx {01 ONx [0, 1] as follows. It also maps the aN itself by identity.
to
x
[0, I]-part
to
TV(y, 0) y so that it is ( Pt (y), t) for y E M. In particular, by Tf'(y, t) defined on the union N U8NxjOj aN x [0, 1]. Since one can choose continuously of 0 and 1, TV (y, t) can be the isotopies 9t and !Pt to be constant for t in a vicinity assumed a smooth diffeomorphism. a diffeomorphism straightforwardly Using a collar of N one constructs and 6 aN x [0, 1] =-+ N, such that 6: i9N x I 1} -+ 9N is the identity, N UONx 101 1 o Tf o 6-1 of ONon N. The map Tf E outside of a collar vicinity is also identity + x Dk+1 -j, !91 E ETnbd+ (Si-I o9N) and Diff (N) has now the property TV o!go itself
=
=
=
=
is
TV
supported : Ng. =
a
collar
Ng,
with
in
The notion
vicinity
of ON. Hence, TV
addresses
of surgery
be extended
can
to
a
smooth map
properties.
the desired
now
question
the
how
description
a
of the other
from a V, C aN, can boundary component, namely, horneomoris which of 1. The image a piece of N as in Lemma1.5. !9, presentation with a part of the boundary of the j-handle and, phic to Sj-1 x Dk+l-j is identified aV0 Lj Sj-1 X Sk-j' hence, it will no longer appear in V1. Wehave 9 (Vo im (9)) where the latter component is from aim (9). Yet, the remaining piece _2--' Di X Sk-j is now added to the total boundary. This is done by in the boundary of the j-handle Sj-1 X Sk-j a(Di X Sk-j) to the extra boundary gluing it along its boundary component created by removing im(g). We, thus, have the homeornorphism:
be derived
the k-manifold
=
-
=
V0
V,
5 for
The relation
i9(Di OVO
X =
Sk-j)
with
aV, again.
-
9(Si
-1
x
Dk+1 -j)
quotient space identifies corresponding point!9(x)
U Di
the
the
Wesometimes
also
Vi
use
=
a
E
the notation
(VO)
g
X
Sk-3.
point
a(Vo
x -
E
(1.5.1) Sj-1
im(!9)).
X
In
Sk-j
particular,
C
1.5
for
surgered manifold. boundary pieces
a
The
aek+l give, hence,
of
about Handle
SomeFacts
rise
submanifolds
to
53
Decompositions
in the k-dimen-
as neighborfollowing notions for these spheres themselves in the center of the respective neighborhoods: 'Yk-j submanifold 0 X Sk-j C Di X Sk-j C V, ascending critical 11 3S3_1 X 0 C Sj_1 x Dk+l-j critical submanifold C V0. descending Sdes For these sphere-embeddings have a natural of their trivializations we clearly unit normal bundles in the respective with metric. some k-manifolds, equipped 1 = sk-j X Sj-1, 01 : V i (sk-j) :=4- Si-I X Sk-j. 00 : V1VO (Si-1) d
sional.
level
manifolds
hoods of embedded
Vo
and
spheres.
V,
Let
that
us
can
the tubular
both be viewed
introduce
the
=
=
SC
es
it follows from basic tubular Conversely, neighborhood theorems (see again determines a tubular neighborhood up [Hir76] Chapter 6) that any such trivialization to an ambient isotopy. By Lemma1.5.2 it is, thus, enough to consider only the isotopy classes of trivialization [,Oj] for the embedded spheres in order to characterize
handle attachments all
Since
there
and surgery.
compact differentiable
only finitely
are
cal value,
we
know that
decomposition
11 }
series
of
(((V
(...
expression
x
V
such manifold
every
we
[0, 1])
x
responding
a
above has
as
ej,k+1)
U
ej2k+1)
U
U
...
critia
handle
k+1
ej,
.
.
.
.
,
V'kin
is
t
k
decomposition and descending
Sdj'-
-il sc
1
intermediate
standard
composition Lemma1.5.3.
manifolds
n
level
((V)91)92
of transversality general position
In
handle
a
generic
are
or
whenever
,
as, for
generic
6 -+ 0) is connected to the segment that x IN } (with Sdl,,, x Sl ,c at some point I qk I X IN I
seen
to the torus
we use
of A over
to make the
the segment in the time tk. As it can be is
Isotopy
beforelhe
the deformation. A
the
Sectional
1.13.
or
along
ribbons the
instead
framing
of
of the full
tori
to describe
1 SaLSC or given by splitting
surgery, the ribbon
1.5
S,',,,,
is, dividing
Some Facts about Handle
Decompositions
61
and
using one component R. The ribbon slicing original framed. also implies the case, where we isotope a line segment At, which is by itself that a framing of a segment is, in This follows, for example, from the observation of itself so that turn, determined by a push-off sliding the two copies one after the it over. other determines the framing of the segment after sliding If two ribbons Lo and Hi are in In summary, a 2-handle slide is given as follows. a tangential position as on the left of Figure 1. 14 then we can obtain the ribbon conribbons Lo along its longitude after the slide by cutting into two parallel figuration Lo' and Lo", and form the connected sum between Hi and the one component Lo' is depicted on the right side of Figure 1. 14. to Hi. The result or Lo" that is closest R
x
as
the extra
that
loop
and
the interval
identifying
R into
the other
CHi Fig.
The uation
exactly
only type
is between
1-handle
point. In a pair through passes tion consisting of the one
contractible,
1.14.
ISOtOPY
of Smale cancellation, a
parts
the
/X
and
surgery of surgery open strand a
a
Of
SaIsc
as
over
Sdles
described
2-handle,
diagram balls C,
Awl
Lo
>
Hi
Lo,,,
two
with
in Lemma 1.5.4,
manifolds
whose critical
this
manifests
and
Cl' exactly
C2, with the
itself
by
once.
Since the
two balls
attached
a
in
ribbon
to it
our
sitin
intersect
C2 that
configuraat the
ends,
isotope it into a position as indicated on the left hand side 1 -slides 1. 12. In this situation of Figure 1. 15 using, possibly, as in Figure we may still have ribbons a, b, c running through the surgery pair C, and Cl'. They can be slid over the ribbon C2 and removed parallelly from the configuration. Hence, after the ribbon C2 is cancelled against the pair of surgery balls these strands are connected in the way indicated hand side of Figure 1. 15. on the right and surgery in Chapter 3. For We shall come back to handle decompositions and properties of bridged links see [Ker99]. more details is
we can
...
1.5.3
3-dim Handles
In the construction
and
Surgery
of the extended
categories as in Section 1.6 as well as for the in which tangle presentations subsequent chapters, we encounter situations 3 dimensional 1-handles and we carry out a to a given 3-manifold, are attached data running through the 3-disurgery, as described in Section 1.5.2, along attaching in the
mensional
handles.
Weconsider
in this
subsection
some
situations
for
which
such
62
Category
The Double
1.
b
a
a
equivalent As which
a
we
*
simpler configuration
attach
MU
manifold
a e
3
8
consider
we
consider
3-manifold 3
an
index
el
--,-
Da. attaching
with
2 surgery
2
x
W to
boundary
Mwith
[0, 1]
ine
following properties. of the 1 -handle preimage L-1 (e 3) 1 C C S1 is a closed, connected piece
On the combined torus
f-
:
S'
x
,C: C where p, c-
[0, 1]
C =4
'62
is
2
C Da is
a
i nterior
precisely
to be
torus
segment
On this
segment
a
of the circle.
of the map C be of the form
let the restriction
x
a
a
2
[0,1]
D8
2
Da
=
3 eI
embedding
of discs,
(1.5.5)
)
manifolds,
of 1 -dimensional
diffeomorphism
fixed
x
such that
one
disc
and lies
jD
in the
of the other.
This condition
by slightly
means
3 e1
shrinking
The manifold
after
that the
into
itself
that its
boundary
OM'z
is
=
3
image of the surgery torus in e 1 is precisely given hand side of Figure 1. 16. on the right as depicted
handle attachment
M` so
a
1-handle
the
DS2, where
x
D28
are
operations.
3-dimensional
we
3 and 4
in mixed dimensions
and surgery
3 with the MUe 1
2
Werequire
*
attachments
of handle
between 1- and 2-handles
Cancellation
1.15.
much
first
C
C
Fig.
to
b
a
C
b
combinations
3-Cobordisms
of Framed, Relative
=
and surgery
(M
W -
10, 1}
x
D2 a
as
3),C
U el
given by the usual index-
is denoted
-
1 surgery
U
j0,Ijx8D.2=8CxS.1
on
W:
CX
S!
,
..S
Soiu,
63
Decompositions
aboathaudie
!,'aas
-3
_-L
dM8'
C l
dM?
4;z?
M?
Fig. 1.16. Surgery with 1-handle
where
we
ian
SJ1,
this
situation
of
identify e
the
3and aC with 1
10, 1} by
2
merid-
with the short
manifold
consider
to
in
is
M? where the part
2
D.,
COPP x
`
intersecting opposite" index
1 surgery
OM?
9M
as
im(,C),
M
-
COPP
with
S1
=
-
C, of the embedded surgery
boundary
Its
given similarly
is
by
an
follows:
10, 1}
-
=
Mis removed.
the manifold
torus 41
Another
above.
as
p,
D,,
disc
of the cross-sectional
boundary
x
U
D2 8
COPP X SiX
j0,1jx,9D.2=,9C*PPxS.1 the D2a holes
By shrinking
the DS2 holes
into
out
C=4 COPP, which is the identity
on
diffeomorphism P,9
:
the
a
restricts
is the on
identity
on
identification
obtain
a
natural
now
that
also the 3-manifolds
M? and
diffeomorphism MIZ
which
an
we
'9MIZ =4,9m?
unique up to isotopy. The claim is Mk themselves are homeomorphic. There is
choosing
c n Copp
endpoints
which is
Lemma1.5.7.
and
Moutside
the boundaries
to
M?
=
of
a
of the
vicinity
union
L U el,3
and which
the map p,9.
3 Proof The first step in performing the surgery along L on MU e I is to remove the CU COPPthe partition image, i.e., the attachin 9 torus !--- (CU COPP) x D2with S1 =
8
the two segments as defined and surgery torus removed can, thus,
of the circle attached
into
(MUe3)
_
iMp
=
(M
-
L(COPP
x
above.
The manifold
be written D2))
with
1-handles
in the form:
U
Cx
(D
2 -
a
D32).
OCx(Da2-D2) The closure the second
of the first
piece
piece
in this
is, clearly, nothing else but M?. In presentation 2 the boundary collar of D unique isotopy
the set Da2- D28 is up to
a
64
so
The Double
1.
that
S'
with
identity
the
Da2
=
of Framed, Relative
Category
from above
(M
can
U
e,3)
D2
write
we can
im(L)
J
=
S
be rewritten
-
D2
-
a
3-Cobordisms
x
S,'
[0, -F). Hence,
where J 5---
,,
as
U
M?
=
C
x
7
x
S,',.
Wx7xS! Before
we
that
note
a
complete
by regluing
closed
of the removed torus
[O,s]xCxOD 2.
The collar
8
7 and by radial
with
of the deleted NU
(C
x
To this so
the surgery collar vicinity
that
we
7
assignment
torus x
S,,')
(C
=
we now
we
glue
U COPP)
opposite
the
opposite
in normal direction
identify x
7
1-handle
D02
torus
S,,,
X
let
us
in Mis of the form N
of the torus
S,,'.
ODS2
N and the attached
Hence,
a
is identified
can
be identified
combined
with
vicinity
the thick
torus
x
along (CU COPP) x 10}
torus
x
S,"
09DO X S1 2
find
NU 2 where DT
interval
the
=
(C
(C
x
x
U COPP)
S,,
2
U D0
X
S1
2
DT
x
U(CUCoPP) x f0j=8D2
X
D2 is
X
a
S1, with
disc
annulus
an
Sx1 now atboundary, piece M N the COPP x Sx'. Clearly, complement along boundary piece 2 this attaching operation can also be seen as the attachment of the disc DT to a local surface along a connected boundary piece COPPmultiplied with Sx1. cross-sectional This attachment is trivial with the circle. Since the 2-dimensional so is its product M?. shows that the total result of surgery is diffeomorphic to M N attached
and, hence, again
to its
tached to the
a
disc.
2 DT
This
is
X
-
-
A
A
A
A
X, ---
-------
OPP
N
N
Fig.
The situation
Figure
can
1. 17. On the
also
right
1.17.
be summarized
side the manifold
cross
sectional
with
1-handle
in the
(M
U
cross
ell)
-
sectional im
(L)
is
picture depicted. 7 x 11,
strips through the 1 -handle C x N around the removed torus vicinity also of the form COPP x J presentation
upper arcs are the cross sectional 1 E S1. In the lower part the
which is in the
Surgery
is x
given
in
The two -
11
with
depicted,
f 1,
-
11.
1.5
configuration
This
Cx i
cylinder
OC x i
annuli
right neighborhood on
C)
the
x
J
areas.
The
in the
cross
naturally A
through
run
S,1,
x
to
the
1}
-
---
Do
regluing sectional
diffeomorphic slightly
x
of
more
attached
11,
Do
S,,'
sectional
view
combine
with
which in the
we
changing this yields
the thick
let
attaching the picture
the
of the
the components
precisely the space (CIPP U surround two correspondingly, in the picture as the patterned are indicated thus, fills up these areas surgery procedure,
It is then obvious
view. to
1},
-
x
cross
strips
annuli, which x T The annuli,
"U-shaped" S1 x i U S1
where
equivalent to the one, CIPP x 7 x S,',, without
The two
1.17.
N to two
discs
two
obviously parallelly
C OM. In
Figure
of
11,
x
"U-shaped"
is
S,,'
x
65
Decompositions
about Handle
Some Facts
are
results
this
that
in
a
manifold
that
is
M?.
involved
situation
1 -handles.
is
when
given
This situation
a
is described
L passes torus surgery in Figure 1. 18. The two
e3are attached to manifold at discs in disjoint neighborhoods UA, UB, Uc, UD C OMin the boundary. The link L also splits up into four pieces of the form with e3n im(L) are given by the intersections D2 x [0, 1]. Two of them L.,,pli, handles
=
the 1 -handles,
sections
Figure
and the
of the surgery 1.18.
remaining with
torus
two
pieces L 1
U
L2)
M. The situation
=Mn
depicted
is
im(L) on
the inter-
are
the left
of
side
3
el
'
UA
UA
UB
ZI
Z2
M?? UC UC
UD
,
Fig. 1.18. Surgery with 1-handle
The manifold
by
with the two 1 -handles
M8'8'
Its boundary is given by removing cylinders so that
=
discs
e3 and the
sur
9 ery
performed
on
it is
given
e 3),C (MUe3U 1 1
D2 A
C
UA,...
D2 D
C
UD and regluing
66
The Double
1.
3-Cobordisms
of Framed, Relative
Category
OM&&
=
SAB X[0) 11LJSC1DX[0i 'I-
(OM-(D A UDB UDCUDD 2
2
2
2
AIB:--SAlB
OD2
10/11,8D2
x
=S1CDX10111
CID
homeomorphic boundary
as follows. can be obtained First, we additional obtain it that two Mfrom so we C L, UL2 cylindrical parts -5-- S' x OL2, both of which are naturally OL, and Z2 boundary pieces ZI with M?? this from obtained is manifold The then Z2 so Z, identifying by [0, 1].
A manifold
remove
with
a
the
=
=
boundary piece ending
that the circle in
UA is identified
in
with the
corresponding
one
UB: U L2)
(Li
M
-
??
m
Z1
Z2
=
is obtained by removing the end boundary Correspondingly, 2 2 discs which the the from of are boundary, precisely DD, and DA pieces theCj gluing the respective endings of the Zj pieces together, which are just the circles
of this
the
2 0 DA
manifold
2
aDD am??
Comparing the expressions pieces push the cylindrical in OMand thereby construct
M
-
=
___
use
=
SABICD
Lemma1.5.8.
Proof Figure that
on
The
1.16,
we are
am??,
we see
of the discs
that
we can 2 DD
2 DA
=-+,gm??.
to show that
There is
a
this
extends
to
a
homeomorphism
on
difteomorphism
m??
m&& :::4
restricts
and
collars
itself.
the 3-manifolds
which is the
into
OD2 D
diffeomorphism
:'qm"'
Lemma1.5.7
=
aMSS'
[0, 1]
X
natural
a
.
__
2 OD2 ODC B;
for the boundaries
P'98 Wecan in fact
(D A U DB U DC U DD
,OD2A
identity
on
Moutside
the boundaries
configuration if
we
a
vicinity
above is
a
special
Rvr the 3-nnmoifold
here with the lower
1 -handle
of the unionC
Ue
3
U e 3, and which
the map poo.
from.
substitute
considering
To the added upper apply Lemma1.5.7.
to
of
case
from the
Mthere
1 -handle
one
described
the 3-manifold
in
MU el3
between UC and UD attached.. torus L we then
between UA and UB and the surgery
also closed tubular let us introduce neighbordescription such that the inclusion and L, C N, is L2, pieces C1 2 2 given by [0, 1] x D C [0, 1] x D and L, C N, by [2,3] x D' C [2,3] x D' that of L as already in the proof of Lemma 1.5.7. in a parametrization extending For
a more
precise
hoods N, and N2 of the 8
a
S
a
Swrie, r'
1.5
Moreover,
and the
where each of the four
2
U N2 U e31 U e 1
3
pieces corresponds
Da
e,1namof naturally
is
pieces ODa x cylindrical The image of )C in the above
identification
of
respectively.
identified
construction
of discs
as
2
D.
with
Also,
in Lemma1.5.7.
express
2
as
X
ST1,
before,
segment of the circle SIT. In particu3 D2a X [0, 3]. The write N, U N2 U e 1 Da2
x
[0, 1]
parametrization where we
2
D.
write
(Ni U N2) by aNj and aN2 of N, U N2 U e3 U e 3 is
[0, 3]
and aD2a
with the upper 1-handle
the manifold
ST1
X
to a
lar, if the upper 1-handle is omitted we can manifold MU e 31 can, thus, be given by gluing
we can
67
with
N,
by
attaching discs neighborhoods
with the
Hence, the union of the 1 -handles
the 1 -handles.
identified
coincide
n amwill
Nj
the end discs
Decompositions
abcoutllui e
6
C
D2
.62
a
a
-
with
is the fixed 2
-
M
into
[2, 31
x
DS
-
[0, -r)
and the surgery
embedding
S,1,
x
torus
so
that
removed
as
(MUe3)
-
i7n(L)
=
M
-
(Ni
Li
[0, -r]
N2) x[0,1j;,9N2={0jxS.1
8Nj=j0jxS.1 A
cross
of
Figure
sectional
view of this
construction
is
given by
the
x
S,,'
x
[0, 3].
X[2,3]
diagram
on
the left
side
1. 19.
[Ojx[2,31
toix[0,11
Z2
V-1
Jx[0,31
Fig. 1.19. Surgery with 1-handle
Obviously, the attachment can again be understood as a gluing of 2-dimensional with S.1. Namely, the region [0,,r] x S,1 x [0, 3] is given by the "Umultiplied hand side of Figure 1. 19, and it is attached either or right on the left shaped" pieces, and x the line jo} x [2,3]. Since this is topologically [0, 1] along segments 10} it with a square, such that identifies is there but disc a a diffeomorphism nothing that the two line segments become opposite edges. of the square it is obvious that coordinate the horizontal However, by shrinking the two edges it is glued to. In to identifying the gluing of the square is equivalent the circle product this means that instead of gluing in [0, T] x S,1, x [0, 3] we may as well identify the pieces 10} x S.,1 x [0, 1] and 10} x S,,' x [2,3]. This is, hence, the boundary piecesDN, with ON2 in M (NI U N2). equivalent to identifying of the pieces CI But the tubular pieces N, and N2 are nothing but thickenings in the desired way to and L2. Hence, the manifold that we obtain is diffeomorphic M??. surfaces
-
68
Category
The Double
1.
1.6 The Central
of Framed, Relative
04
Extension
physical
Most of the relevant
db
-+
topological
3-Cobordisms
Gob
-+
quantum field
theories
of the category Gob. The largest class are (projective) derstood as a representationnot cif the ordinary cobordism
by
thereof
class
Specifically,
Z.
of 3-cobordisms
ditional
with
pairs (M, n),
consider
we
and
corners,
n
E Z is
a
double
are
but
category
where Mis
an
not
TQFT's that
tors
a
truly
funcbe
can
homeomorphism
which indicates
integer,
un-
extension
an
ad-
an
structure.
In this
section
we
extend the definitions
make the Z-extended
3-cobordisms
into
of the two types of
compositions
00b.
category
to
so as
The constructions
of the
involve and the integers are naturally compositions bounding 4-manifolds, Thom-cobordism the in dimension four determined by the as 04, interpreted group of the 4-manifolds. between the categories Hence, the relation can be signatures viewed
as
central
extension 1
remaining
In the
Gob
egory
6bB,
category contains
a
2-arrow
which
&7b are
COB-4 d7b. In
ings
and handle
modulo surgery
on
a
with
Gob
chapter
we
classes
&iB
the 4-manifolds
are
We obtain
Wso that
classes
are
a
of aW form.
both
The
projection
defined
via
glu-
axioms
category
double category
strict
Cob preserves
cat-
large
of the 3-man-
natural
a
the double
we obtain
of the
sets
standard
compositions
shown to fulfill
dib--+
the restriction
prescribed
homeomorphism
and horizontal
They
a
double
One 3-stratum
corners.
have
of the 4-manifolds.
vertical
the extended
of the 2-arrow
Wwith
manifolds
(1-6.1)
1.
--+
construct
the definition
given by taking
then
attachments.
Cob. Furthermore,
it is
of this
four
are
dib
)
from Gob and the others
and cobordism
ifolds
r-
We start
cobordism
sets of
f24
---+
sections
explicitly.
follows:
as
compositions,
for
i.e.,
double functor.
The cobordism
appearing
categories
in this
section
are
summarized
in the fol-
lowing diagram:
ebB mod surgery
&
COB
homeornorphismISg d4b
Let
such
as
us
first
the almost
double
cobordisms. Section tion
justify
Chem-Simons
1.6.3.
1.6.3.111
category
the first
7rc
Gob
the particular extension we choose from physical models Theory in Section 1.6. 1. Then in Section 1.6.2 we construct
Compositions In particular, -
(1.6.2)
mod homeornorphism
66B
of 4-manifolds
and the double Section
1.6.3.V
bounding
category describes
step of the horizontal
d4b
3-dimensional are
then
the vertical
composition,
relative
constructed
composition; in
which
a
series
in
Secof
4-dimensional
composition show, using
dib,
COBis also
on
auxiliary
db
the final
-+
attach-
2-handle we
verify
that
in Section Finally, interchange law
that
this
1.6.4
we
holds
the
COB. This proves
category
69
Gob
on
Gob
that
double category
a
2-Framings,
1.6.1
on
-+
1.6.3.H3
0bb.
on
attachments,
the
-
and in Section
defined
well
handle
it is not valid
1.6.3.H2
Section
composition;
5-dimensional
although
is indeed
attached;
is
the horizontal
for
ments
1-handles
f24
Th Canuai Extension
1.6
of 3-Cobordisms
and Closure
of the Chemexciting and important observation of Witten's in the construction it is as Simons theory [Wit89], see Appendix A, is that, although purely topological of the structures a classical theory, the quantized theory will depend on additional of In the quantum group constructions 3-manifold, namely framings or 2-framings. data additional encounters a dependence on Reshetikhin and Turaev [RT9 1 ] one also matrix of a representing of the linking in this case the signature surgery diagram, These two extension of a bounding 4-manifold. which is at the same the signature Indeed, the possible set of topological are in fact quantum field theories equivalent. that do not insists reduced if one on ones is dramatically depend on framings or since these are merely controllable is rather At the same time their nature signatures. of out TQFT's. Z -extension has to be restricted functional when SCS is quantized, integration Specifically, has This means one to impose a "gauge of gauge classes. to a set of representatives of a metric p. Witten [Wit89] which in turn requires a choice condition, fixing" computes that the dependence on this metric is given by an overall phase factor obtained from the 71-invariant 77(p, 0) of an associated Dirac operator, see [APS75a]. The fact shown in [APS75b] that Scs(wp) 377(p, 0) mod Z, where wp is the An
-
=
Levi-Civita a
on
candidate
bundle
the tangent
for
a
term to the
counter
keep
One needs to
Chem-Simons action
ordinary
mind, however,
in
M, makes this functional
TM of the 3-manifold
Scs(B)
although
that
values
B on TMany lift to a functional for any connection that is an isotopy class of trivializations choice of a framing,
with
That is,
[Wit89] no
we
a#
:
for
can,
an
of
a
S s (B)
branch
on
E R. As a result
on
R/Z
in
R depends
R3
TM: 4
the invariants
-+
on a
x
constructed
M. in
H'(M, Z/2) [M, SO(3)],
-+
and thus
trivialization
counts
defines 4-manifold
sign(W)
the
[Ati90]
in
from
while
canonical
Wby the fact 1 =
6
pi
a
corresponds spin structures a
the
1. A class
sequence 1 -+ H'(M, framings is (non-canonically)
exact
of
2-framing to
and
in
Chern-Simons
For non-abelian
in
H1 (M, Z/2) then
but, by the relations
framings,
itself
be understood
example,
element
Atiyah bounding
such
the metric
on
[M, SO(3)] by
a
[Wit89].
[APS75b] [APS75a] theory the invariants fact, depend only on a 2-framing, meaning an isotopy class of trivializations and 2-framings between framings RI M. relation The =4 x TM E) TM
depend
longer
do, in
obtain
a :
in
is defined
an
only
sees
element
in
Z)
-4
given
the mapping degree The term H3 (M, Z) .
in M.
2-framing
for
that the Hirzebruch
(7W (D TW, a#) W
a
3-manifold
formula
with
M = 09W
Mwith
a
70
The Double
1.
holds, the
of Framed, Relative
where p, is the relative first It turns out at the boundaries.
tions
one
that lifts
SCIIS(wp)
i.e.,
Category
the action
on
77(p, 0)
in R,
=
Thus this
p extends.
framing
Pontrjagin [Ati90]
3-Cobordisms
class
that
the Levi-Civita
with
respect
canonical
this
connection
to the
given that Wis a 4-manifold precisely the desired one for
is
to given trivializaframing is precisely originalq-invariant, to which
the metric
the renormalization
of
the Chem-Simons functional. Instead of 2-framings to bounding 4-manifolds,
suffices the signatures of to consider framings and metrics can be extended. We still need to extend the choices of canonical 2-framings from closed manifolds to manifolds with boundaries and comers, as well as explain the composition structure of the extended data. Especially the latter will occupy the larger part of this section. To begin with let us consider an ordinary cobordism, it thus
metrics
or
which
such
zg_
M':
Eg"
---+
(1.6.3)
surfaces.
Whenwe impose a 2-framing on every such cobordism., we have compositions give us again 2-framed cobordisms. This forces us to impose boundary conditions, i.e., we can allow only such framings on M, that the restriction to every boundary component of closed
to make
sure
that
g is
a
fixed
trivialization
ializations
by
a
:
TM,,
2
that
I Z'
depends only
simple tangent embedding of Z.
in the
standard
of the Euclidean
space
For each g > 0 let body of genus g into
us
as
indicated
consider
the
bundles x
[0, e] in the
an
R
--+
x
[0,S] on
Eg.
If
Zg we
[0, e].
x
consider
framings)
(ordinary
a
for
simplicity
choice
can
triv-
be made
V and using the induced trivialization.
C
figure
unknotted,
below:
untwisted
embedding
of the handle-
3-sphere, W+ 9
=
e
3 0
U e31 U
3
...
...
U e1 C
S3,
(1.6.4)
I
9
where g
ejd
--
Moreover,
with OR
x
Dd-i we
shall
denotes use
Zg. Obviously, framing depicted above
g
of collar of R3
Di
or
=
S3.
the
a
j-handle
of dimension
d. We identify
complementary handlebody lig framing and associated 2-framing g
S3
the
also extend to the standard
Zg H+
, _
9
for the choice
handlebodies
as
subsets
The Centrai
1.6
for
Of9S2
of
of the g, and the restriction also assume that the standard
every Let us
.
embeddings
we may choose
Moreover,
W-4-
J24
Extension,
spheres 2-framing sphere
S12
-+
c--+
on
S+2
is inside
6ib
Obb
-*
SI,
71
S2
such that
C
coincides
with
the initial
0-handle
that
i.e., S+2 C el.0 This allows us to extend the (1.6.4), decomposition to the connected sum of two such handlebodies, standard framings and 2-framings for which the cut out balls are bounded by and one complementary, one standard connected them naturally as the standard write We S2 can and respectively. S+2 in the handle
cobordism:
,H9`9
=
9-
to
compatible
be
(M,,)
sure
that
their
restrict
common
have the
(M.)
=
a
g,
with
M,,
by gluing
tained
we
requiring
of
Now, instead
may
as
which
I I
g.
general
ones
the
on
by
(1.6.5)
Eg_
-*
consider
is thickened
identification
wgtg
zgg
cobordism
M,,
2-framings handlebody
on
corresponding.standard
the
surface,
following
on a
well
the standard
to
:
gtg
2-framing
we
to
W+ #?C g=
cobordism
an
Egg
-4
This
pieces. from
[fo, fl].
interval
Eg,.
:
the standard
clois ob-
(1.6.5) along precisely,
More
space:
(_ Eg-
U
ZT91)
X
VO f 1 1
x,
I
(_Z9_ uzgtg)
(_Z9_UE9tg)X-(fOI
X
(h 1
(1.6.6) which :
we can
consider
13-cobordisms
as a
map between classes:
formal
surfacesl
of closed
-+
Iclosed
3-manifolds
with
2-strata}
X
-+
(Mo).
properly to ligg!9- are, thus, in one-to-one 2-framings on (M,,) that restrict correspondence with the 2-framings on M, which yield the prescribed ones at the boundaries. Using the correspondence from [Ati90] this structure can thus be further substituted by the signature of a 4-manifold that bounds (M,). This will be our point of view in the following. of the the 2-framings of surfaces on collars Since we have fixed 2-framings the Correspondingly, extend to the gluing of the manifold over surfaces. 3-manifolds if closures standard we glue framings extend to 4-manifolds bounding the respective that appear in their boundary. over the surfaces these 4-manifolds along cylinders into the gluing of framings can be translated and extension Thus, the composition The
of 4-manifolds.
MObetween closed surfaces
Now, the cobordisms set
of the 2-arrow
tween are
M,,
obtained
sets
surfaces
connected
of the category with boundaries
by restricting
MU,,Sl
,
[0,I]
surface
2-framing
a
nD2
boundary pieces between from This is justified pieces --- S1 cylindrical than the horizontal
are
only
Cob. For the relative
X
[0, 1],
the holes the x
[0, 1]
the we
very
special
subMbe-
only 2-framings that ordinary cobordism corresponding have "filled up" the c- ` S1 x [0, 1]
shall
consider
in the surfaces.
picture of Chern-Simons theory, where the r6le play a different boundary Z_,;4 5--- W "Wilson-lines" "currents" or to They correspond
geometric
pieces.
on
where
we
a
cobordisms
in the
72
1.
The Double
inside
the
serted
in the
Category
three-manifold,
physical
that
can
tion
of the field
and the holes
state
be evaluated
of Framed, Relative
against
space.
have the
3-Cobordisms
meaning
Hence, they correspond
"charges"
of
to a
choice
that
are
in-
of observables
given quantum field theory. But the construcdepend on this choice. Thus, the 2-framing needed for the construction should extend over tubular of cylinders, such fillings that it is compatible with the standard framing of D2 X [0, 1]. Finally, we need to explain how the 2-framings will extend under the horizontal Note that part of the horizontal compositions. composition is the gluing of neighborhoods of holes in corresponding surfaces to each other. This needs to be done such that the two standard 2-framings extend to the glued surface collars and yield the standard 2-framing of the resulting surface. One way of achieving this is to align two given (collars of) surfaces in RI and attach a tubular piece A x [0, 1] to the annuli -5-- A x 10, 11 around corresponding holes. As indicated in the top piece of Fig. 1.20 the embedding in R3 then induces the desired framing. In the horizontal composition this procedure is applied both to the
source
theory
itself
and the target
states
should
in
a
not
side of the cobordism.
Fig. 1.20. Standard 2-framings
We can consider
mensional. then
1-handles
remove
a
torus
the to
D2
of surfaces
where resulting space as the 3-manifold, the cobordisms, in which the tubular pieces x
SI along the
curve
T
=
W1 U C1
U
we
attach
are
W2 U
filled,
C2.
3-diand As the
The Cen'tral
1.6
extends
Wj it, thus,
lines
compatible
is assumed to
framing
framing
induced
But this
of
means
a
we
opposite
the
torus
For
give boundary parts we bounding 4-manifold
a
X
67b
-+
be able to extend
can
now
x
S3
D2
-
S1 Hence,
x
.
the
one
along
4 and surgery
extend.
in di-
attachments
1-handle
translate
in dimension
the
framing. by a surgery along T. precisely the gluing of the gluing.
original this yields
that
the
with
S' into S3.
in and still
torus
73
[0, 1] framing along
where it coincides
embedding opposite
to the horizontal
Obb
-+
of D2
rise
attachments
1 -handle
mension 3 into
S',
x
from the 1.5.3
in Section
that
cylindrical
D2
the
to
then differs
obtain
from the results
Wefind
unknotted
it also extends
that
glue
we may instead The 3-manifold
over
standard
D2
the standard
with
the said torus
S?4
Extension
into
a curve
also the
that
respective framings we bounding 4-manifold with this surgery description 2XS1 D the torus normal the on with is note that the framing framing "incompatible" sections we shall define the and refer to Lemma0 in [MK89]. Thus in the following extended the a structure as with horizontal composition of 4-manifolds composition the gluing procedure as described above. we obtain with handle attachments so that attachments,
2-handle
spin
1.6.2 In this
section
we
closure,
structures
closed
surfaces
a
precisely,
ing of
a
closed
surfaces
of COB(a,
2-arrows
as
W,
manifold
smooth four
bedding b,
W,
M', by four-manifolds,
Moreover,
we
introduce
from the cobordism
to construct
us
with boundaries
of surfaces
cobordism
relative
on
as
they
are
ad-
M,, of used in
of Gob.
the definition More
see
6UB (M,,).
aW
i.e.,
W, which allow
on
of
2-framings
the admissible
replace
which bound the standard ditional
and 2-Arrows
4-manifolds
Bounding
To
to the
extends
structure
the
to which
an
M, a surface coordinate M,,O, p, b), where p
(W,
b)
chart
(M,,)
:
tuples
C COBwe consider
oriented,
compact,
consist-
cobordism
connected
of
homeomorphism p, and an em=4 aW, 0, : Zi U Z , =-+ 19M,,
0,
a
-
and 0
a
b:1 ID X[P2)P3]Ul
ID
2
comers
of W.
Z9t9
fo,
x
(M,,),
of
The 2-strata
Zg,
The condition
or
=
More
precisely,
[po,
:-
p,
bt
M'.
are
the four connected
surfaces
Z,,_
x
fo, _Tg_
x
fl,
fl.
embedding
the
[po, p, ] the ends of the cylinders I
with
these
x
on
X
become, via the homeomorphism p, 2-dimensional
thus,
Specifically,
and
2
I
(a
+
b) copies
[P2 i P3 ] into the 3 -manifold into the boundaries.
=
we
b of
that bt
require
(Ub
D2
of the full
M, is that b
po)
C
holes
on
x
Oo (Z9j,
cylinders
D2
maps the discs
I
X
at
such that
b _1
00 maps the discs
Similarly, target
we
holes Of
into want
the
that
Zqtg,alb,
obt:1
O
o
but with
2
X
P0
of the target
positions 1
ID
bt
:
[jb
order
D2 Xpl
r
permuted by
)
Zg-,a/b in E,,,, maps
the vertical
the
given order.
b discs
arrow
P
E
onto
Sb-
the
74
The Double
1.
Category
of Framed, Relative
3-Cobordisms
The
mapping properties of b' are now completely maps the a discs D2 X P3 to the source hole positions the discs D2 x p2 to the source hole positions in Z,_
Finally, discs
we
want that
j*,,
to the interior
[P3 P21 components
permuted by
in order
f
X
a
without
bl
o
order,
and E
Sa
10.xfo
Yx ff,
f
xf,
x
in the cobordism
0
,
D2X go, f, i
t
and D2X
p1
(b)
C
MOhave the
of the Wilson
n_
mean-
observables
line
D2x IP21 P31
M,,
D2x[ PO"'J]
in
In theory picture. picture they fill out [0, 1] boundary parts of the
the quantum field
topological
the
the S'
x
relative
cobordisms
that
previous
ered in the
consid-
we
sections
of'this
chapter. As
we
already of O:ob
cobordisms
M,
More
in the
formally,
sense
of the double
, A:
6&B-
sketched are
in the end of the
recovered
we define
defined
1.6.1,
Section
2 away the D
following
map A from the 2-arrows
double
without
category
the
previous
by cutting category
structure
so
X
the relevant
[PO/3 P1/21 pieces
from
i
far,
sets
B,
of
and the 2-arrow
sets
Gob:
Gob:
Q
=
(W, M, 0,
p,
b) -+zA(Q)
r
=
[M,
-
im(i;)_,
0]. (1.6.7)
Here
definition
denotes
the
of Gob in Section
homeomorphism 1.2, and
class
is obtained
of cobordism
classes
used in the
in the canonical by extending'O, way to the new S' x [0, 1] boundary components that are created by removing im(b) from M, Thus 0 is, in fact, defined on Z-,:;4 as in (1.2. 1). Clearly, if a relative cobordism, M5--' M, im (b), is obtained by cutting away cylindrical pieces from a cobordism M,, of closed surfaces, then M, can be recovered from Mby simply regluing the D2 x [0, 1] pieces as indicated already at the end of the previous Section 1.6. 1. The assignment 0 : M -+ M, can be described in formal language using the terminology of fill functors as in (1.3.5) or, alternatively, the elementary cobordisms from Proposition 1. 1. 1. The resulting map is denoted as -
follows:
-
the end
of the 3-manifold.
of im
i
ing
in natural
1
I component.
x
that the D2 X [p
Recall
Zg_
00
That is,
to the
picture
D2
one
in
part of the tubes D2
b maps the inner
right the way the pieces of (M,) fit together to form the is depicted schematboundary of OW ically as the surface of a cube (all depicted dimensions are one lower than the actual ones). Also the embedding b with all its boundary conditions as for at least given above is indicated In the
analogous.
The
1.6
0
Oob(a, b)
:
notation
we
O(M)
9fill
=
construct
filling can
following
then have the
Mof Obb is =
O(M).
be obtained
structure
from
a
Since the the 2-arrow
Obb(a, b) Let
us
The map A from onto the extended
g:
us
gested
also
have 7rc
give
to the
framing filling
of the
closure
2-framings
of
classes
of the extended
sets
every
extends
any such
[Ati90]
and in
precisely,
that
are
one-to-one
eWb without
category
Obb(a, b)
x
projection
by
:
can
d-lb---+
Obb
Va, b
Z
:
be, thus, modified
(M, n) to
E NU
(1.6.10)
M.
-+
give
101.
corresponding
the
projection
the
o
precise
=
Sg
(W, M, V),,
=
A
reason
so
the
that
all
that
p,
b)
(A(Q), sign(W)).
-4
diagram
in
indeed commutes.
(1.6.2)
of these maps
are
(1.6.11)
onto
as
already
sug-
in the notation:
Lemma1.6.1. 2-arrow
=
&7b : Q we
More
2-framing
is to
corners
category:
66B
Clearly, Let
(1.6.7)
isotopy
with
as
denote also the obvious 7rc
1.6.1
a
Wbounding the standard
four manifold
(O(M)).
c--W
of 00b.
with
in Section
(1.6.9)
manifolds
four
extension
addition,
in
signature of W, are simply given
of M, i.e., the
As
equipped, explained
identity:
[M., 0,,].
=
these
1[Ua. 1[Ub e tr' OhM01.
=
obvious
b))
p,
the purpose of introducing double category (2-framing)
a
M,
with
Slili(m)
-
that
cobordism
75
Obb
-+
(1.6.8)
0 ( A (W, M, V), Recall
&-b
-+
Obb(O, 0)
--+
M-+ In this
Q4
Extension
The maps Afrom
and,5gfrom
(1.6.7)
(1.6.11)
are
surjective
on
each
set.
Proof. For a given Min Obb we construct a preimage of A in 66B as follows. O(M) and denote the filled in tubes by b so that M Clearly, we choose M,, the closed 3-manifold b. Given now M, we construct M,, (M,). By Rohlin's 1-- (M,,). W such that W Theorem [Roh5l] a 4-manifold we can find Together with the appropriate homeomorphism this, thus, defines a tuple that is mapped to
=
=
-
M. In order
[M, a]
for
tuple (W, M, b) tive. The signature we substitute
of the tive
surJective we have to find a preimage in 6bb of a Suppressing the homeomorphisms we first find a to mapped Mby.A, using the fact that this is already surJec-
to prove that Min Obb and
that is
Sg
is
E Z.
of Wmay now differ
from
o,.
Hence, if
m= a
-
sign(W)
#Cp2, W#Cp2# plane Cp2. As each Cp2 has signature 1 and signatures connected summing we find that sign (W') sign(W) + m Wby W'
=
...
where
we
have attached
projective
under
=
=
> 0
copies
m
are o,.
In
addicase
to
Iml copies
CP2' the projective plane with opposite orientaTherefore, tuple (W', M, b) is mapped by Sg precisely signature desired that have found the we preimage. [M, a] so
m
tion
3-Cobordisms
Category of Framed, Relative
The Double
1.
76 < 0
we
connect
and
-
of
the
1.
the 2-arrow way of viewing that the smooth 4-dimensional
Another
of
sets
db
be extracted
can
the
from
group 04 --- Z is given by More precisely, two four manifolds, the signature. W, and W2 with i9W, 5--- 19W2 that is there is a 5-manifold W, LJOW (-W2), if are 5-cobordant, Q with o9Q this .6. 1) on the level of In and only if sign(Wi) (1 explains particular, (W2). sign
observation
cobordism
:--
=
sets.
In this
dib
the 2-arrows
sense
with
4-cobordisms
structured
topologically
be characterized
can
smooth
as
from basic
It follows
modulo 5-cobordisms.
comers
5-mantheory that if W, and W2 are cobordant via a compact, differentiable 4-manifolds. the other from each obtained be on can doing by they surgery
Morse
ifold
Thus, the 2-arrow
sets
also be viewed
can
as
follows:
66B (a, b) surgery & homeomorphism' A
Gob(a, b)
Compositions
1.6.3 Let
us now
of
66B.
on
the
as
well
of
a
define
These
fulfill
classes
horizontal
as
2-arrows
the axioms of
a
not
composition
make
6bB
into
operation a
on
the 2-arrows
double category.
of Gob they will,
However, defined
be well
in fact,
Webegin with
double category.
product on WB. Vertical Compositions
p and
phisms
7P, entering
of two cobordisms
where
M,,
cobordisms
Clearly, so
that
:
Z,,
-+
Zi,,t
tuples
the 2-arrow
o,
to declare
(W, M, b) and N,,
simplicity
For
:
thus, have
we,
(V, N, d)
this
a
will
used
a7b
definition
the easier
vertical
1.6.3.V)
V,
and
and
vertical
compositions
equivalence as
6bB
in
a
4-dim
:
=
(V
of
let
us
66B.
what o, is
W, N,,
o,
the homeomor-
For the vertical on
o,
where
Z*t
suppress
each part of
M, d
o,
b),
Z"
0
0,
composition a triple,
are
ordinary
of Obb.
the
Zgi.,
we can
x
define
[fo, fl] the
3-stratum
composition
appears in the boundaries of the 4-manifolds simply
of both Wand but
gluing
vo 'W
=
V Z'i t
W. X
(1.6.12)
[fo, fi I
yields again a 4-manifold with corners. The f, -stratum of its boundary obviously given by N,, Lj_r,,,i7, x f, M, which is nothing but the composite N, observe that cobordisms in Oob(O, 0). For the fo-straturn of representing This
over
part:
is then o,
M,'
The Central
1.6
?&9 U-P gint gint
9-
H+ #,Hgint 9..
#Wuzrgild git
gtg
S24
Extension
&b -+
-+
0bb
-H+ #S3#-H9. gtg
=
77
=
-Hgtg. 9'.
(1.6-13) The other which we
are
remaining boundarypieces of Vo, Ware Zg_ x [fo, fl] and Zg" x [fo, fl], just the cylinders over the source and target surface of N', o, M, Hence,
have, indeed,
ages meet in
[Po"Pil
N,
M.
o,
or
UPI =pO, [po",
pi ']
required
properties
gluing
S'
of the
(1.2.3).
the vertical
to
[0, 1] pieces
im
(d
b) im(d)
within
o, has the
Moreover,
for the vertical
a
of their
(P)
o,
vertical
A (Q)
(N, n)
4-manifolds
[Wal69]
glued
the
(M, m)
o,
A
more
ordinary
of
consider natural
H, (M,,
o,
W.+ )
the
cobordisms.
relative
as
defines
can
operation'if
binary
the
in Gob we have
images
on
Q).
db
is to choose for
P and
Q in
are
6&B-
obtained
with
two 2-ar-
L5g(Q)
=
from each other
so
are
is
well
P o, Q and defined by
by using the Wall 2-cocycle signature of two 4-manifolds when already been applied in detail to the sit-
be obtained or
the
M, N E Gob(0, 0) homology V,= H, (Z") of form w. Am., AN,, C V denote
first
H, (W-gtg
a
Q).
construction
symplectic
function
cube with
Q
if P and P
cobordisms
the rational
im(b)
-
homeomorphism, then, clearly, product of (M, m) and (N, n)
or
vertical
5g (P
=
explicit
the four
completely parallel to the composition for Gob in the result is homeomorphic Mo,
A (P o,
2-arrows
that expresses the non-additivity with comers. In [Ker99] this has
uation
a
on
Q. Hence,
=
composition
=
surgery
monotonous
imin the
properties.
and, thus,
(M, m) and (N, n) corresponding (M, m) and y5g(P) (N, n). Note, that o,
their
p"), 0
x
coincide.
rows
P
[pO,pi '], d(D 2
and
of the vertical
P 01,
-+
composite
,A
via
p')1
X
b embeds
d and b is
following
6bB-
of
sets
1-arrows
One way to describe
if then
'
x
operation
the 2-arrow
horizontal
intermediate
2
from N,, o, and M,,
o,
-
-(P1 Q) closes
b(D
embeddings
of N,
gluing
W)
in
in the definition
we remove
The
component
a
example,
For
D2
1.6.2.
of the
composite product
o,
M,,)).
o,
straightforward.
going hole of Zg, Thus, any [po, pi ] yields an embedding of
=
In summary, the
Lemma1.6.2.
o9(V
C
in Section
x
Hence, if
and d
out
Note, that the gluing
((N,,
c---
also
b is
[po',pi]
X
incoming
respective
W)
o,
of d o,
D2
component
a
a(V
that
The definition
N,),
A:i:
of closed
surfaces.
the intermediate the kernels
Here
surface,
we
with
of the maps into
corresponding kernels for the maps into H, (W- ). If for a an element A E U' Am. + AN. we denote gi t A + Aand, similarly, by A Am. + AN. a corresponding decomposition, for the A:L decomposition, bilinear form v on U' by we can define a symmetric v(A,,q) w(A+, 77-). The Wall cocycle p(N,,, Mo) E Z is then w(AM.,,qN.) o,
and
o,
and
C V the
=
=
=
-
78
3-Cobordisms
Relative
Category of Fratned,
The Double
1.
given by the signature of the form v. Its value is precisely signature of the composite of two bounding 4-manifolds individual Thus, we find: sum of their signatures. Lemma1.6.3.
If
we
(N, n) then
we
o,
define
a
vertical
(M, m)
=
(N
o,
Q)
=
M,
+
n
0:)b
on
which
from the
is different
by
p(O(N),
m+
have
Sg(P
0"
operation
gluing
the horizontal
Sg(p)
5g(Q)
0"
1-Handle-Attachments:
Compositions:
Horizontal
1.6.3.Hl) scribe
composition
anomaly, by
the
the
6bB
in
Qj=(Wj,Mj,o,bj)E6&B(a,b),
Let
us
now
de-
for two cobordisms,
Q2=(W2,M2,o,b2)ECOB(b,c), (1.6.14)
with
AM)
Zgi,-,a/b
MI:
=
in Gob and
coinciding
The first
-+
intermediate
vertical
in the construction
step
A(Q2)
Zgi,t,a/bi
M2 :
Z92,-b/c
_+
Z92,tg,b1c
arrow
Of Oh is
a
binary
gluing
operation
Oh of the
1-handles e4 beW, and W2. It consists of attaching 2b 4-dimensional handle is of each end such that one two the glued to W, and the 4-manifolds, tween b of the handles e4 = (D 2 X [fO i h 1) X [PI) P2] are attached other to W2. Specifically,
4-manifolds
at
the
the
Zgi,tg,alb
X P1 x
[fo, fl] stratum of W, (D 2 x [fo, f, I) x pj
of W2, such that the
tum
Z92,tg,blc
identified
X P2 X
with the
[fO f1l 7
cylinders
Straover
The other b of the two target surfaces respectively. outgoing In summary, the attached in the analogous way at the source surfaces.
incoming are
*h-gluing
is the
identification
following
W20hW1
=
space:
e4
W1
U
bjpo,pjjx(fojj]
W2*hW,
boundary following.
Of
0-1)
[fo, fl]-Pieces:
The
The Z
are
holes
and
1 -handles
Since
and the
x
the 1-handles
sponding piece
in
...
U e4
2b
natural
has three
by
an
X
(1.6.15)
[f0JI]
we
shall
discuss
pieces of W1.
and
W2, the
pieces,
have been added at these
W20 h W1 is obtained
W2. bIP2,P31
index-
which
in the
corre-
Moreover, since the the result can describe
1 -surgery.
x [fo, e e fl], we cylinders, over and then taking the cylinder on the surfaces, also by doing index- 1 -surgeries the discs, where the are done by removing The surgeries on the surfaces the result. e3 are attached, and then gluing in copies of S' X [Pl) P21 0. Wemay, c the surface from Figure 1.2, with a As a result we obtain the surgered Z, x [fo, f, thus, use the homeomorphism. from (1. 1.5) to identify pieces with the cylinder over a standard target surface as follows:
1-handles
themselves
are
4
=
3
=
=
The Central
1.6
(Z92,b/O
U
b(S'
-Pgl,Olb)
U
[PI P21)
X
i
[gj,O1b:92,b1O]XVOJ11
of tile the
are
0-2)
homeomorphism
to this
form
as
X
6b
-+
Obb
-+
79
V-, Al
b(S1 Xpl)
b(S' xp2)
Thug, up
124
Extension
required
for
the Z
[fo, fl]-pieces 66B (a, c),
x
2-morphism Q1 and Q2
a
Z91+92+(b-1)
3'
in
[fOJ11-
X
(1.6.16)
W2*hWj
in
already
are
whose horizontal
1-affows
of those of
composites
7he W-Pieces:
the Up to here we know how the pieces in a(D2 X [P1 P2 1) x [fo, f, ] C ae4in 1 2 boundary of a 1-handle are glued and placed in W2*hWI. The other pieces (D X
[P1 P21)
c9[fo, fl]
x
7
cobordisms
D2
X
[Pl) P21
The result
i9e4l
C
fo
x
added to the handlebodies
will'be
aW, and aW2.
in
To the handlebodies
at the standard
of the handle
D2
discs
addition
x
simple
for
the
ies.
Weassume the handlebodies
to be the interiors
ure
1.2. It is clear
as given by'deformations combined handlebody. We, thus,
be extended
that the definition
the interior
to
morphism, unique
up to
aw, b,92 gi
This allows the
V1
us
:
of the
isotopy,
W+ U e 3
spaces.
Ue
...
which
obtain
a
in
Fig-
1.3
can
homeo-
11
(1.6.17)
92
identifications
for the 1 -handle
attachments
to
3
UW92,tg
W+
'H-
92,-
91,t9
91
Ue
3
U
...
,W#(7i+gl,tg+92,tg+b
attaches
-1
specifically.
To this
the holes
W+handlebodies
as
9
from the
is trivial
wish to characterize
we
to
+ (-H 91,tg+92,tg+b-1
)#W-92
homeomorphism
However, more
U e 3Uli+
g) #-H-
92
92,-
b
the last
morphism.
9
U-H+ =4,H+91+92+b-l'
U e3
...
b
Abstractly,
W+handlebod-
depicted in Figure
components:
Ig e3U 'H91 91:. U
91
el
follows:
following
make the
to
U
1.1
91
of the surfaces
of a,
as
3
1-handles
boundaries.
pj in their
particularly
is
and 2+1 -dimensional
add the
we
with
Figure
in
92
# properties underlying
for connected the homeo-
the deformation
end note
label
91
of
1, and lt
1.2, plays
a
special
the first
1-handle
respectively,
in the
that
rOle.
efil"
c---
gluing
3
el,
of the
efil"
Wemay assume that
is
handlebody pieces W+ [1/21 91/2,tg decomposition as in (1.6.4). Since the sphere S+2 used in each W+ 9V within the balls to connect the corresponding opposite handlebodies lies entirely D3 ,,/2,', the W; are already attached to the piece attached
ters
directly
3
to the D
e
3 0
of each
C
that
en-
the handle
D3 U efi"t [1 ]
3
UD
[2 ]C
'H+,v 91
U
efi"t) (e 3= 1
U
...
Ue
3
UH+
92, tg
b
into
Assuming the efil't
that
a
handle
deformation is
entirely
of the within
S+2-attachment this
simply
of the
connected
W9part,
handlebodies we see
that
it
80
3-Cobordisms
Category of Framed, Relative
The Double
1.
will also the isotopy uniquely given up to isotopy. Furthermore, alignment of the handlebodies with the first handle as in Figure change. Nowthe W- are complements of the standard handle bodies
be such that
is
1.3 does not
the
S'
W+ with
-
a
g'
neighborhood
with that
3-ball
U#71;9-
phism, -Hgl
Ig
91:_
U also
U
=
U e3 U
a
'Hg- .
-
w
unique
which is
removed is
3
D'
W1
-
In connected
.
3 is removed from U and D
3-ball
Hence, in summary,
up to
Ue
...
really
in
the
we obtain
S3
sum
glued
g_,
following
so
of
that
71;9, in
so
homeomor-
isotopy:
U'Hg2,tg
92,a.
b
H+
U
91,t9
(efl-t
-
(?i+,_ 91
U
ld,)) 92
Ue
3
U
Ue
...
3
U?j+
92,tg
b-1
The
picted
region inside efi"'t figure below.
W+ 91'_
the handlebodies
with
LJ'H+,_). 92
91'_
91,tg+92,tg+b-1
and
W+ 92,se
in
(1. 6.15)
(1.6.18)
cut
out
is de-
in the
In order to consider
to obtain
the
complete li-part
the other b e 4-handles
that
Of are
W20 h W,
attached
as
to the bottom
we
also need
side of the four-
to the picture of Section 1.6.2. The restriction but this b time W-part of the boundary implies again an attachment of e 3-handles, 3-handle is a e in the two lig'9w components. Analogously, to the W- -handlebodies 1 9.,c disc attached at a target disc in Z;, -071-91'_ and the corresponding source b e 31 of the W-part after gluing the first In the presentation in Z;2,_ 92
folds
if viewed
handles
as
in
as
in the schematic
these
(1.6.18)
surfaces
appear
when
we
cut
the two
out
Wg-
from
Let us denote efil". Hence, this is where we have to add the remaining e 3-handles. 1 3-handles added as follows: efi"' -piece with the W+ cutoutandthebe the resulting 9-
F
=
efirst
91
U
jj+,w) 92
UE91,-
LJE92,
e3
U
...
Ue
3
b
The
dles that
depicted
piece
F is
are
attached
to
in the
the inside
diagram above, including surfaces
of the cut out parts.
the
2) (b Obviously, =
added hanF cannot
1.6
The Central
f24
Extension
6b
-+
Gob
-4
81
3-space so that the dashed parts of the latter handles have dimension. lying in an additional Of W2*hWl with all 1-handles F is now also a piece in the complete'h-part is given by the foladded. Specifically, the way it is connected with handlebodies lowing extension of the homeomorphism from (1.6.18): be embedded in Euclidean
thought
to be
of
W91 Ig Ue3U 1 91:_
as
3
U e1
...
jj+
UjJ92,1g
92,-
(Id 91'_
-
91,tg+92,tg+b-1
li+,_)Ue
Li
3
U
92
Ue
...
3
b
2b
3
3
+ =W
UFUe,U
-
91 t9 ,
U el
...
UH+
W+
C_
#F.
91,tg+92,tg+b-1
92,tg
%
(1.6.19)
b-1
homeomorphism.
The last
by
is here
the connected
that
assume
sum
virtue
of agi,b,92 the 3-ball
is such that
and
(1.6.17),
as
in
in
W+
we can
-handle
91,tg+92,tg+b-1
corresponding standard S2 -sphere is replaced precisely by F. Thus, the only reason that we are not able to canonically identify the handlebody tg+92,tg+b-1 of F the last b e3I -handles is that in the construction (1.6.19) with Wg' by
bounded
in
the
were
gl,-+92,-+b-1 added to instead of cut away from
0-3)
The
M,,-pieces: M,,-part Of W20hWl)
The
Again,
the
e
4-handles
3-dimensional
to
form
e
3
D2
label
with
3-handles
to the
jt
D2
the other
_-'
Z
9M2,,,
C
92, tg
x
Mj,,
ferent
M2,,
the
correct
from the
OW2Oh Mj), and
M2,,,
with
bottom
attached
this
x
one
C
i9M,,,
with
between the
in
the
closed
Oh
of surfaces
as
if
defined
(1.6.14).
For this
an
D2
in the
e
we
this
us
at
p,
that
describe
and, thus, say, of the
target-disc
a
cobordism. to the
3 1
=
can,
and with
surface
target
D2
and
cobordism,
the
let
x
Mj,'
of
cobordisms
reason
M,-parts, e 3-handle, 1
j,
label
filling diagram in Fig.
consider
restrict
the
x
[PO; P31
X
are
M2,,. already
thus,
between them
attached
However,
we
W-part.
[fo, f, ]
on
---
(1.6.16)
in
as
to the
[PO) P31
of the first
same
surfaces
source
surfaces.
obtain
end
analogous X
The 1-handles
the 2b 1-handles
with
completely f 1 ] orD2 P2/31 X 49h
with
Eg,,,
surface
and discs
m2, nim(bl) _
D2
x
pi
nim(btl)
1.6.2mi,
in Section 2
is identified
coincide
coincides
which disc
with in the
source source
which disc in the target
vertical surface
surface.
with
with
Zg. Moreover, the position [po, pi ] are related precisely by
surfaces D2
[PI/O
M, and M2 the relative
required
]
pi
o,
as
is
in
a
iden-
cobordism
general difproduct
of the horizontal
1.6, 0 the
in
as
M,,-part
and,
(1.6.8),
obtained
are
from in
a
Ml,,,
slightly
way.
As
[po,
as
D2X
at a source-disc
one we
correspondingly, different
[f
X
490.1 As for the handlebodies,
is
target pi
attached
and
between
=
homeomorphisms
With the
is obtained
[Pl; P21
in the second cobordism.
correspondingly tify
3
Of
[P1,P21,
X
X
e
pieces
for the last
account
D2
---
efil8t.
the
labels
with
is identified
b
b
copies of D2X [p2)
source
and target
of the discs
discs
at the
is connected
by
a
X
P3 ], and top and on
the standard
1 -arrow,8 the target vertical E Sbof of Q2, the same permutation
M2,,
of D2
top and bottom of
I-arrow
of
copies
D2X
Q1. Since P tells us
[P2,P3]-piece
to
the 1-handles
with
e
pieces,
each type.
Specifically, -12
U
U
2
3
D
(mi,
-rb=
D2
exactly
and
X
[P2,P3]-pieces
results
cylindrical
of these
four
D2
[PI,P2]
X
now
pieces,
in b
one
of
have that
we
u
...
[po, pi]-
X
and e3
[Pl,P2]
X
containing
each
connected
T1
2 in the union of the D
correspondence
This
3-Cobordisms
of Framed, Relative
Category
The Double
1.
82
n
im(bt))
Ue
3
U
U e3u
...
(m2,
n
im(b')).2
2b
(1.6.20) Here each connected
[PO)
.,Zl
P2
P3
Our
full
]-intervals
pj+j
torus,
together
where
we
denote
by (1.6.21)
-
P(j)t P(j),
=
=
j,
=
labeled
labeled
-
-
x on
when we identify
Tj
with
union of these tori
with
pi in the j-th torus component the standard surfaces:
Zgl,_, Eg,,t.,, disc in.Eg2,,,
disc in
labeled
-
D2
labels
given
of
becomes apparent
(1.6.20)
in
discs
disc in
disc in Z
labeled
is to present the M,,-part Wemay define the latter
goal
Of
2
19(W2 *hWl)
as a
by filling or, (1.3.5), holes of M, and the target holes of M2 using the functors from and from source the cylinders cylinders Ml,,, target by removing
cobordisms. the
Tj C Tj C Tj C
X
asserted
structure
the target or source holes D2 XP0 C 7j = jt X
a
i
cross-sectional
each of the four
D2
S14p is
PO 'P4
The connectedness
D2
X
Upi [P1 P21 UP2 [P2 P31 Up3 [P3 P41
P1 I
-4p
D2 Xpi
(pi,
the four
by gluing
obtained
the circle
D2
Tj
component
source
alternatively,
cobordisms
from the relative
either
M2,o:
M.1',
=
2
M1,0
-
M2,,, 2,2 also With this notation, M-1
09M.1,1 and 2
Mi,,
=
Zjt
5
S1 3
3
U el U
'
-
...
X
im(bt) 1 =Stjj(Mj), fi im
1.-
=
6 11 (M2).
introducing
[P2 P31 i
U el UM 2'C' -.1
(bs)2
=
=
Tj
MI'l
2
the n
cylinders
19M-1,21
we
si
Zj, obtain
the
2
UZjt
...
Zbt
(71
U
*
*
'
U
x
[po, pl
=
Tj
n
presentation:
Tb) UZ1,
-
-
Zb.
M-1,2
*
2
2b
(1.6.22) the 4-manifold attachments to obtain the 1-handle 1.21 we illustrate In Fig. 1 using the same scheme in one dimension W2*hW1 for the simplest case b and in Section 1.6.2. Here bi and b2 are short for im(btl) lower as in the picture respectively. im(bl), =
1.6.3.H2)
Horizontal
Compositions:
2-Handle-Attachments:
of a horizontal remaining step in the construction also 2b 4-dimensional to the 1-handles, is to add, in addition The
folds:
binary operation on the 2-handles e4to 2
B four
ThC CCIAI-al
1.6
S24
EXtellSiGn
6;b
-+
4
Cob
83
4
e
e
2
ML 2.,
2j,
Yj Fig.
W2 Oh W1
1-handle
1.21.
(W2*hWl) UYjU
--::
attachments
UYbUr-lLJ
...
4
(e 2
UCb
...
U
4).
(1.6.23)
U e2
...
2b
Here the
Yj
4
e2 along a identification in
torus
glued specify
1, Cj, with i boundary a(W2OhWl). 4 is obtained by writing e2
and
embedded in the torus
S1
=
2
D
x
09(W20hW1)
=
is
(,9D 2)
naturally
W20 h W1 In order
to
.
the location
x
2
D
b denote
=
C
complete
D2
0 (D
identified
to
2b full that
Recall
D2
x
2
with
D2)
a
piece
the definition
that
so
,
x
S,
tori
=
of
D2 that
x
attachment
an
of
there
is
in Oe4along 2
(1.6.23)
canonical
a
o9e 4. 2 Hence,
a
standard
which
we
are
4-handle
a
still
e
4 2
is
need to
of the 2b embedded tori.
lie in the W-piece Of a(W2OhWl) as described in (1.6.19) all of the tori will be contained in the previous section. In this presentation Part 0 2. Half of each torus will run F-part, as depicted in Section 1.6.3.Hl, through the "added handles" ell, such that
Lj
The b tori
all
of the
-
,Cj for the
j
1 -handle
-th
Hence,'Lj and
j,
and half
,ci will discs
be the full with
labels
complete For a picture a
e
intersects
torus
of
3 1
.
the
radius, n
2) (ID 2
e 3= 1
n
Here !D 2
2
[P07P31
X
C D2denotes
surfaces..Eg,,_
and
The second half
efi,st.
-
pd, 91
2
C D
u
of the
the standard
Zg,,_ j-th
[PO; P31
X
! `!
(.!D 2) 2 in F
see
x
torus
id,_)
S2,,
in
(ID 2) 2
92
where
the discussion
S2p
e
3 1
disc with half radius.
in F in the discs
cylindrical piece added to the handlebody j, and jt, only that we halved the diameter
Lj theCj
=
=
of the
with
labels
jt
F, namely the piece x
in
[qo,
q3 1)
Figure
of each.
[PO) P31 Upo
-
1.2 between
Thus,
q0,P3
W-boundary part
-
we
q3
below.
have
[qo, q31
The other
b tori
Specifically,
above.
Category of Framed,
The Double
1.
84
Yj
lie
the
j-th
M,,-piece Of 19(W2*hW1) as given in (1.6.22) Yj is simply defined as the j-th torus Tj with
in the torus
Furthermore,
halved radius.
we
2) (ID 2
have that the inclusion
S14p
X
3-Cobordisms
Relative
Yj
c
Tj
2
D
_--
X
S41P
2 by the standard one of (I2 D') C D Let us next describe again in more detail how the various pieces the after attaching differ from the respective ones in 19 (W2 0 h WO
is described
o-1) The Z
ite surface o
e
in
-2) The W-Pieces As
4 are 2
explained
x
[fo, fl]-parts
canonically
identified
and still
Of
with
a(W2*hWl)-
the correct
S1
4 a e 2 -handle
attaching
1.5.2,
to
a
T
torus
_--
D
X
C N
OWresults
=
in
x
S'
In
o9(N
our case
responding
obtained
means
in
that the
(9(W2*hWI)
of these lie in the F
all
W-piece Of 19 (W2 0 h W1) is obtained by b 2-surgeries along the toriLl....
it suffices
piece,
to
Here each surgery torus passes parallelly 6 this in Section 1-handle. As explained be described
and W+
Figure 1.2. 91, 91"": Hence, with this homeornorphism surgered F-piece the following: FL .
....
as
..Cbefil-st
exactly implies
the effects that
1-handle
Lb. only
cor-
Since
there. of surgery 3-dimensional
through a the surgered
once
from the
as
well
manifold as
a
can
neighbor-
picture it becomes apparent that with our precisely those full cylinders in efil't
following cutting
away
inserted
of the 1-handles
in the locations
are
W+
describe
both the 3-dimensional
by removing
hood of the surgery torus. In the in choices of the L1, this results that
x
T).
-
this
piece
along T. by D2 along
N
an
=
S'
Walong
4-manifold
on index-2-surgery 4 is manifold the where U NT surgered W), O(e N-r Specifically, 2 cutting away the torus 7- from N and regluing the opposite torus S1 a
compos-
:
in Section 2
from the Z
disjoint
all
unchanged (1.6.16).
are as
O(W2 Oh W1)
[fo, fl]-Pieces:
x
The 2-handles
Hence, they
of
2-handles
between the standard
handlebodies
in
(Wg+1,
-
_
U
e3l
and the
U
...
U
one
in
(1.6.17)
we
for
obtained
the
e3l Ulig+,_)
b
efl-t
91,-+92,-+b-1)
-
gl,-+92,.+b-
(1.6.24)
complete W-piece of 9(W2 Oh W1) is now obtained by applying the surgin (1.6.19). eries along the Cj to the identifications Combining this with (1.6.24) of the canonical obtain a we finally W-piece with the correct stanhomeomorphism I and with genera gt', dard connected cobordism from (1.6.5) 91,tg + 92,to + b The
-
9sc
=
91,sc + 92,sc + b
-
1:
Fig. 1.22.
'H91
gl:-
9
Ue
3
U
From added to cut out handles
3
U e1
...
92:-)'Cj'. .
U-H92
along Cj
via surgery
+
g
1)
#N_ gl,-+92,-+b-
91, g+92,tg+b-1
Lb
2b
jj91,tq+92,tg+b-1.
(1.6.25)
gl,-+92,-+b-1
In the
construction
be considered
as
Ml,,
3-manifolds but in addition dles
can
be
some more
WO
tangle
of
4-dimensional and
M2,,.
we now
interpreted detail
In order to
Wgl: 91
we
still
have the
also have added the 4-dimensional as an
here.
from the handlebodies
cobordisms
composite
For the
U e 3U
...
4-cobordism,
additional
Ue
3
simplify
let
we us
WO
and
92,-
glued
2-handles.
which
the notation
U'Hg2,tg
W2 and W, will
the 4-manifolds
presentations relative
=
to
the
W2 and W, These 2-handescribe
want to
in
write
qj91,tg+92,tg+b-1
91,..+92,w+b-1
2b
that
is now W.. The extra 4-cobordism. (1.6.25) becomes by considering a collar of WOin W20 h W1, homeomorphic to the product the handlebody with an interval [0, 6], and add the handles e 24along the tori in 7t* x boundary part: so
ob-
tained
of
the
-
P
:_
(710
X
10, 61) U(Cl
X6)U
...
U(Cb
X
IF)
(e42
U
...
4).
U e2
1
(1.6.26)
b
By definition
of surgery
W*
(WO),cl
X
0 and
....
this ..
c,,x
implies E 5_--
now
that
W.. As
P is
we
a
relative
indicated
4-cobordism
above,
between
in the cobordism.
1.
picture
for
handlebodies
Hence,
as
the
source
manifolds.
For
POFV = -P In
an
4-cobordism
opposite
handle
4 e 4-k
serving
as
=
the
4-k
D
x
part
the
to
switched.
the target manifold are The boundary handles e 4. 2 full
tori,
one
e
Dk, since the r6les
attaching
of which
:
k handle
a
our
Dk
=
k
of
a
with
is identified
x
piece
manifold,
case
Cj
manifold.
target
Wo.
--+
D4-k in the
is reverted
boundary
and the
new
is
to
a
If
attachment.
k
-
piece added
composed made up correspondingly
in the
4
of the handle,
of 2
POPP is still
2-handle
j-th
of the
W. is the
cobordism
W, 4
the standard
need to consider
P, however,
source
In
we
opposite
in the
interested
really
we are
presentation
tangle
of
the derivation
3-Cobordisms
of Framed, Relative
Category
The Double
86
we
=
4
-
to
2-
of two
denote
the
2 4 where Lj -5-- D x S' lies we, thus, have o9e 2 Cj UCIF, opposite part by LOPP 3 3 2 in'h* andCjPP -- S' x D is the opposite torus in W. by which Cj is replaced =
in the process of surgery. From it is apparent 1 -handles,
our
through
handle in NO. The tori
are
depicted
in Section
discussion
that in
Cjopp
is
Fig.
1.23.
a
torus
1.5 of
cut-out
From cut out to added handles
Fig. 1.23.
In Section
1.5
could be substituted
and add
a
1 -handle
we
by at
an
the
opposite
via surgery
2-surgery the j-th
handle
along
passing
cut away
bodies
C3PP
LOIP,
which of surgery along the annuli is to fill up the cut away cylindrical piece Thus we do obtain side of the bounding surface.
showed that the effect
index-
a
surrounding
1 surgery,
d2.
-Val
the
(Wo)'Copru
homeomorphism.
1
'
the
cobordism.
opposite Po-'P
4
(e 2
=
U
%
87
of
is thus: 4
U e2)
...
U(LojPP
---I
IV
-
Gob
-+
The handle structure
W* as expected.
uLopp b
...
Cob
--
XO)U
LJ(Lb
...
o' 'P
X
Xo)
[O'E])
(1.6.27)
b
o-3) The M.-Pieces: the
As for
the tori
Yj
the tori
Tj,
by performing index-2 surgery along contained in Since they are parallelly of every Tj. As of surgery in a vicinity the surgered full torus is homeomorphic this situation S' and D2 factors exchanged. Specifically, we have a
W-piece, this part is obtained M,,-piece Of 19(W2*hWI). is enough to study the effects
in the
it
explained
in Section
to another
full
1.5, in but with
torus,
homeomorphism
(Tj) which restricts
Zj.
Zj,,
cylinders
to the
1.63M,
Section
=4 S'
Yj
[P3, P2]. The surgery allows us to give the
Ml,,,
X
C
OTj,
defined
0, P11
Zjt
i
identifications,
factorwise
with the obvious
common
Zj.
[0, 6]
P,
],
at the end of Part
0 -3)
from
as
S1 X [PO, P1] =-+ S1 X 0 X
=
,
[Po'
X
along M,,-piece
the
Yj of
=
and
a
O(W2 Oh WI)
U e3 UM 2 '0
...
[PO, P11,
isomorphism [po, pi] (1.6.22), thus, presentation gluing of the M.I,j along their
in the as a
2
cylinders: U e3 U
X6 X
monotonous
the manifold
on
S1 X [P2) P31 4 S1
)Yl,---,Yb
2b -
MI 21
uz t
1
b(S'
zt
...
M -1,1 2
46 11 (M2)
Oh
t
-6fil (M1)
--
x
UZ1t
16 11
0
[0, 6]
-Z1
Etfi
it
.
....
x
[pol pi])
Uzj.
M-1,2
.-.Zb.,
2
M1,2
.Zbt -Zb.
A Oh M1)
O(M2 Oh M1)-
(1-6-28)
identity is obtained by inserting the form of (Tj)y, for each j. We with J -+ 0, so that the intermediate line by shrinking cylinder, the corresponding boundary pieces of Mi 1 and MI 2 can be thought of cylindrical P of the This is, obviou Sly, jUS t & definition onto each other. as being glued directly in defined in Cob as horizontal composition M-1 2 Qh MI 1 of relative cobordisms in of the M.I,j Section 1.2. Wecontinue to use the presentation terms of the partial fill functors, in (1.6.8 . and the definition their functoriality, 4 4-cobordism let us also denote the relative given by the b last e2Finally, Here, the first
arrive
at the next
2
attachments
R=
on
the
M,,-side
(M2,o*hM1,o)
X
21
'
of the
composite
[07 61 UY,
Xe,...,Y,
of the 4-manifolds 4
xe
(e2
U
...
4)
U e2
as
:
b
M2,o*hMl,o
--+
O(M2 Oh MI),
(1.6.29)
88
The Double
1.
where
of Framed, Relative
Category
3-Cobordisms
the notation
we use
M2,o*hMi,o
=
Ml,,,
Ue
3
%1
U
e3UM2,o111
U
...
1--
2b
1.6.3.H3)
Of Oh for the four-manifolds
operation
the 2-arrow
on
from the
Our observations
The
Lemma1.6.4.
tuple ofmanifolds
x
(W2
':
Oh
previous
composition
as
to
us now
compatible the operation
define
a
binary
Wesup-
1-arrows.
Q,
for 2-arrows
and write
W1
x
now
with
a(W2
[fo, fl]
summarized
and
the I
WI)
Oh
as
follows.
compositions.
-arrow
with
have been identified
Zg,,_+g2,_+b-1
and
(1.6.30)
in (1.6.30) yields again a binary thereby a horizontal
defined
1. 6 2) and
in Section
of
Zgl,tg+92,tg+b-1
are
of manifolds
Oh
66B- compatible
[fo, fl]-pieces
Li bl). O(M2 Oh MI), W2 2 1
section
characterized
in the class
The Z
Proof.
allows
vertical
with
as
Q2 Oh Q1
operation
ebB
of
sets
homeomorphism,
the
press in notation Q2 from (1.6.14)
Composition:
the Horizontal
Factoring
The construction
X
[fOi f1l
0 1 above. These are precisely the surfaces associated to the composites of 1-arrows, [92, b1c] o [gi, a/b] [92 + 91 + b 1, a/c], corresponding horizontal in (1.6.25) also shows that the W-piece of The identification in (1.1.2). as defined 09(W2 Oh W1) is also the correct handlebody associated to these two composites of with 1-arrows. Clearly, the 1-arrow structure of O(M2 Oh M1) that was identified of O(W2 Oh W1) is also given by the corresponding the M,,-piece composite square. Putting all these parts together in the standard closure as defined in (1.6.6), we fiin Part
-
the
=
nally
-
obtain
(O(M2 as required cylinders
in the definition over
the
a source
holes of the
composed
the
holes
c
target
Oh
of Section
surfaces
previous lemma only particular way in which it A from (1.6.7). the using map In the
The
Zg,+92+b-1 as
.
required
the 1-arrow
In the
structure
Wl)i
ends
Zg.,
precisely
which
are
in the
also
the
[fo, fl]a source
way bl 2 is shown to bound 1.6.2.
same
in Section
is constructed
mapA : 6&B-+
Oh
Now b,
1.6.2.
of the surfaces
holes
inZ9,+92+b-j
19(W2
M1))
of
O(M2 Oh M1)
is summarized
next
was
more
relevant.
formally,
A
Lemma1.6.5.
The
tal
Oh:
composition
, IM)
Oh
Gob isfunctorial
A(Ql)
--",' I(Q2
with
Oh
Ql)-
respect
to
the horizon-
1.6
Proof.
only thing
The
that
we
M2 Oh MI however,
It is,
need to realize
still
from the constructions
obvious
-+
dib
-+
0ob
89
here is that
O(M2 Oh MI)
-`
04
Extension
C'entral
T
(bt2 2
-
U
b1f).1 of the manifolds
and definitions
of the bt 2 and bl 1 fill
Mj,
precisely the target and images F'J' vertical 1-arrow of also the of source source M1. M2 Oh identify They cylinders 1-arrow 1-arrow a of M1, and the target vertical M2 Oh M1 with the source vertical of Of M2 Oh M1 with the target vertical M2. 1-arrowy Mi
M,,,,i
and
-
Recall
that
that
the construction
product
Lemma1.6.6.
on
the map Sg
Using again
composition.
the operation constructing Gob to a product on b. completely analogous to the
be done
can now
b
77ie category
in
of
the main purpose
the horizontal
extend
the
6bB6-+
:
admits
a
b
With
to
Lemma 1.6.5
for the vertical
one
from (1.6.11),
horizontal
66-B is
Oh on
we
find:
that
composition
is
defined
by
59(Q2) Proof.
'; -g(Q'j) a
at the
4-manifold
precisely
is
by
surjective
5g(Ql) andSg(Q2)
=
explained
As to
Sg
Since
by
-59(Ql)
Oh
6 9(Q2), 1.6.2
end of Section collection
4'VQ2
class.
only thing
think
we can
Hence,
Ql)-
0`29(Q'20hQj)
thenalso
of 5-dimensional
Oh
the
Lemma 1.6.1,
=
4+ 1 -cobordism
its
some
` _-
we
'5g
to
check is that =
1.6.4
Double
It is trivial
surgeries
as
in their
interior.
to
Category Properties
verify
However,
of the manifolds W2 Oh W1 leaves the interior union. the of their boundaries Hence, W21 Oh along
of the
law from Section
"59(Q20hQl)-
the map that assigns know that W, and Wl' differ of
composite unchanged and only adds handles Wl' is obtained from W2 Oh W, by combining the surgeries on each follows that their signatures and, hence, their image under Sg are the the construction
if
that
of
Gob satisfies
B. 1. Hence,
it is
It
dib the
a
4-manifold. same.
associativity
double
category,
axioms and the since
the involved
interchange composi-
orders. boundary pieces, only in different fulfilled. However, the interevidently composition is defined by gluing of change law is not obvious, since the vertical 4-manifolds whereas the horizontal by hancomposition is defined quite differently the different In fact, 4-manifolds with corners. dle attachments to corresponding tions
For
are
dib
given as the same gluings axioms the associativity
over
are
also
However, the followproduct orders in 6&B- do in fact yield different 4-manifolds. related 4-dimensional lemma that shows a are by ing they surgery and, hence, yield classes in eWb. the same 2-morphisms as equivalence Lemma1.6.7. the
compatible categories.
double
4-dimensional
W1, W2, V1, and V2 be 4-manifolds structure as in the assumptions -arrow
offour 2-arrows that have of the interchange law for VI) can be obtained by doing b
Let
I
Then
index-2
(W2
surgeries
ov
V2) Oh (WI o, (W2 Oh W1) Ov (V2
on
Oh
Vl)-
90
The Double
1.
of Framed, Relative
Category
(W2 Ov V2) Oh (WI ov Vj) Equivalently, Vj), where the 5-dimensional cobordism is (W2 Oh WI) Ov (V2 Oh VO X [0) 11-
3-Cobordisms
is cobordant
(W2
to
Oh
WI)
Ov
MOh e'2
b handles
by attaching
obtained
to
Proof. Weneed to show that an index-2 surgery on (W2 Oh WI) Ov (V2 Oh V1) yields law. To this end let us present this the opposite product order as in the interchange vertical for the individual compositions product in a way where the identifications inof the cylinders have almost been made with the exception over the respective tennediate
Thus,
in the surfaces.
discs
the 4-manifolds
defined
equation
in
as a
LUX, let
us
for
introduce
the
product
following
WE)vV
=
W 9i.t
Ub,52'
Z9
where
we
(1.6-31)
V.
E.
Here
product
of the vertical
modification of Section
1.6.12
X
VO, fi I
removed the b open
discs
source
from the
correspondsurfaces for W, ev V1. Two corresponding ing b target discs from the intermediate surface are still discs in an intermediate glued together along their S'-boundary, boundthus, combining to a sphere S2 Hence, in the product we obtain additional S2 x [fo, fl] over these spheres. In particular, we ary pieces, namely the cylinders surfaces
intermediate
products
of the
product w2e, V2 and
for the
the
.
have
Wj
Vj
ov
Wj
=
ev
3
Vj
D LjbS2
[f
X
fl].
0,
(1.6-32)
XVOIf, I
compositions are interchange law, in which the horizontal also by making the vertical carried out first, can now be presented gluing over the the the 1-handles and 2-handles for all two horizontal first Z.*, and then adding each With 2b handles of each type added, in composition we obtain composition. formula: the following The
product
(W2
Oh
order
in the
Ov
(V2
Oh
(w2
ev
v2)
W1)
VO u
:--
(w,
ev
Vj)
U e41 U e 4U 2
4
U e1 U
...
4b
4
...
U e2
4b
=QUe4U 1
U e 1 U e42 U
glued
here that
in the
strict
of the 4b 1 -handles
all
not
of
sense
the W2 Oh W1 composition b target
corresponding S'
x
Q the
[p', p"] manifold
x
a
handle
are
1 -handles
o9e 4
[fo, fl] (W2 &, V2) C
-h
glued
e
4 1
attachment, in the
&,
Vj)
to
[P 1,
X
the b
subsequent composition
vertical
of the V2 Oh V1 in their boundaries.
(Wi
D2 since
which
Above we
...
U e2
4b
2b
Note,
4
4
...
we
have
P
source
X
[fo, fl]
are
composition along the pieces
to
the
K
have also denoted
already
of
1-handles
=
by
added these 2b
`Yhe CenLrai
i.6
1 -handles
that
to the intermediate
attached
are
composition
thus, combines in the vertical 4
4
el UK el so
that
Q
=
S2
-
=
d?4
Extension
Gob
-+
to
[pl'pll]
X
[fO fl],
X
,
bs2x[Pt,P11ix[f0j,1
(W2e-,V2)
(wie,,vi)-
U UbS2 X (pill
X
D2
x
I
UbS2 X (pi
[fo,fl]
each S2
Observe next that
S2
=
91
I -handles,
pair of these
Each
surface.
i
-+
V, p"]
X
[fO
x
X
f,
+ -,
[fo,fl]
-]
-
in the
lies
d of
Q for some small enough e > 0. Wecan use this to perform an index-2 2-handle attachment described in corresponds to the 5-dimensional surgery, 2 in Section 1.5.2 this means removing each S2 x D.Lemma1.6.6. As described 3 piece and replacing it by D X S81, along the commonS2 x S' -boundary. Wedenote the surgered manifold by interior
which
Qk
S2
Since the removed
(Wi S2 D2 k
Vj), it 2 D,,_pieces
E) V x
=
(Q
UbS2
_
2 Dk -pieces
x
begin with,
to
ULjbS2XSI
UbD
well
the D3
as
S811.
X
added to
by adding only
also be described
as
3
part of the 1 -handles
are
Q& can
that
is clear
D2)
x
S&"-pieces.
X
=[pl,p"]x[fo,fl]wehavethatS'&=Ip',p"}x[fo+e,fl-
I fo
fi
+ e,
written
(e41
e
-
I
so
(w2ev v2)
to
Note that
since
JU[p',p"]x (w, ev VI) can
u
U
be
as
4
UK e 1
S2
x
S2
X
u 3
D
ole
where
we
3
[P1'P11]
D3 X
X
X
S1&
[fo, fo
x
[PI, Pit]
X
1P p"]UD 1
_S2 X [fo, S2 X [f,
Ole
and D
1,6
=
The D3x
V2)
fo
-
3
E)v
D& 2) U D
3
+
ffo
s]
3
UD
+ 6,
fl
X
_
jP1, Pit}
61
U S2
X X
[f
0
+,6, f1
[P" P"I
V,
X
xfp',p"}x[fo+E,fl-,-]UDl,ex[pi,piI
_
-
6]
U
'-,
3
fil
11
denoted
D3
(W2
piece added
that the total
(W2 E), V2) 4 e 4UK e
-
+
E]
fl]
6,
3 US2 x Ifo +,) D X Ifo 3 US2 If, _,6q D X Ifl
+ -
Ej 6}.
as
subsets
e] -pieces in the above formula are glued of the corresponding pieces in (1.6.32).
Ub D3
X
Ip 1}
[f
W,
V1
fp', p"} x [fo (Wi ev VI)
U
+ e,
f,
-
into
We
have, for example,
(WI
G,
VI)
_V
X
0
+ 6,
fl
_Ub (D 3 x[fO,fo+6]UD
3
X
[f
I
_
fl])
E,
W1 0" V1. The last
W,
ov
Vi
homeomorphism
results
is removed from the collar
from the fact that with
3
D
X
If O}
a
4-ball
D3
C a (Wi
o,
[f0 fo+ 's ]C Vj) so that the
X
,
92
The Double
1.
Category
of Framed, Relative
3-Cobordisms
by the removal can be "pushed ouf '. In the course of this the boundary pieces Do,, are pushed into the positions of the D3,S in M, o N, C o9(Wl o, VI) that are obtained by removing a 3-dimensional neighborhood around the holes in the intermediate of 3-dimensional surface of composition cobordisms. Note that Q& is now presented by adding the remaining DI', x [p',p"] with UP(DI x [fo, fo + -] U D3 X [f1- 6, fl]). 0, 1 to (W2 o, V2) U (Wi o, V,) j gap caused 3
=
--
then becomes an "push out" homeomorphism this presentation M X [P1,P111 to (W2 Ov V2) U (W1 Ov V1) ordinary 1 -handle attachment of *e 4= 1 3,0 surface. We, along the D3,S in the boundary around the holes of the intermediate thus, obtain the following homeomorphism, which is canonical up to isotopy: With the described
Q& 4- t'(W2
V2)
ov
U *e
4U
...
U
*e 4
U(Wl
OV
Vi).
of the surgery on the interior of Q is depicted schemati= 1. b shaded The case on the left area simplest corresponds
Below the mechanics
cally again only to
the S2
replaced
the in
x
D2 D3
for the
XS&1-part.
and bm 2
Ml,,
combine to
is removed,
piece that
C
b-141"
The
denotes,
and the shaded as
before,
area on
the full
the
cylinder
right
indicates
D2 X[PO,
PI]
corresponding piece in M2,,. In the horizontal gluing they which runs transversally the 1-handle *e 4. bm, through Analogously, 1 1,2
for the thickened
the
-
strands
N
bl,2.
V2
W 2
W,
U
Q is a part of (W2 Oh W1) Ov (V2 Oh Vj), the surgery on Q also extends product of the interchange law in the way implied by the statements in Lemma1.6.6. The general form of the surgered product manifold is now: Since
to this
total
((W2
Oh
(W2
o,
WOOv MOh Vl))&
V2)
Li
(Wi
ov
Vj)
U
*e 4U 1
2b
the 2 b
new
of the e4-handles 1
...
4b
U e42
Ue41
U
...
U e41
2b
of this proof we show that 2b of the e4-handles can be cancelled 2 and 2b e41 handles, such that the remaining 2b of the e4-handles 2 of the are attached precisely in the way prescribed by the definition
In the remainder
against
U *e41 Ue2 U 4
...
*
The Central
1.6
horizontal
composition.
of Section
1.6.3.H,
Off,
and
on
"M,,-pieces"
(M2
N2)
given
as a
U
interior(Mj
(Mi
Nj),
o,
gluing
the
Ni).
o,
neighborhood
ti'cal
"M,,-pieces"
in the
and the
db
-+
are, using 'W-pieces"
Gob
-*
terminology
the of
93
NW2o, V2)
Vi).
o,
For the o,
of these handles
The attachments
all either
d24
Extenis.1on
of
of
*e 4-handle
a
corresponding
source
in the intermediate
for the e4-handles 2
tori
in
an
index-1
which
over
we
surgery
do the
for either
"M,,-side"
the
on
*e 4-handle. corresponding 1 4 U e (where we have U
the
izontal
results
of surgery balls S' and S" is discs D2 C Egi. I C or target
Egint
surfaces
attaching gluing. through composition run transversally surgery diagram on (M2 o, N2) U (Mi o, N1) The
pairs
Each of the b
U e'
...
ver-
horIn the
added
2b
the
1-handles
for
at the
4-handle e2
a
total
source
(M2
of the
Oh
of the
and target
Mi)
sphere S'
appears
as a
composition)
surgery
data
attaching
the
bm,
ribbon
which enters
N, and emerges at the partner sphere at the very source. In S" C M2o, N2 and, furthermore, passes through the e 4-handle I the picture below its pieces in M2 and M, are denoted by b2m and b1m. Analogously, in the (N2 Oh Ni) -composition, b N for each of the b e 4-handles we have a ribbon 2
transversally
which
runs
spheres
1
...............
over
the
S' and S"
summarized
M
surgery
one
/b /b
on
the
same as
right
IM
*e 4I -handle
bm-ribbon
the
o,
and, hence, through with
side of the next
bm
M
M,
C
the
same
the
same
pair
The surgery
labels.
of surgery diagram is
figure:
M2
41 MI
2
..................
M b
2
MN
MN
b
I
2
................
..........
...............
......
bN1
bN b
N
2
N, N,
2
2
that moves one surgery sphere S' an isotopy we can apply configuration bm -handle from Mto N until it is right 1 the added through piece along S". situation In this move next to its partner apply the cancellation surgery sphere the ribbons bm and b N are in Figure 1. 15 of Section 1.5.2. As a result, as described replaced by their connected sums, and the surgery spheres disappear. Thus, instead of the and the 2b attachments of the *e 4-handles of the intermediate attachments 1 b e 42for with data e4-handles U consider we can attaching (M2 o, N2) (Mi o , Ni) 2 To this
the ribbon
handles locations
bjM
with the respective up the strands of the vertical surface in the interiors of the intermediate
given by joining
is, however,
composition For the
exactly
the
"M,,-part7'
(W2
V2)
Oh
OV
"W-parf
union of handlebodies
'the
(WI
of the Ov
attaching
at the disc
composites.
prescription
This
the horizontal
for
Vl)-
boundary piece of (W2 g, Hgl91:_, ." Wg,:,g lig"91,tg =
bjN
ones
o, t
V2) and
U
(Wi
tg H92 92:-
ov ,
=
Vi)
is
92 ia
H92:
-
given by the tg H92 92:int as
0
V
94
in
The Double
1.
As before,
(1.6.13).
results bined
index-1
an
along
located
at
a
pair
of surgery
the intermediate
(W2 Oh Wj) -composition
From the
3-Cobordisms
*e4I -handles
of the b intermediate
the addition
surgery
handlebody,
of Framed, Relative
Category
we
surfaces
also have the
Sj'
balls
on
Sj"
and
the W-side this
attached
com0
Zgj,,,
in the interior
e3, -handles
on
C
'h9tg.
9-
Z,_
to the
LbM. Since the boundary pieces and running through these surgery ribbons Lj, for the horizontal original prescription composition was that the ribbons LjM should handle of close through thefirst they will all run in the composite through the first pair of surgery balls Sl' and Sj". N the surgery ribbons LN, Similarly, L, for the (V2 Oh Vj) -composition run I that are attached to Hence, in the total composition through the e 3-handles I of another of them runs one pair through surgery balls Sj' and Sj". In summary, every tg 1 W" IJ92 where the extra e 31U e 3, U we obtain a surgery diagram on 91:. 92:- U e 3U 1 1 .
.
.
,
...
2b
handles
those attached
are
to
Zg_
and
Eg,,.
It is
depicted
following
in the
figure.
picture it is now easy to see that every pair of surgery balls Sj' and Sj" cancelled against a ribbon LF as described in Section 1.5.2. The resulting the one for the horizontal is then precisely composition (W2 Ov V2) Oh (WI Ov
From this can
be
picture
Vi)
with
the
Hence,
omorphic
(W2 as
We obtain Lemma 1.6.3,
definitions
741
handlebodies.
Ov
V2)
4-manifolds now
Oh
(W1
with
Ov
comers.
the main result
Lemma 1.6.6,
V1)
and
((W2
of this
chapter
Lemma 1.6.7,
of the maps between
Oh
W1)
completes
This
2-morphisms
Ov
the
(V2
Oh
are
home-
proof
by combining
the identification from Section
Vi))&
in
1.6.2.
the results
(1.6.29)
from and the
1.6
Theorem 1.6.8. into
a
strict
Cearil,
and vertical
Exieux;ityn
composition
L14
--+
66
defined
--+
OA
above make
95
dib
double category.
The map irc
categories.
The horizontal
1
:
dib--+
Cobfrom (1.6.10)
is
a
strict
doublefiinctor
of double
of Cobordisms
and Presentation
Tangle-Categories
2.
and TQFT's of 3-manifolds of quantum invariants rigorous constructions combinatorial finite manifold in a a way, and then given proceed by, first, presenting basic The data. most combinatorial this functors to example of applying algebraic of a manifold as a simplicial. such is a presentation complex, taken modulo so called that these types moves. It is not hard to imagine Alexander or Pachner subdivision cobordisms between when considering become quite complicated of presentation All
known
and
surfaces
closed
cobordisms
tive
The types constructions
S1,
of are
in
different
two
presentations ones,
which
links
of links,
modifications
results
3-manifold, the same manifold,
yield
of these
succession
we
arise
we
also
want to describe
imply
would
the latter
rela-
that
we
groups.
exclusively
therefore,
will,
what is called
from
on
surgery
of
our
manifolds.
It
in
use
all
and Wallace,
in the
same
the
could
He extracted
3-manifold.
01 -Move and and, using Cerf-theory,
called
the associated that
things,
Among other
(see [Lic62] and [WA1601), that any already without boundary can be obtained by doing surgery along a framed link, when surgery solved the question, the three-sphere. Later, Kirby [Kir78]
3-manifold
along
cumbersome when
to Lickorish
known
L C
more
corners.
encode the framed braid
simplicially
was
even
with
be obtained
02 -Move,
the
that
he showed that one
types of
two
do not
change
any two
from the other
links,
by applying
a
moves.
and modified will have to be substantially presentations maniclosed to Calculus For only apply example Kirby purposes. fold as surgery on the simply-connected space S1, but we need to describe manifolds handlewith boundaries and corners obtained via surgery on non-simply-connected for the relative cobordisms, we bodies. And not only that: besides the presentations the two types of compositions rules that translate also must find simple and efficient level of presentations. the into defined on in in b, as operations Chapter 1, that we need for our purcalculus of Kirby's Another necessary modification The described
extended
for
surgery
our
of TQFT's proThis is because the construction poses is to make the moves local. that verification the for local data to Hence, ceeds by assigning pictures. algebraic
algebraic structive
have local,
For cobordisms
fit
best
L C S'
for
our
yield topological elementary moves.
invariance
indeed
relations to
between closed
needs,
we now
has
consider
the type of surgery developed in [Ker99].
surfaces
already tangles T
been C
R2
x
[-1, 1],
T. Kerler, V.V. Lyubashenko: LNM 1765, pp. 97 - 172, 2001 © Springer-Verlag Berlin Heidelberg 2001
and in-
it is both convenient
or
rather
that
presentations, Instead their
of
a
link
projections
98
Tangle-Categories
2.
into
the
strip
R
and Presentation
[-1, 1].
of Cobordisms
framework
In this
the composition of cobordisms over simply stacking tangles on top of each other. Also, the Kirby there by a set of local moves. They arise from the "bridged linle' moves are replaced calculus moves at the bounddeveloped in [Ker99], complemented by additional aries of RI x [- 1, 1]. In summary, cobordisms in [Ker99] are presented as a functor from the ordinary category dib(O, 0) into a special subcategory of a natural tangle category, similar to the ones introduced in [FY92], but taken modulo additional equivalence relations given by the moves. The restriction to invertible of the mapping tangle classes yields a representation class group of a corresponding closed surface. If we use manifolds with comers for surfaces with boundaries. we also obtain For a surface analogous presentations with one boundary component we reproduce the tangle presentation of Matveev and of the mapping Polyak [MP94], which was obtained via Wajnryb's presentations
surfaces
class
x
translates
into
groups.
Following the Obb by constructing
same an
The purpose
category.
and 1 -arrows
are
as
principles,
invertible
of this
those of
want to
we
functor
chapter
F_bib,
represent
the entire
double
category
it that maps it to a combinatorial is to define and describe the latter. Its
but the
on
2-morphisms
are
now
double
objects of generalizations
elements. tangle spaces with additional is defined as before composition by stacking tangle diagrams that of a horizontal comrepresent given classes on top of each other. The construction of representing position starts with the juxtaposition tangles but is then followed by two further operations on the tangle diagram. The two compositions will turn out to be compatible with each other and, thus, define a tangle double category. the
previous
A vertical
Summary of Content In Sections
2.1
and 2.2
we
develop
the notions
and conventions
needed to define
This shall, eventually, tangles represent the of strands and that in can occur a diagram, specifying types coupons the possible local pictures, the 1-arrows that define the boundaries of a diagram, and the allowed global properties of strands. From this we define in Section 2.2 the 2-arrow sets of the double category 7-gl as equivalence classes of admissible The notion will defined be a number of tangle diagrams. equivalence by introducing which of for all set other as a serve elementary moves, equivalences. generators The purpose of Section 2.4 is to show that the classes of planar tangle projections in 7-gl are in natural one-to-one correspondence with classes of tangles that in three dimensional are embeddings examined with the help space. It is carefully of transversality methods how the process of projecting tangles in the plane gives rise to the so called TI-Moves of 7-gl. In this section we also relate another group of of auxiliary to the addition strands that have no relmoves, the TD-Moves of 7'gl, the admissible
that
cobordisms.
use to
we
included
evance
of the a
variety
for
a
surgery
TQFT functor. of
presentation In the
tangle classes
of
course
for different
a
cobordism
of this
but
types of
are
useful
following tangles and
and the
in the construction
sections
different
we
sets
introduce of
moves.
example,
For
Ingredients
Local
2.1
2.5
in Section
of
we
and Horizontal
Tangie-Diagrams
that
the formulations
also introduce
99
I-Arrows
correspond
to
of Kirby and Fenn Rourke as well as the "bridged link calculus". the surgery calculi from the double to pass gradually Moving between various types of tanglesallows double of cobordisms to an equivalent to the category category of tangles equivalent of
category
tangles,
in terms
changing
do it in many steps, rather than to do it in stead of
a more
in the table
categories
whose
theorem,
one
detailed
Weprefer to of which TQFT's are easy to construct. the classes of tangles and sets of moves gradually,
we
survey
although
tangle
equivalent
introduction.
the end of this
at
would occupy dozens of pages. In-
proof
summarize different
and horizontal vertical 2.6 we introduce compositions o, and Oh for bicombinatorial It is shown that corresponding explicitly. tangle categories of admissible on the level tangles do indeed factor into equivalence nary operations law so that fulfill an interchange Moreover these compositions classes of tangles. In Section
the
(7-91)
Ov)
Oh)
does indeed
form
double
a
category
as
In the remainder
desired.
of
of tangles that are of decompositions how the Wealso explain further relevance for the horizontal technical composition. and braided tensor categories are naturally of braid groups of surfaces structures represented within the double category 7'gl. N. B.: In the table on the next page the 2 -arrow sets of all the listed tangle catebetween the respecin the sense that there is natural bijection gories are equivalent, which then also extends to a double isomorphism between tive classes of tangles,
chapter
this
special
consider
we
double categories. For each category
and
forms
the tangles that we consider by the space they types of ribbon strands and other by the different the tangle according to their local and global properties, components that constitute On these sets an and by the constraints on the allowed embeddings or projections. equivalence relation is then generated by the listed moves. For convenience we also provide on the next page the reference in Chapter 2, where the complete definition is to all the other categories of a given category is given and where its equivalence
embedded
are
or
we
characterize
projected
into,
asserted.
2.1
Local
of
Ingredients
and Horizontal
Tangle-Diagrams
1-Arrows In this
section
we
tangle diagram, include
i.e.,
several
shall the
more
is called
compile projection
types
context,
2.1.1
Horizontal
I-Arrows
combinatorial
terms
In
a
of all
list of
a
of ribbons
"Kirby
the
other
a
as
well
tangle as
so
in
R2
called
x
needed to make up [-1, 1]. The list will
coupons,
which,
in
a
an
circle".
dotted
and Intervals
on
the horizontal
arrows
namely triples of numbers, [g, a/bl, formally defined for b >, 1 as in Section
ingredients
local
ribbon
Rx are
111 the
same as
g, a, b E N U 1. 1 by
with
10}.
the
The
ones
of
&Rb,
composition
is
100
2.
Tangle-Categories
Category
and Presentation
Space
Compo-
of Cobordisms
Constraints
Moves
Reference
nents
7-gi
R.
111
x
Strands: external internal:
R'
x
[-1,
1]
TIl-TIll, TDI TSI
projection height fct. Coupons&joints: well positioned projected All embeddings wrt.
(top, bot, thru, clos) auxiliary: Coupons
7'91R2
S-ft-i0s: in gen. pos.
As for 7-g I
-
-
2
R
X
[-1'
1]
TDI
As for
7-g
Coupons&Joints well positioned
I
TS4
Isotopy -
TS1
,rglwell-pOS R2
Section2.3.4
TD5,
TD5,
Section 2.4.1 Thm. 2.4.8
TS4
-
Isotopy,
Section
T16, TI 11,
Lemma2.4.3
2.4.1
TD1 TD5 TS1 TS4 -
-
rgldec-proj R2
R2
x
[-1,
1]
As for
'rg
Coupons&Joints & projectable well positioned
I
IsotopyT16,
Section
T17, TIl 1,
Lemma2.4.5
2.4.1
TD1 TD5 TS1 TS4 -
-
7-g 1plan R2
R2
x
[-1, 1]
As for
7'g
I
Strands:
Isotopy,
Section
in gen. pos.
T13, T16, T17, T110, TH I,
Lemma2.4.6
projection Coupons&Joints: & projectable well positioned All embeddings w.r.t.
S'
7-gls2 7-gi
1, 1] As for
x
rg
92
_
X
S2
-
I
without
the
-
-
-
Isotopy, -
-
TD5,
Section 2.4.2 Lemma2.4.9
TS3
Isotopy, TDI, TD2, TS1 TS3
auxiliary
2.4.1
TD1 TD5 TS4 TSI
TI)l TS1
1, 1] As for 7gi All embeddings
x
TIl
Section 2.4.3 Lemma2.4. 10
-
strands
'r go JS;* S2
1, 1] Strands:
All
embeddings
external internal:
(top, thru) Coupons
7'9 1 JS32L
S2
X
[-1'
1]
Section
2.4.3
TS2*, TS3*
Strands:
All
external internal:
(top, bot clos)
thru,
I Ball-Pairs
Isotopy,
Lemma2.4.11 TD1, TD2, TD3*, TD4*, TD5*, TS1*,
I
embeddings
I sotopy,
TDI-6, TD2*, TSI*, TS26, I TS3*
Section
2.5.1
Eqn. 2.5.1
S2
Ki
I S2
[-J,
X
of
1]
and Horizontal
I -Arrows
Constraints
Moves
Reference
embeddings
All
Strands: external
S2
[-1'
X
1]
embeddings
All
Strands: external internal:
[-1, 1]
x
r.-Move,
wrt.
Now we wish to describe between horizontal
positioned projected
[gl, a/b]
[91
---*:
[gc, a/b]
arrows
+ 92 + b
-
1, a/c].
(2.1.1)
Wx [- 1, 11 that represents a 2-arrow [gt,, a/b] (see, e.g., the squares in Secon specify a + 2g"' + b disjoint intervals
tangle diagram
a
TS3
well
ns
o
-
-
projection height fct. Coupons&joints:
auxiliary:
[92, b1c]
TIll, TDI TD5, TSI, TS2, TH
in gen. pos.
bot, clos)
in
and
have to end we, first, intervals R line x 11} and a + 2% + b disjoint the upper boundary associate each to we R x I [g, a/b] 11. Specifically, boundary line To this
1.2).
tion
on
a
-
a
2g
+
as
closed,
+ b
disjoint
mutually
connected,
2.7.1
Section
Strands:
internal:
C
2.5.2
Section
Lemma2.5.6
Ribbon-TS3
Strands: external: last removed
(top, thru,
Lemma2.5.5
Isotopy, Signature,
(top, bot thru, clos) R.
Section
Hopf-Link, Ribbon-TS3
(top, bot diru, clos)
fg- I kS' 2R
2.5.2
Isotopy, 02-Move,
internal:
_
101
Tangle-Diagrams
Components
Space
Category 7-Ig
Ingredients
Local
2.1
intervals
the real
on
line
the lower
sequence of R denoted
follows:
J IS6
,, exactly paths t i-+ ps (t) of the the after before and local ribbon the shortly Thus, diagram shortly passage which include will however, one 7r-twist, deformation will, parameter through h. change its orientation. conthe more explicit Next, let us investigate consequences of the transversality after end 0 this To assume ia we ditions at a particular h"' point (h, ia). may The requirement variables. R(O, 0) E X2 then translates to the conditions shifting 0 and Po (0) 54 0. in -440 on vectors (0) dt
im
in
(R, aR) at
TX2 X1
n
particular,
=
0 for
all
This
a.
that these
=
=
=
2R
The condition
parallel independent. to
(0, 0)
at
to (0).
TR(O)
E
X1 then
implies
Thus, transversality
that
this
for
means
these
two
d2 d7
that
case
vectors
: O (0) is
linearly
are
aR the vector R X2 implies that 9S and LRI Moreover, the condition at complement c_- R4 in the form this is RI. X2 coordinate In on vectors a condition T C space R(0)
plane
d2
projection: Now, as long of
framing
to the
only
at
as we
(s, t)
=
(0, 0)
transformation
coordinate
vectors
and
ad Osdt
some
order
a
; , (t)
framing of the
vicinity
any two
linearly
depending
on
move
ex and e.,
we
apply
may
will
be close
parallel
vector
origin. Also independent
the relative
deformations
small
d
to
2
'9d asdt
to
, o (0),
we
: o (0),
not
dt
always apply
we can
vectors
to
the unit d
2
of dt2' &0 (0)
orientations
rescaling
of vectors
and parameters
we
may summarize the situation
follows:
d: O(O)
dt in
but also in
d dt
: o (0).
After as
d
s (t) Y_X It
make the
to
to
J o (0) span R2.
over
Since in lowest
vector.
linear
a
do not pass
the above condition
can use
ad Osdt
: o (0) and
dt7
a
sider
vicinity in the
=
d
0, of
dt2
: O (0)
=
2ex,
(0, 0). Among the
following
only
the
four
positive
0d
, o(O)
Osdt indicated
signs.
=
cases
The other
ey,
(two cases
and for
v,(t)
each
follow
sign)
=
e,,
we con-
via reflections.
Tangles
2.4
Note, that from the above formulae the 7r-twist
(t (s)) flips
dt
condition
this
there
will
order
we
be
no
: , (t)
on
pure powers of
we
0
t2+S2
have to add
s-parameters
t2
so
and forth
third
: , (1)
and
If
b,
write
(and
conditions
that
where
: , (t) are
x
even
the
vicinity
of
such that, E
as
R2 will
(s, t)
R2
' (0)
:
at
its
0.
: , (0) thus, of : , (t) -
0. Weshall,
=
,
through y F-+ y
in
a
stey
02 (S) t)
+
to
find
to fix
be distinct
=
useful
two
move as a
: o (0) before,.: ,
a
local
points
(0) and
=
on
ey and below for different ex +
t2 =
+ y
e,,
we
of
(- 1)
Y(S' t)
ex
values
ey, The
-
of
=
t3)
S(t
paths
t
_
, ,(t)
=
a
s.
(s, t)
and
all that
wish to add
further,: s
independent
for
diagram
a
precisely,
0 and,
move
the strand
move inside
0. More
this
model for
that fulfill e,,, the easiest functions and odd in t for the x or y variable, respectively)
(S' t)
X
depicted
passes
-+
maintains
power expansion From the above we find that to second
in order
consider a
&, b =
+
e,,
terms
we can
outside
following: that
R2
have :
we
Hence,
0. In order
order
order terms,
=
we
(s, t)
as
higher
unchanged
remains
(t))
as s :
P, (t (s))
t-+
have:
02
where
V),
isotopy
occurring.
s
s
orientation
its
hold.
to also
: s (t)
so
change
Wemay further apply an s-dependent that for the isotoped path : ' , (t) = 0,
assume
whereas
direction,
direction
that the strand
explicitly
we can see
its
does
Hence, the 7r-twist
direction.
so
-+
s
129
Three-Space
in
these are
the
[-1, 1]
are
t3' for
t
E
s.
Y
..........
A
.......... ...........
S >
0
S
(t)
the
twists, lines
lines
are
the
push-off
's of each
the ey, which then indicates sequence of paths is redrawn as =
over-
in the
orientation
0
along the framing in the case of corresponding ribbons. Below of ribbon immersions, a sequence including the crossings between solid and dotted and undercrossings. Particularly, these twists above graphs become 7r-twist, change and, as explained,
The dotted vs
S
S
Figure
Figure
Fig.
splitting
the
to define
of
a
2.1.
can
easily then
identified
apply
as a
collective
TH Moves
top-line
at the
only
7r-twist
given
all the way up. The substitution the TH Move, since two of the braids
for
a
since
7r-twist
the two
on
any such twist is also compatible
can
be
with
given in Figure E above parallel strands, to which we
strands
of internals
27r-twist
D
map S
splitting
The
isotoped are
C
S_
E
Figure
B
S
individually. for 'rglS2,
namely TD1, TD2, TD3, TD4, TD5, TS1, into the Moves TD1, TD2, TD3*, TD4*, seen to translate TS2, and Hence, the operation. TD5*, TS1*, TS2*, and TS3* for 7-gl';* S2 under the splitting equivalences TS3, are easily
The other
map S is well defined. of The construction
pair an
K.,
As
as
an
K.
Kj+
the
interval
and insert
between the
and the
K_j
the
larger
new
interval
shown.
The
reasons
that
the existence
diagrams auxiliary identity.
the
at
each other
ribbon
ribbon
ensured the
strands
to
auxiliary
internal
Kj
I+
S-1
inverse
an
depicted on connect we simply
straightforward. right hand side pairs of internal is
at
S-1 is well
the top line. in order
ribbons
analogous to those that immediately clear from We only need to use some obvious isotopies along to see that the opposite composition also yields the defined
of S. The fact
that
are
S
o
completely
S-1
=
id is
2.5
A
Alternative
Calculi
Equivalences
and Further
143
Composite Correspondence: In
previous original
sections
have shown that
we
the
tangle
classes
of
S2
are
the
same
paragraph planar tangles. ribbons and still tangles over S2 we may omit the auxiliary allows second the obtain the same category. us to consider Moreover, paragraph internal bottom and closed well ribbons as no as auxiliary planar tangles'without find there is that of these all ribbons using the splitting we maps map. Combining that both contain between no two one-to-one a tangle categories correspondence the in S2 other with the with and over ribbons, one plane tangles tangles auxiliary R,, x [- 1, 1]: as
the
know that
we
Corollary S2
x
for
in this
From the first
of
section
the
2.4.12.
[- 1, 1]
of tangles
classes
The composite
and R,
x
[- 1, 1] yield
of 0, S-1, and the relations between tangles in classes between equivalence a natural bijection
asfollows:
TgIS2 The 2-arrow
sets
Note that
of
the
aside from ambient
both
categories
7gl are
in
bijection
over
S2
also
tangles for the category namely, isotopies,
S;*
are
with
subject
those to
of 7-gl.
only five
moves
TD1, TD2, TSI, TS2, and TS3. The
moves
for the
planar category
are
1, TD1, TD2, TD3*, TD4*, TD5*, TS 1 *, TS2*, TS3*, and TS4.
TIl-TI1
7'91'S2
of a will represent a surgery presentation presentation in the next section, in which also The remaining the Moves TD1 and TD2 will disappear. surgery moves TS 1, TS2, in the proof of our main theorem on tangle presentations and TS3 will be interpreted and as a "aof cobordisms in the following chapter as handle trade, cancellation handle decompositions. Move" for representing it contains The planar category involves a lot more moves. However, only top ribbons and external and through internal ribbons, but no bottom or closed internal the allow us to construct ribbons. This form will ribbons as no auxiliary as well TQFT functor in a systematic way. In
applications
cobordism.
Wewill
2.5 Alternative
a
tangle
discuss
Calculi
from
a
modified
and Further
Equivalences
are almost what we consider a surgery tangle diagrams used to generate T91n2 S of the 2-arrow and the moves used in the definition diagram for three manifolds, A well known sets Of T91'S2 give rise to what is often called a surgery calculus. Its calculus of links, is Kirby's see [Kir78]. purpose is to establish surgery calculus of a bijective links, which are tangles correspondence between equivalence classes
The
X
"X
144
boundary that
without
only
consist
manifolds
of three
classes
of Cobordisms
and Presentation
Tangle-Categories
2.
of closed
ribbons,
interior
and
homeomorphism
boundaries.
without
calcuis deduced from the so called "Bridged Link! Bridged Link calculus is nonetheless equivalent to a of the original Kirby Calculus, as shown in [Ker98a]. Another varigeneralization from peculiar which results is the Fenn Rourke picture, Calculus ant of the Kirby and has technical reduction combinatorial applications. advantages in particular between the menIn this section we will either show or review the equivalences relations in the context of admissible tioned calculi tangles. The relevant bijective can be inferred tangle classes with no referentirely on the level of combinatorial derived
yet,
ence,
Of
description
The
lus,
to
We will
791'SX2 The
[Ker99].
in
three
manifolds.
also
discuss
how
go from tangle diagrams over S2 back to one the TS4 Move, but instead by eliminating
we can
R2 without
introducing
of the external
strands.
This reduction
the horizontal
compositions
tangles
over
2.5.1
From
Coupons
to
for the
Bridged
[Ker99]
in
and
we can
smaller
an even
inside
S2
S2
11}.
x
X
an
build
a
an
map OB : OB=4 for The condition
ends in
PI
can
orientation
=
p and
an
of descriptions [_ 1, 1]. Following
p'
are
c
tangle
for
7-g 1BL S2
can
-
end in the internal
of
admissible
the calculus
contains
extemal
and extemal
and intemal
intervals
ribbons
at the
o9B'.
attaching
ribbons ribbon
OB' under the identification
thus
in the construction
have
p C o9B another
interval.
OB(P)
tool
categories.
boundary pairs of surgery balls embedded in S2 X [-1, 1] in DI - -- B', we also end. For any pair of balls (B, B'), with B their between bounding spheres, i.e., reversing diffeomorphism that
Wecan also
which ribbons have
essential
-
set of
admissible
[- 1, 1]
an
Links
precisely. As before,
be
double
tangles can be substiof "Bridged Links" with these ingredients a tangle starting category, 7-g 1BL, S2 of 79 IBL definition the summarize Let more us moves. S2
The coupons used in our previous tuted by pairs of balls inside S2 X as
will
tangle
considered
part of the
to
the balls
is that
if
some
type of ribbon
has to emerge at the image of this interval map. The two ribbon pieces ending in same
component of the tangle.
With this
of components we can then use the original strands. tangles. However, now the tangles contain neither coupons nor auxiliary include also Here of we course, isotopies. Among the moves are first of all, 09Y is the the intervals of the attaching as long as isotopy on a sphere isotopies allow also and 59B OB. Moreover, we composite of the isotopy on the partner sphere class oriented the Since mapping group isotopies of the identification map OB itself. B to any other. of the sphere is trivial move from one 0 we can therefore three moves: the following Besides the isotopies we introduce notion
Definition
2.2.2
of admissible
2.5
TS 14 Move: Two partner
Calculi
B and B,
balls,
between the balls.
right
are
Ribbons
are
entering
replaced
be
can
in the form
of
an
by
internal
an
P2P3
the
...
ribbon
f
in-
at
of balls
A
closed
A. The ribbons
annulus
passing through the balls are now passing through the disc bounded by this annulus as depicted. TS2* Move: Consider a pair of balls with that
were
only
internal
one
pair
ternal
be separate
ribbon
and the closed
of balls
eB
through
passing
ribbon
them. Let the
in-
from the rest of the
/-7
can be elimiconfiguration all together from a tangle diagram.
diagram. nated
TS3*
Such
a
Move: On the
right
we
have
p'
such
that
consists
ponent
B
a
ribbon passing through top internal pair of balls, B and B', at intervals p and
145
intervals
at
OB and emerge at corresponding Pl P21 tervals pl, p2, at (9B'. For this configuration
pair
Equivalences
such that the map OB coincides reflection at the plane in the middle
with the miffor
...
and Further
other,
to each
next
Alternative
i,
a
the
entire
com-
of two
small
strips
i,
lio
IV
from the top line to these intervals. another bottom internal Furthermore, the same is passing through pair of balls at intervals q and q', but ribbons run through B and no other
ribbon
B'.
Wecan eliminate
pair depicted,
nent
and the
nect,
as
and
q'
the top compoand con-
of balls,
the end intervals
the top line so that it becomes of through internal ribbons.
Remark2.5.1. on
q
of the second bottom ribbon
the
right
however, ribbon
In another
belong
redundant
next
to
a
since
to each
other,
a
version
to
pair
of the
top ribbon
TS3*
Move we may assume that the strands
of
instead
a
pair of through
balls we can move the surgery and then apply a cancellation BL
Of7glS2
ribbons.
around this
This
connected
is,
top
move.
is equivalent to all of the other definiin particular rgln' S2 we introduce an intermediate tangle tangle categories, for admissibility as category 7-glr"S2 It contains the same elements and conditions that for every pair of balls (B, B) there is a r' 7 -9 1BL we require but, in addition, S2 combination ribbon denoted as r B This is a usual ribbon piece --:-- [-L, L] x [0, 1], In order to
tions
see
that the definition
-
of
,
.
,
.
146
R2
embedded into
endpoints.
There
the identification The a
we
moves
and
We shall
definition
disjoint
[0, 1]
tangle diagram except at the f L} x [0, 1] C W, such that only maps the two end interval
from the
C Mand
-
of
the
between the balls
is introduced is
the
maintain moves
planar
flat,
a
Moves
piece
same
are
BL
7glS2
in
pictures projection
in the
plane
of
for
these
moves.
two
additional
the recombination
They both concern TD1* and TD26.
[0, 1]. moves
in the
notation
there org 1BL S2
in
7'91'S'2'.
Of as
x
OB between
which
three modifications below
JL}
require
ribbon
TS3*
illustration.
which is
spheres not on the interval but, moreover, is the identity of the modified in 791'S'2' consist, firstly,
recombination
TS24,
[- 1, 1],
map
each other
onto
x
of Cobordisms
and Presentation
Tangle-Categories
2.
and
where
TS14, of each
Besides
moves
ribbons
,
for
these
that enter the are
depicted
TD14:
TD2*
M
guarantee that we can change every recombination given balls to any other such ribbon. The crossing move TD1* allows us to change the path of the center of the ribbon in any given way, since the paths are in a simply connected three space and every homotopy can be between deformed into a differential by transverse intersections isotopy interrupted Notice The Move TD2* allows us to change the framing by any integer. strands. of the r B, s so that 7r-twists condition at the end points that we have an orientation The two additional
ribbon
r
B
moves
between two
would lead to
an
immediately
recombination
inadmissible
Since the choice
of these
There is
a
bijection
natural
,r g Iree S2 which is induced
by the
omission
coupons to those with
Lemma2.5.3.
gories,
arbitrary
There
is
a
natural
--+
between 2-arrow
the
following
is
-
included,
find
surgery
bijection
of the categories,
BL
ribbons.
are
sets
7gIS2
of the recombination
ribbons
Once the recombination with
is, thus,
implied:
Lemma2.5.2.
classes
ribbon.
ribbons
additional
we can
maps that relate
tangle
balls: between the 2-arrow
sets
of the
cate-
2.5
R:
by replacing
which is induced
Alternative -
rglS2
nx
coupons
and Further
Calculi
--+
I
r9VS2
Equivalences
147
'-.
,
by pairs of balls.
A tangle is quite straightforward. representative ribbon pair of balls with a straight recombination below, the in and out going strands are entering of the plane through the balls along the equators that are obtained as the intersection the coupon and the two spheres. The extra ribbon also lies in the plane of the coupon. The fact that R factors into tangle classes is immediate since the moves TS1, TS2, TS3, TD1, and TD2 in Tg'I n'2 are, with the given positionings, readily implied by the moves TS14, TS241 TS3 TDl*, and TD2* in Tg' Irec S2
Proof.
The definition
of R on
a
coupon is simply replaced by a between them. As in the illustration
,
4r
VICE%
i
9C
An inverse
map R-1 is defined
aligned
an
The strands attached to the spheres as follows. isotopy along an equator (or rather a pair thereof) that also conribbon r B As depicted above we tains the attachment point for the recombination B the ribbon r by parallel strands that continue the incoming and can, then, replace the equators to which the incoming outgoing strands at the balls. More precisely, We can find a and outgoing strands are attached bound discs DI inside the balls. 2 2 [0, 1] x [-L, L] such that the identification homeomorphism D U rB U D map are
by
.
OB between
[0, 1] x I-L} as the natural identity to [0, 1] x ILI. along disjoint intervals Fj C [0, 1] x IL}. The parB 2 allel strands are, therefore, generated by replacing D2Ur U D by (OjFj) x [0, L]. Finally, a coupon is introduced right across these parallel strands by slightly expanding a rectangle [0, 1] x [-6, s] C [0, 1] x [-L, L] classes well defined on equivalence Now, in order to show that R` is actually in the above construction are taken care of let us first check that the ambiguities B 2 2 by equivalences in TgIS2. To begin with, the homeornorphism. D U r U D C--is not unique but all such homeo[0, 1] x [-L, L] (with fixed attachment intervals) morphisms are isotopic to each other. An isotopy between two different homeomorphisms can be lifted to an ambient isotopy in three space and, hence, to an isotopy into intervals for a tangle from Tg1 S2. Moreover, two ways of moving the attaching position along equator on the sphere may differ by a braiding of the strands in a I of the recomof the sphere, S2 _I 1--- int(D 2), with the end interval vicinity bination ribbon removed, and a corresponding opposite braiding for the outgoing R-' we will, thus, the recombination strands on the other sphere. After applying The strands
the
are,
spheres
thus,
maps attached
148
Tangle-Categories
2.
have
right
braid
a
on
of Cobordisins
and Presentation
top of the coupon
well
as
opposite
its
as
at the
bottom of the
coupon. But if we apply the TS I Move in 7 gl S2 to this coupon we easily see that the braids can be pushed through the annulus from the TS 1 Move and cancelled against
equivalent tangle with no B identification homeomorphiSM 0
obtain
the TS I Move we, thus,
Reversing
each other.
the at the coupon. Changes in choosing between the two spheres are dealt with in the exact braids
remains
It
under R-1
show that
to
equivalent
also
are
parallel expressed by
in
way. "
2
.
The Moves TDI*,
from the substitution.
strands
of the
same
tangles are equivalent in 7-glIS2' their images rgln SX Isotopies in 7' 91'S2 also include deformabut those can be expressed as collective isotopies
two
ribbons
of the recombination
tions
if
an
the Moves TDI and TD2 inrglS2,
if
we
also
use
and TD2*
the fact
that
can
be
we are
place the coupon along the parallel strands or recombination and for TD2* we place the coupon ribbon. For TD14 we put it right at the crossing, outside but right after the 21r-twist. of tangles in the moves TS 14, TS2* and TS3 4 leads exactly The recombination for the Moves TS 1, TS2 and TS3. to the pictures is the it is obvious that the map R-1 that we have, thus, constructed Finally, of classes. inverse to the previous map R on sets tangle free to choose where to
In summary,
we
of
tangle categories,
by
three
between 2-arrow sets following fundamental bijection given by planar pictures with twenty moves and the other surgery data with only three moves:
have the one
dimensional
7-gl
correspondence
This
can
I-
I
I
I
-
be used to find in later
a
few further
computations
'Tgl, following three: Let a coupon C have 9 O-Move (03 -Move): strand internal one passing through it, exactly be very useful
which will
which is of the alent
closed
we
closed
a
tangle diagram. to the
ribbon
Then this
(2.5.1)
with
equivalence relations tangles. Let us discuss
component R
tangle
is
equiv-
where both the coupon and the component R have been removed
Move
diagram.
the coupon with an annulus via the TS 1 -Move we obtain precisely and O-Move in [FR79]. It is easily derived the (93-move in [Ker98a] where the coupon is replaced by a pair of equivalence in the category -r g 1BL' S2
substitute
what is called as an
balls.
picture
in
the
one
internal
from the If
part of
a
BL
7glS2
The
resulting
such
as
is, clearly, configuration TS2*-Move.
in the
contractible
to an
isolated
cancellation
2.5
0
General
have
a
Cancellation:
Here
internal
ribbon
closed
Ali
Calculi
adv
and Fuither
Equivalences
149
we
A,
coupon C exactly once, which may have also other ribbons passing through. As indicated
which enters
and exits
diagram
in the
bon A looks
the
for
except
on
like
a
right,
the rib-
isolated
annulus
the
an
piece running
c
A
through
C. The cancellation
move
is
given by
re-
both the annulus
moving (or adding) A and the coupon C.
shown in [Ker98a] in -r g 1BL is explicitly or an equivalence S2 diagrammatic proof starts by replacing the coupon C by another annulus A* using the TSI.-Move. Now, A surrounds only A* and we can use the TS 1-Move again to replace A by a coupon that is placed in a piece of A* only. The P-Move from above then allows us to remove A* together with the extra coupon. 0 This move Connecting Annulus: The fact
that
[Ker99].
The short
considers bons the
two
is
separate
R, and R2, which
same
there
this
are
internal run
' RIM2
rib-
through
coupon C. We assume that ribbons no other running
!j#R2
through C. equivalent to the one where R, and R2 are replaced by their sum R, #R2. The connecting operation is performed in the plane of the itself is removed from the diagram. C the on right. coupon. See the picture One way is to replace the coupon There are several proofs of this equivalence. all of R2 until it reaches its move one of them along by two balls as in 7-g 1BL' S2 them both the cancel and then generalized using partner ball on the same strand cancellation. Another proof replaces the coupon by an annulus via the TS 1 -Move followed by a P-Move. and then applies a 2-handle slide as in the next section, that As they are, in both proofs we really assume R, and R2 are closed internal the be ribbons. cases where R, and R2 are easily generalized though to They can the TS3-Move, or this end ribbons. To internal one applies top, bottom, or through This
configuration
is
connected
rather
its ribbon
version
described
in Section
2.5.2
below,
at
the ends of the ribbons.
previous arguments for applying backwards to the resulting applied Connecting-Annulus-Move intervals attachment has the that as R, and same boundary so R, #R2 configuration this is that in condition observe before the the to move move. Thus, R2 only global ribbon and different to R, R2 belong components. This turns the
R, and R2 into
closed ribbons.
After
the TS3-Move is
the
2.
2.5.2
Kirby
of Cobordisms
and Presentation
Tangle-Categories
150
and Fenn Rourke Moves
Although the present versions of setting up equivalence classes of tangles will be it is inand construct used to present cobordisms TQFT functors, predominately calculi the that also introduce the versions original generalize directly teresting to of closed 3Rourke of Kirby [Kir78] and Fenn [FR79]. They give presentations classes of links in S'. In this section we shall in terms of equivalence manifolds K which are equivalent 7-f g I BL to the category describe categories -rg 1 S2' and r g 1FR, S2 S2 situations. in for closed 3-manifolds link calculi the and specialize to corresponding and [FR791 pertain only to the special case from [Kir78] the results Specifically, of trivial
with gi
1-affows
we have to add
boundary
do not contain
decorations
uses
non
yield
the
=
moves
equivalence same equivalence
b
=
=
0, when there are no boundaries. Hence, andr g 1PR of,r g lKi Also, they S2 S2
coupons
as
and
moves,
one
Finally,
classes.
without
themselves
a
to the definitions
such
local
3-manifolds
92
`
.
or
and at least
balls,
surgery
[Kir78]
needs to prove that both type of calculi and [FR79] consider only both [Kir78]
framing
the additional
or
signature structure equivalences
that from
our categories. FR of 7 glK ' and Tg 1S2 [Kir78] and [FR79] need to be relaxed in the definitions S2 Ki lKi in admissible of of The notion set The T a A) tangles Category 7-g S2: g 1S2 we
include
This
cobordism
in
means
same as for -r g IBL or T91'S2 except that S2 ribbons. but in the only tangle coupons TS 1, TS2, and TS3 are replaced The equivalences
precisely or
the
In each in 7-g lKi. which generate all equivalences S2 in either -r g IBL that they are already equivalences S2 1) Hopf-Link Move: A Hopf link con-
of
sists
a
Hopf
depicted.
as
link
In this
in which
one
components has 0-framing moved In
fact,
framing that
this
is
2)
2-Handle
an
bon R, and
a
the
following
balls
three moves, to see
7'91'SX2:
0
re-
component has framing
boundary boundary
slides
2
we
closed
strip, that
with
start
distinct
auxiliary
tween them so
the
be
changed by by equivalence in 7-gln'S2 is For Slide (02 -Move):
slide
on
surgery
it is not very difficult
move
can
be
the 2-handle
on
the
allow
of the two
that the other
we can assume can
R2.
by
not
is
added.
or
An
we do
case or
some
linking
with
unknots
two
number 1
that
s,
one
is
end of
a
rib-
ribbon fit
be-
s
ends
over
the other
immediate
either
O-framed
0
or
annulus.
1, since the The fact
from the TS2-Move. Rj#RA
:D: C A.
R .,,#RA
_C RB
R1, and the other R2 Wethen slice
of of
-
R2 down the middle into ribbons R,, and Rb
two
parallel
-
As in the
diagram
boundary
of R, to another
on
the
right we then also cut the strip s from one point on the point on the boundary of RA and then continue this
cut
in both
directions
to
also
Aiw ma fvz
cut
R, and RA
Equivalences
and Further
Calct !
2-3
point.
at this
the connected
As
one component R, #RA, basically the component RB, which is another ribbons, and, in addition, R2-
which is
sum
result,
a
151
we
pushed
obtain
original
of the two
off copy of
proof that the 2-handle slide can be obtained from the defining moves in an The basic idea is to introduce a coupon C with given in [Ker98a]. cancellation the move via the of in A the s annulus general auxiliary strip place from the previous section for 7-gln'S2- In 7-glILS2 the coupon is replaced by a pair of balls, one of which we can drag along R2. The annulus A then becomes stretched into the ribbon RB, and R, is extended along R2 to R, #RA. At the same time R2 to eliminate which is, then, used for an opposite cancellation is shrunk to a strip, details For cancellation. via see balls [Ker98a]. or coupons again any 3) Ribbon-TS3 (a-) Move: For a pair of inIP I! IP I! ternal. through ribbons we can always insert A ribadditional of link an a top configuration The
IBL gS2
is
bon and
picted
an
on
bons
we
right.
de-
as
through
rib-
a
easily
is
with
the coupon
Remark 2.5.4.
replaced
Besides
where
one
annulus of
Instead
bottom type ribbon. recognized as the TS3-Move
have
This
the
internal
additional
the
we
have
via the TS 1 -Move.
this a
of the Ribbon-TS3
version
C, instead
top ribbon,
The move, as depicted, annulus A, but the top ribbon
would
with.
again
of
a
introduce
would be turned
into
slide
T
pair
however, is redundant,
all
Since
gories, a
we
equivalences
three
have
a
representative
since
Of
tangle
define
a
'i
791S2 -
natural
well-defined, from
we can
of
also consider
we can
through
ribbons
top ribbon
T, and
internal
ribbon
closed
R, which yields
over
the
to start a
closed
R. This
original
A and R are, thus,
of ribbons
as
be removed.
equivalences in the other catelKi which takes 7-g S2 __, 7gBL/nx, S2
also
are
map
' and g 1K S2
a
a
top ribbon C and is no longer linked to A. The and can, therefore, in the #-Move configuration,
move,
Move
pair
:
maps it to its class
in
BL/nx
7_91S2
__+ 7-g lKi, on the representative tangles, map IC : r g jnx S2 S2 the 1 -Move. annulus TS in an a using diagram by tangle replaces every coupon in 7'1gIn In order for IC to be well defined we need to check that all equivalences S2
Wealso
can
which
can
be
expressed by
is true
for TSI,
slightly
more
After
resulting across
a
both sides of the TDI -Move
on
simply
this
only
the coupon is found to be
a
2-handle
a
27r-twist
slide
by
an
of the extra
annulus strand
A the
running
A.
be created
we
remark that
by 2-handle-sliding
See below the Fenn-Rourke Move is
obvious.
subtle:
move
over
The fact that moves of 7-g lKi S2. The Moves TD1 and TD2 are
of the three
replacing
For the TD2-Move can
combination
TS2, and TS3 is quite
replaced
under )C
them
Move for
by
a
over
on a
collection
unknot
of
strands
parallel
A' with framing
1
or
-
1.
Moreover, the coupon in the TD2the TD2annulus AO. Hence, in 7-g jKi S2
details.
0-framed
an
152
Move
means
of strands, seen
that
we can
if these
add
so
that
([Ker986]).
Lemma2.5.5
bijection
a
The
proof
in
the inverse
structs
also the
are
coupons with
identifies
A' around a collection A'. But this is easily
annulus
A0.
over
on the equivalence tangle classes.
of each other
correspondence
of
classes
map
7-giKiS2
___+
BL/nx
T91S2
classes
equivalence
unknot
0-framed
of tangles.
from our outline [Ker98a] is different only in so far that it conof on 7-glnxS2. But by Lemmas 2.5.3 and 2.5.2 on ,r g 1BL instead S2
know these two
we
the
on
:
a
inverses
The natural
j
defines
by
A'
by 2-handle-sliding are obviously another bijective we obtain be true
to
Now, IC and J
1 -framed
a
or remove
also surrounded
are
of Cobordisms
and Presentation
Tangle-Categories
2.
same.
The
correspondence
referred
what is often
to as a
from Lemma2.5.5
"dotted
circle"
in
also
Kirby's
lan-
[Kir89].
guage
of Shortly after Kirby introduced his calculus [Kir78] Category -r g 1FR: S2 that smaller of set out moves a [FR79] singled equivalence still generates the same equivalence classes of links and, hence, also gives presentathe general 2-handle slide or 02-Move is retions of three manifolds. Specifically, ribbon over which we slide is a I -framed closed which the in a placed by special one, unknot A'. If we apply this special 2-handle slide to all strands running through the unknot A' the resulting equivalence is what is called the x-Move in [FR79]. in the to the tangle The Fenn-Rourke Calculus category situation generalizes of admissible The Calculus. the 7 as glFR same straightforward tangles Kirby way S2 the the same those for -r g lKi. moves are are precisely given by equivalence S2 Only the following three: Here the Cancellation: 1) Signature and isolated two separated unknots, number one with +1, the framing other with framing number -1, can be cancelled against each other.
B)
The
links
Fenn and Rourke
00, OC)
If
one
with
of the unknots;
a
1-framed
equivalence
is slid
over
component and
in the
Kirby
the other
parallel through
of
are passing framing +1 be separated
or
if at the
applied opposite
strands a
ring
-1. The unknot
from
same
time
the a
to the collection
direction
configuration component,
turns
into
a
Hopf
link
and, hence, this is also
an
H
0
Calculus.
Fenn-Rourke Move (n-Move): 2) Starting point of this move is collection
this
0-framed
a
0
other full
a
that
A' with A' can ribbons
27r-twist
is
of strands
in
of the A'
-framing.
JA!
The
for
case
framing
one
opposite framings has to original case the isolated 3) Ribbon-TS3 (o,-Move):
depicted
is
exactly
is
7-glKiS2For each of the
factors
follows
the
_+
any 2-handle slide can be obtained as the same for tangles in S2 x [-1, literally The natural
([FR79]).
bijection
a
Obviously,
2.6
Compositions now we
rules
sition
However,
only
7-gl
as a
defined
classes
for the we
of
definition
in the
combination
Hence,
--+
7-glS2
proof
The
of K-Moves. obtain
we
is
following.
the
Ki
of tangles.
also in
a
discussed
correspondence
bijective chapter.
natural in this
Category
Double
7-gl already
namely the easy compotangles. which were the same as for the 2-cobordisms. 1-arrows of 7'gl, need to explain the two composition operations for 2-arrows in
have
equivalence
of
and
a
1].
classes
is implies that 7-glFR S2 tangle category we have
every other
sets
equivalence
this
with
Until
the
on
same move as
map
r g IFR S2
1:
defines
with in the
also equivare clearly equivalences in Tgj1R S2 admissible on representing tangles identity Fenn Rourke have proven for links in S' r g jKi. S2
that
Lemma2.5.6
case
unlike
the
that
'rg IFR S2
:
the
that
above all
moves
in 7-g 1 Ki. It S2 into a map I
alences
as
also
unknot is not discarded.
1 -framed
This
right. In addition, an equivalence. Note,
153
Category
Double
as a
the
on
be introduced
Igi
and
Compositions
2.6
still
of
the 2-arrows
of
Wehave
but sets,
nothing
as
described
7-gl. In this
section
compositions classes
so
of
7'gl.
set
category
We start
defined
with
for vertical
they tangles. binary operations on compatible these operations we prove that satisfy and Appendix B. 1. in the introduction Weverify
that
as
factor
well
into
elements
two
as
the
horizontal
equivalence
of the 2-arrow
the axioms of
a
double
Compositions two
tangles
1-arrow
the T,, and T, that are admissible for 7'gl. Furthermore, the horizontal 1-arrow be the should same as source T.
of
T1.
Specifically, of Section 1-arrows
[gtg, a/b] The
[%, a/b] this
both the rules
introduce
shall
we have
Finally,
as
horizontal
target of
that
Vertical
2.6.1
we
of admissible
as a
let
2.2. 1, of a. and P.. and vertical
composite and vertical
composition
the square, 1 -arrows
that
us
assume
T.,,
has horizontal
Correspondingly, 1-arrows
tangle
T,
1-arrows of squares
T,,
al
of the
sense
[g, a/b]
and
let T, have horizontal
a, and o,
in the
o
similar
diagram (2.2.1)
[g", a/b] 1 -arrows
and vertical
[git,
a/b]
and
01. has then
a,,,
and as
01
horizontal o
1-arrows
#.. Schematically,
for cobordisms
as
follows:
[g, a/b] we can
and write
154
of Cobordisms
and Presentation
Tangle-Categories
2.
[9-,a/b]
a
b
a
t'l -T,
al Oau
a
T.
0"
Ou
'3,0'au
For two succession
of T1,
as
Hence,
we can
final
gle
is
at the are
then
intervals,
exactly 2g*t
the
we
that
notice
the
Tu and at the topThat is, from left to
of
same.
internal
(i.e.,
intervals
top of T, such that external
on
exactly bounding at these intervals come together shown in the diagram in Fig. 2.2. match
lines
bottom-line
g"
pairs
intervals.
external
place T.,,
of the two
strands
2. 1. 1,
in Section
external
initial
right we have a of such), and then b
intervals
and external
described
b
[gt,,alb]
Tu and Tj with these numbers
tangles
representing of internal
1,81
t'111T,
al
b
[gtg,alb]
(2.6.1)
b
[gi,,t,a/b]
a
a
line
b
-
in succession.
that
are,
thus,
intervals
and internal
and external
The internal
connected.
The
resulting
tan-
To V T
U
Fig.
In addition
auxiliary K1,
...,
to
strands.
KA,,,
The
ones
composition
The vertical
strands
and external
the internal
(see Section
2.2.
from
2. 1.
1)
as
Tu
are
already
connected in
we
have to find
at the
top-line
Tu. The A, auxiliary
a
rule
for
the
at the intervals
strands
from
T1,
Compositions
2.6
however, intervals
will
KA.
Now, this ones.
in
141
and
in parallel over T. and the at top-line. KA. +A, +1, of how to create is only a prescription
be extended
as a
Double
connected
155
Category
in their
order
to the
of two
given
...'
tangle out composition
a new
need to make sure that it does in fact define
Westill
a
of elements
Tgl.
Lemma2.6.1.
(TI, T,,)
The composition
-+
Tj
o,
T. oftangles,
7-gl. tanglesfor into the equivalence it factors Furthermore, hence, defines a composition on 7'gl. closes
within
as
described
above,
the admissible
classes
of 7-gl,
of tangles
and,
Proof First, let us prove that the resulting tangle is admissible again. The fact that of external tangles leads to a combined permutation of the a initial the composition I-arrow the final for the similar and composition. permutations to al o a,, is clear, strands of that arise when the g'W pairs of internal combinations The possible line be intermediate the other can each at connected to each tangle are easily identiand is combined with a pair that befied: if a pair in T. belongs to a bottom ribbon, longs to a top-ribbon in T, then their combination in Tj o, T,, clearly yields a closed from T." combines with a through-pair ribbon. Furthermore, internal a bottom-pair in T, in T. and a top-pair for from T, into a bottom-pair Tj o, T., a through-pair This another connected to and are two through-pairs through-pair. give a top-pair, attached to ribbons no auxand are the that new top through implies, in particular, have ribbons and bottom one closed the and new auxiliary exactly iliary ribbons ribbon
each.
The counts
(2.2.2) internal
of the ribbon
from Section ribbons
of
2.2.3.
types For
T., by C,
can
by using by C,' the
be summarized
example,
we
denote
the number of closed
ones
in
T,',
number in T, o, T, If N is the number of newly created closed identities: the following of the composition we obtain a result
the notation
as
in
number of closed and
by C the total
internal
ribbons
as
T--T,+T,-N, B=B,,+BI-N, C=C,,+Cl+N, A=A,+Al. H=H,+Hj+N-gjt, ribbons, follow from the other The last two, for through ribbons and auxiliary ribbons number A of auxiliary the that fact the In of ones by virtue (2.2.3). particular, ribbons implies that we do not the sum of the already present auxiliary is precisely have to add any more ribbons of this type. The tangle as given in the above picture, is already admissible. therefore, is well defined on the equivalence It remains to prove that the composition That is, we need to show that if T1' classes in 7'gl. T, via the moves then also of T, this is T, o, T, For all moves that can be localized in the interior T,' o, T,, move. obvious because for the composite we can use simply the same equivalence Thus, we only need to consider the TD3, TD4, TD5, TS3, and TS4-Moves. can be easily or twists. For the TD-Moves the crossings pushed up (or down) of ribbons of the extensions the Ti. auxiliary parallel along If we apply the TS3-Move to T1, the result in T, o, T. will be that the strands of to each other by an arc. are connected the internal pair of TI, ending at the top-line, -
-
156
and Presentation
Tangle-Categories
2.
of Cobordisms
ending at the bottom line. Moreover, is, however, easily recconfiguration ognized as a case of the Connecting Annulus Move at the end of Section 2.5. 1. Applying this we recover the picture where the pairs of intemal ribbons are connected of which is the original the application to each other directly, one for T, o, T,, before the Connecting Annulus Move only applies when the the TS3-Move to T1 A priori, if involved ribbons are closed intemal ribbons. This, however, is easily generalized the move with TS3-Moves at the top and bottom of the diagram for we conjugate The
the
same
happens for through
pass
arcs
the
pair
of T.
of strands
This
a commoncoupon.
-
T,
T.-
-"
of a is to tum an overcrossing on T1 in T1 o, T,, parallel strands in the middle section into an undercrossing. moved to either the upper or lower boundary line of a strand can be easily T., where we can apply the TS4-Move to this composite tangle.
of the TS4-Move
The effect
strand
over
Such
T,
o,
all
of the
composition
vertical
For
the
that
for
find
identity tangle T
any horizontal
it
we can
such id[g,alb], [g,,, a/b] -+ [gtg, a/b] o, T. id[.qv,a/b]
:
id[g_,alb]
clear
is
[g, a/b]
tangle
an
any * o,
rules
I-arrow
that we
id
[g, a1b]
for
have
=
gint
tangle is simply given by paralrepresenting external and through strands lel, straight vertical for every interval on the right. as depicted *
.
.............
b
a
7'gl forms an ordinary category, composition Already with only the vertical 1 -arrows and whose morphisms are the 2-arrow sets objects are the horizontal of a quotient given by the equivalence classes of tangles. This category is, naturally, a special subcategory of the naYve category of isotopy classes of ribbons tangles in of three space (see for example [FY92], a slice [JS9 1 ], and [RT90]). Furthermore, 7'gl thought of as an ordinary category under the vertical composition 7-gl (a, b) for which the number a of initial o, decomposes into subcategories whose
external the
and the number b of final
strands
decomposition
in the
following
(7-gi, 0')
=
external
strands
are
fixed.
Wesummarize
identity:
U
(7-gl (a, b), o,).
(2.6.2)
(a, b) ENOx No
2.6.2
Horizontal
Compositions
composition proceeds in a similar way as for the rule for admissible a composition one. First, tangles and then classes. the into equivalence prove that it factors beThe composition rule for the admissible tangles will, however, go slightly the small that other. 1 Due the each next to sets to -arrow tangles yond simply putting 1.2 which already in the situation of cobordisms in Section we have chosen here, the cobordism also define the in form of natural transformation the we a. required horizontal composition of tangles in two steps. For two given admissible tangles T, The definition
vertical
of the horizontal
we introduce
and
Compositions
2.6
7'91
as a
1 -arrow, 0 E Sb, of T1 coincides and T, such that the target vertical This tangle 1-arrow of T, we first define a tangle T,,' IhTl. in the
In order
in that
2.2.3
of Section
Y and
braids,
special
of the intervals,
the order
correct
to
added at the top and bottom line in the second step of the construction. of the braids from (1. 1.7), where chosen to be exactly the presentations For
a
mapping
of the
elements
as
of
intervals
the internal
source
not
some
the top and bottom line are not next to each other. The order of the of the standard one proposed in Section 2. 1. 1. is a permutation
intervals
internal
sense
the
with
is, however,
vertical
quite admissible of the pairs at
157
Category
Double
larger
class
group of the sewn surface. admissible tangles
class
are are
they
appear
of the
the class
and, thus,
of 1-arrows
Y-1,
These
com-
and the spealso well defined as classes. The two steps of the construction cial braids are clearly of the horizontal composition can therefore be summarized in the formula:
TAhT1 only depends
posite
TrOhT1
Schematically, diagram of 1-arrows
(2.6.3)
in the following composition is illustrated composite TAhT, may be thought of as the diagram:
of the
The
and 2-arrows. of the
of the factors,
classes
YOv(TrAhT1)OvY_1-
-:::
the structure
two boxes in the middle
equivalence
the
on
right
C
a
91
[91,-+9r,-+b-1,a/c1
a
C
b
a
TI OhT1
I'
11T
a
tl-IT'
13
1
(2.6.4)
b a
[gl,
+g,l
t,
tg
+b- 1,
Y
C
a/c]
a
simplicity, dealt
let
with
in
The first that
the
omit
us
similar
a
TAhT1 tangles auxiliary
the definition
with
Westart
way
step is
to
as
juxtapose
and internal
the external
of
in the
case
of
the two
ribbons
SUC
[g1,tg+g,,V+b-1,a1c]
of
representing in our discussion. They the vertical composition. tangles into one diagram.
on
the level
T, and T,
tangles.
of
will
be
the
on
can
This same
For
easily
be
means
top line
Let us source T1 by bottom line respectively. b. intervals external 1P with j 1, a and the target by 'Jkt with k 1, with b and k with of intervals the external 1, 'Jkt T, are Uk' Similarly, of T1 at the top line are given intervals k c. In the same way, the internal 1, if by 1I1',1I1',1I2i, 1Ig','_,1Ig1,' _, and those of T, are 'I, If, .Ig",
denote
or
the
external
=
=
.
.
.
,
=
=
.
.
.
of
intervals
.
,
.
.
.
.
.
.
.
,
.
.
.
,
158
2.
Thus,
and Presentation
Tangle-Categories obtain
we
the
following
of Cobordisms
sequence of intervals
at the
top line
of the
posed tangles 1 l 0.
braiding
=
(XOY)OZ c
(VX)
-+
between them
category c
(Y'(O Z) a
(onefor
X
n
of the
in detail
junctorial
a
Y)VV),
&
C by
the category
denote the iterated
vX (n times)
Y and ' -
that the functors
V(XV),
-+
the definitions
4.1.4.
Definition
(X
(4.1.5)
categories
review
are
Pt,
'+
-L * 1VV),
1V
this
morphisms. Several basic relations of examples of braided categories asitriangular Hopf algebras and relations
XVV, f
-+
XT2-+ 4
0
by replacing isomorphisms
Finally, V
=
X
of C.
X
become
(YV
(1 -4
we assume
achieve
the canonical
such that
=
self-equivalence
monoidal
a
(C, 0, 1),
-+
YVV 24
d2
2-categories
X
hold
10
[Sch92]
X),
X0
1).
Y
[Lyu95a]
In a rigid braided category we can define ing again the conventions from Figure 4. 1:
P
2 U
C/
-2 U
X
'U
O
-2
WX
WX
XVV
XVV
us-
X
\,\'D
U2
223
isomorphisms
functorial
X
X
categories
Ribbon monoidal
4.1
(4.1.6) meaning of these isomorphisms in the case explained in Sect. 7.4.4. of monoidal functors There are isomorphisms The
(Id, C-2, I[,) (Id, C2, 1[1)
2 U 1 2 U
In
particular,
(see (4.1.5)) 7
)
XOY
tul
2
VV
The square of the monoidal
(-VVVV, h (14) where
d4
an
(C
Nowwe define
balancing
an
4.1.5.
isomorphism
U 0 and called
a
Xvvv
also
a
X
X,
v
:
Id
such that
use
ribbon
canonical
exists functor
1),
(g,
4 u : 0
functors
some
Y)VV
X
(XVV
(0
(4.1.7)
2
-+
XVVVV ,
At
YVV)VV 2
f
,
(X
4
ftttt,
-+
0 2
Y)VVVV)' U21 is, in
isomorphism.
uO
(Id, 1, 1)
(-VVVV h, d4).
-+
of balanced
properties
=
U-1
0
,
(ribbon)
categories.
balancing flx : X -+ Xvv (Id, 1, 1) -+ (-VV) j2, d2), such that #2 Xv. The category C equipped with a balancing braided
category.
A
is
is
balanced.
Wesometimes
-+
(C'
and recall
rigid ofmonoidalfiinctors V
is
1VVVV). The natural
Let C be
#1X
PV, j2, d2) )
of monoidal
categories
(0
YVVVV __L4
1VV __. 2_+
isomorphism.
Definition
1)
0,
dtt
Ribbon
4.1.3
1
YVV -24 (X
functor
(XVVVV
:--
(1 -
=
fact,
j4X,Y
:
diagram
of the
UJOU1
X
7
7
X(&Y_ 2
is
(-VV) j2 d2)j (-VV j2 (12)
)
commutativity
the
implies
this
Hopf algebras
quasitriangular
of
-+ c
2
Id is =
the notation
twist a
v.
2 0
A ribbon
self-adjoint
(V-1X Ovy-1)
u
In any balanced
twist
(vxv
ovx(gy.
=
It
category RT90, Shu94] v
[JS91, v1X ) automorphism.
can
be determined
=
vX
identity equations
of the
from the
there
Monoidal
4.
224
2 U 0
0-1 In
and monoidal
categories
U21 OV-1
=
-2 =U 0
=
U-2 1
U2 1
=
0
2-categories
V-1
0 V:
=
X
U-2 -1
0
XVV'
--+
V:
WX.
X -+
UJ-2
its square is given by the canonical 0 U2. Vice isomorphism v2 = 1 in any rigid braided category with a ribbon twist (called ribbon category) 2 exists a canonical balancing u 0 given by the above formulae. Thus, ribbon
particular,
versa,
there
and balanced
categories In the
The
following
Definition
any
(c) for
any
1
:
have that
are
used to
we
have vX
strictly
rigid
book.
if
object
X
we
have evX
Xv, Xvv
=
=
X, and OX
Ix
=
:
X
For any ribbon
we can
of all,
Set Ob D
=
D((X1
ObC
1
1
morphisms C
I
:
(X, 0)
-+
X, (X,
given by A(X,O)
I-
replace C by
=
C(X,
=
C (XV,
and the
functor
C there
category
an
10, 11. Morphisms
x
0) (Y' 0)) NX, 1) (Y' 0))
=
:
X 0 Xv
-+
1, and
coevx
exists
a
ribbon
strictly
rigid
cate-
equivalent
category
satisfying
(a),
so we
for C.
(a) holds
that
ev' X v
C.
Proof.
First
=
Xv 0 X.
-+
to
D(L, M)
I[,.
=
1;
.
X
Theorem 4.1.7.
Unit
1v
object
gory D equivalent
assume
-+
=
D is called
category 1
:
d2 : I -+ I" and v, 01 simplify notations through this
X.
=_
coevxv
11
=
synonyms.
are
we
A ribbon
1, d,
=
categories I
=
results
4.1.6.
(a) I' (b) for XVV
of X
case
as
D((Xl 0) (Y' 1)) NX, 1), (Y' 1))
Y), Y),
composition
of D are defined
1
in
D
inherited
are
=
C(X,
=
C (XV,
from
YV), YV).
C. The inclusion
functor D, X -+ (X, 0) and the projection g : D -+ C, A : _1 o 9 -+ Id-D to each other, 1) -+ Xv are quasi-inverse Ix, A(x,j) 11xv. Therefore, D is equivalent to C. Notice that )
=
=
C(9L, 9M).
The tensor
product 0
in D is defined 1
:
D
X
D
-L
as
C
X
C
--L4
C
D.
is chosen as aL,M,N associativity a9L,9M,9N for L, M, N object is (1, 0), and the corresponding isomorphisms are rm A` o Igm. The functors IM
The
=
unit
E Ob D. The
=
A-1
o
rgm,
=
(-E, 11, 11) (g, 11, 1) are
monoidal
equivalences
: :
(C' 0' 1) (D' 0" (1, 0))
(D'O"(110)), ) (C' 0, 1)
between C and D. The
braiding
C9M,9N for M, N E Ob D. The functors 1, g preserve Wechoose the following rigid structure of D:
in D is chosen
braided
structures.
as
CM,N
V
(x, 0)
:(X, 0)
ev
coev
ev
(X, 1)
:
M1)
=
Ax' 0),
(X, 1)
0
(Xv
=--
0
V
M1) (X
_
(1, 0) coey) (XV
:
(X, 0)
0
=
0
0
M0)
=
(X, 1)
(XV
X, 0)
V(X, 1),
ev, (1, 0),
XV, 0)
X, 0)
=
225
categories
Ribbon monoidal
4.1
0
(X, 0), ev
XVV' 0)
(0
(1,0),
)
(4.1.8) coev
(1, 0) coey)(XVV
:
xv, 0)
(D
(X
xv, 0)
0
(X, 0)
=::
0
(X, 1). (4.1.9)
Calculating
u21
u21
2 U -1
obtain
=V,
M-+ M, M E Ob D with =V
-1,
where
v
I (X'O)
=
vx
9
I (X,I)
v
=
vx
we
Indeed,
v.
xv
xv
(X, 1)
(X, 1)
morphisms
duality
the above
x
2 U
\--,
\_1
and, similarly, the
xVV
in other
xv
A'v
(X, 1)
(X, J)VV By
T V
U2
cases.
general theory 2=
U 1
4.1
from Section v
(Id-D, C-2, I[,)
:
j2, d2)
(Idl),
isornorphism of monoidal functors. Hence, diagram (4.1.7) and the properties =V-1 OV 11 in D. Also u40 11 and d2 twist yield i2 U21OU2 I D in to reduces an of definition isomorphism the balancing Therefore, general Mv -+ Mv, and 1 : that such 0-1 01 -4 p2 (Id, 1, 1) (Id, 1, 1) MV M these requirements. 1 satisfies Hence, D is a ribbon M E Ob D. The choice 0
is
an
of ribbon
=
=
=
-
=
=
=
category
and
compatible Finally, (b) and (c).
Xv
equation isomorphic
the
the left
v
to 2
uO ones.
coevaluation
replace
:
vX is
=
X. The
vX
-+
In that
D with
from
not
same
C
another
(4.1.8),
:
C C) D and!9
:
D
C
are
well
as
-+
=
category
where (a) is satisfied
as
H-mod, where H is a ribbon Hopf algebra, Xv is canonically satisfied. Nevertheless,
necessarily
holds in any ribbon
Xv. This allows r6le
I
The functors
structure.
In the category
Remark 4.1.8.
via
the ribbon
we can
twist.
ribbon
is its
v
with
right (4.1.9),
the
6v:xv(gx-
duals called
xvop
us to
the
right
equipped with flipped evaluation are
xv
0
xVV
these
Weidentify
category.
use
ev
dual
objects
the left
in
evaluation
and coevaluation:
1,
objects place of and
226
4.
Monoidal
6oei v
Often they will replaced by 6-v the action in Sect.
of
a
Recall
coey
1
:
be denoted and Coev in
group-like
16-10XV
XVV 0 XV
simply ev applications.
and
X 0 XV.
in the theoretical
coev
In the
part and should be
Hopf algebra context#
by
Drinfeld
[Dri90].
is
given by
This is discussed
7.4.4.
that
gory by which has
in braided
the category a
natural
Hopf algebras monoidal
in
of one-holed
it has
Hopf algebra study. Werecall
a
structure.
our
and introduce
category, shall
categories
of cobordisms
Lemma 1.3. 1. Moreover,
Later
2-categories
element introduced
Hopf algebras
4.2
and monoidal
categories
first
surfaces
distinguished This implies
the
the notion
is
a
balanced
cate-
torus a 1-hole object the, importance of braided notion of a Hopf algebra in a braided of integrals for such Hopf algebras. -
-
that the
integrals are related to surgery. The relationship between the inproperties of integrals. and the Radford tegrals antipode (the formulas) are recalled in the braided setting. Then we prove several lemmas, which give a practical recipe how to find the objects of integrals. we
see
Weformulate
4.2.1
Algebra
standard
in
a
Let C be a braided
monoidal
monoidal
category
(H
(9 H -Ln+ H
(HOH AOA
-'-A+
Recall
category.
object H E Ob C together with and an associative comultiplication.A an
that
H E C [Maj93] is a Hopf algebra m : H (9 H -+ H multiplication H 0 H, obeying the bialgebra axiom
associative
an
:
H -+
H)
H0
HocoH
m2m, HOHOHOH
HOH(&HOH
HOH). (4.2.1)
Moreover,
H has
and the inverse
a
unit,
antipode
in the classical
77
:
1
-y-1
-+ :
H, H
a
counit,
6 :
H. The
-*
H -+ 1,
defining
an
antipode,
relations
for
-y
:
these
H -+ H are
the
that the unit is also a morphism. particular, of multiplication, as well as coassociativity of comultiplication, is forAssociativity mulated with the use of associativity isomorphism (in the non-strict case). We use the graphical language of [BKLT001 to represent the operations of a Hopf algebra as listed above. The elementary pictures are summed up in Fig. 4.2. To distinguish such tangles-operations from the previously defined one we draw same as
them with For
case.
Notice
in
fat lines.
example,
the
bialgebra
axiom
(4.2. 1)
can
be drawn
as an
equation
H
H
H m
Proposition in
4.2.1.
braided
a
an
antipode
category,
I
H (9 H
)
c
-y
Hopf algebra operations
"n
)
following
statement.
of a Hopf algebra H of the algebra H: 72
(HOH
H)
H
---!-+
H)
H0
H).
H:
H0
H0 H
)
antipode
e
7 is an anti-automorphism that is, an anti-automorphism
of the coalgebra
227
H counit
i?
prove the
one can
HOH-M HOH
c
anti-automorphism 'a
(H
The
unit
of
notation
language
monoidal
(HOH and
graphical
the
A
categories
H
H
H
comultiplication
Fig. 4.2. Graphical
Using
H
H
H
multiplication
in braided
Hopf aigelbras
4.2
(H --!--+
H)
H
Let A, B be Hopf algebras in C with antipodes 4.2.2. Proposition -yA, -1B. A --+ B be a bialgebra 0 : homomorphism (a morphism preserving multiplication, Then it commutes with the antipode: unit and counit). comultiplication,
(A -14
f
*
to
'A
(A
(A
Dual
With each
its
in C. In
B).
The via
is also
o
BOB
The element an
inverse
is
+
a
Hopf algebra (H, Hopf algebras
m
monoid with
associative
an
B).
The unit
B of Mhas
Weconclude
that
product
convolution
0
of this an
-yA
o
m, 77,
A,6, -y)
in
a
rigid
(Hv, At, et, mt,,qt,
braided
-yt)
and
-yB
re-
is
monoid is
inverse
=
category
o
-YA.
o
:
relationship Hopf pairings. a
between
Hopfpairing
(HOHOA
w :
if it satisfies
are
asso-
Specifically,
the
and the dual
an,operation
A pairing
C
(1H, tA, te, tm, tn, t-y)
category the two dual algebras are isomorphic. vH -+ Hv is an isomorphism. of Hopf algebras.
ribbon
u20
A is called
-2E4 B).
B their
A
:
of
B
Hopf algebras
dual
balancing late
=
A0 A
1
However, -1B
ciated
0
(A
,
g
4.2.2
=
C(A, B) morphisms M the convolution product. For f g : A
The set of
Proof. spect
-0-+ B)
A
Let
H (9 A
the
operation is easy to formuHopf algebras H and equations:
1 between two
following
HoHoAoA =
-+
H0 A
(HOHOA-M
"'
)
1)
HOA
1),
Monoidal
4.
228
and monoidal
categories
(HOAOA
2-categories
HoHoAoA
HOA
(A====1OA2%HOA "J)1)=(A
6)1),
(H=--HO1-L%HOA w)l)=(H
")1).
77
ev :
Remark 4.2.3.
1)
-10 4HOA
(HoAoA
For instance,
'0)
H 0 H'
For any
1 and
-+
Hopf pairing
ev
w :
:
'H 0 H -+ 1 H0 A
I of
-+
are
w)1),
Hopf pairings.
Hopf algebras
H and A
we
have
(HOAIMHOA w)1)=(HOA!2!4HOA Proof A Hopf pairing w : H 0 A --* I induces to a bialgebra !w : A -+ H1. This homomorphism commutes with the antipodes: 'YHV
Hence, the required
property
0
of
!W
=
'YHt O!W
=
!W
w)1). homomorphism
O'YA-
w.
if !W : A -+ HI is an Hopf pairing w : H 0 A -+ 1 side-invertible, is A a Hopf algebra isomorphic to its dual. It self-dual Hopf algebra isomorphism. H H is equipped with a side-invertible -+ 1. (D Hopf pairing w : of the bialgebra. axiom (4.2. 1) in a braided illustration There exists a topological Kauffman, Saito and Sullivan [KSS971 defined a category of templates, category. which are a special kind of 2-dimensional stratified pseudomanifolds embedded into of certain 3-dimensional 3-manifold. used models as a dynamical systems. They are Kauffman, Saito and Sullivan [KSS97] show that the category of templates is a free ribbon category, generated by a self-dual object B, equipped with a bialgebra struccoture an associative multiplication (possibly without unit) and a coassociative without the bialgebra, axiom (4.2. 1), and counit), satisfying (possibly multiplication that the isomorphism. it is required a bialgebra anti-automorphism -y. Furthermore, Wecall
a
-
Bv Bvv, objects a : B -- 4 B' induces bialgebra isomorphism. B ribbon twist. the and, finally, -y2is Clearly, a self-dual Hopf algebra H in a ribbon category C, for which the square with the above listed of the antipode is the ribbon twist, is an example of a bialgebra from the of functor it a Hence, templates to C. Such Hopf gives properties. category will be constructed 5.2. in Sect. algebras of
4.2.3
Integrals
Integrals Sweedler
for in
for finite
[LS69].
Hopf algebras dimensional The infinite
Hopf k-algebras dimensional
were case
introduced
was
treated
by Larson and by Sweedler in
in braided
Hopf algebras
4.2
categories
229
for Hopf algebras in braided abelian categories were studied [Swe69b]. Integrals define standard this In section formulate their we [Lyu95b]. integrals, properties, between the integrals and recall the relationship and the antipode (the Radford formulas) in the braided setting following [BKLTOO]. Then we prove several lemmas, which give a practical how find the object of integrals. to recipe An idempoKaroubi studied in [Kar7 I] categories with the following property. e2 : X --+ X, in a category, D, is said to be split if there exists an object, tent, e Xe, and morphisms, ie : Xe -* X and p, : X -+ X, such that e ie 0 Pe and If D 0 in is that is then we a category ie P idempotent IX, split Pe every say with split idempotents. in
=
=
=
-
C there exists an embedding, C -4 e, such that idempotents a given category category eare split. Moreover, C can be chosen to be universal in the sense that for any category D with split idempotents F : C -+ D factors in the every functor For
in the
(C -4 e -24 D) The category e is
isomorphism of called the Karoubi enveloping category of C. According functors. it may be realized with objects Xe to Karoubi as the category [Kar7l] (X, e), where X is an object in C and e : X -+ X is an idempotent in C. The morphisms in C are defined by C(Xe, Yf ) It E C(X, Y) I fte t}. The functor i defined f is a full embedding, that is, we have C(X, Y) by i (X) Xid.,, and i (f ) form F
G is unique up to
where the functor
=
an
=
=
=
=
=
=
CWX)1i(Y))If C is a
(braided)
a
(braided)
monoidal
monoidal
(1, 11), In this
then
case
is
so
i is
C,
Weshall
Xe a
0
Yf
(X
=
monoidal
(braided)
and the dual define
then the category
category,
e can
be
equipped
(f
0
e)
with
structure:
objects
integrals
for
of
&
Y)eof,
Furthermore,
functor.
(X, e)
cx,,yf
are
Hopf algebras
(Xv, et) as
o
C is
if the category
rigid,
(vX,'e).
and
the output
cx,y.
following
of the
propo-
sition.
4.2.4 ([BKLTOO]). Assume that H is a Hopfalgebra Proposition monoidal category C with split idempotents. S in a braided antipode an invertible object Int H ofC and the following morphisms H H
Y f fH1 Hf ,
Int
H -+ H called
H -4 Int H called
left (resp. left (resp.
right) right)
with
an
invertible
7hen there
integral-element integral-functional
in
exist
H, on
H,
such that
H (9 Int ,,Olnt
Ht Int H
H
HO
fH
H
H
H(&H
tm +
H
IntHOHInt
Hoe
t
Int H
f(&H
-- HOH
Hf
tM H
(4.2.2)
230
4.
Monoidal
categories
fH
H
Al
and monoidal
Int H
-+
HOH
Hf
H
tip&Int
H(&fH.
2-categories
At
H
HOIntH
4
IntH
tInt
Hf(&H
HOH
(4.2.3)
Hon
IntH(&H
commutative.
are
Furthermore,
(H -4
any
morphism f
H (& H -1-4
Of
H -+ X of C such that
:
X)
H0
(H -4X
=
1 (9 X
-
01
-27-+
X),
H0
resp.
(H-' '+HOH-f admits
-'+XOH)=(H
unique factorization
a
Int H -Y+
f
0
of
X). Any morphism f
form H
the
X
:
+X!_-XO1-MXOH), fH
2+
Int H
)
X
(resp.
H
Hf
H of C such that
-+
(H
0 X
Of -L4
H 0 H -m-+
H)
=
(H
0 X
(X
0 H
-fo-'+
H 0 H -L4+
H)
=
(X
0 H
01
1 &X
c-_
X
-4 H),
X0 1
c--
X
-+
H
(resp.
resp.
-10-4
f
H),
H
admits
unique factorization
a
-14
of
the
X
form
4
Int H
X
H
Int H
This
H).
proposition-definition
unique isomorphism. phism of Int H. In the
integrals
case
of
an
fixes
The
integrals
abelian
rigid
the
object
of integrals
fH1 Hf IfH, Hf monoidal
are
Int H uniquely
unique
C we can define
category
via exact sequences
Int H
0
fH
Hv
H,
0
H
0
where the
(co)actions
Int "'-'
H '
f
OH
)
TT
n
vH
0 H
H (9
Hv
H
of dual
Hopf algebras
H
iP,-He)j
-
,
L
H0
'H,
Int H
)
Hf, ,
on
up to
Int H H are
0, )
0,
an
up to
a
autornor-
Int H and the
10 H
HvoHoH
(H
H0 1
HOHOVH
at
=
(vHOH
a,
=
(HoHv
H
L f 0
Hv
invertible. 1 and
-+
H0
H),
V),
H),
10 H
HOHOHv
0
Ho 1
==
H).
Y to left or (see Theorem 3.3 [BKLTOO]). Applying the functor in (on) H'. Thefour composite H we get right or left integrals on (in)
right integrals morphisms
all
'HOHOH
231
categories Hv
(H
Theorem 4.2.5
are
in braided
hopf algebras
4.2
,L
Hence,
thefour
YI Hf0f
the
natural
H
H
H 0
Hfo f
,
four natural
-+
Int H
IntH
Int H 0 Int
pairings 1
copairings
:
Hv
0 H
f of
+
Int
H'
H0
Hv (D Int H are
isomorphisms. In
Int (Hv) (Int H)'. particular, between integrals relationship
by the identities antipode is clarified Radford of those [Rad76]. Fig. 4.3. These formulae from [BKLTOO] generalize lemma. via the following straightforward The reader can prove these identities The
in
Lemma4.2.6.
The maps
b, given below
p:
are
Hom(H (9 M,
inverse
to
fact,
only
one
H0
N)
-+
Hom(H 0 M,
H0
N),
each other
b(f)
In
and the
PW
f
formula
=
from
Figure
'Y
f
4.3 is proven in [BKLTOO] obtained from it by replacing
(using
the
Hopf are lemma), and the remaining HOP or opposite comultiH with the one, having opposite multiplication (a), (c) at Figure 4.3 are transformed into HOP. Notice that the identities It preserves the plane of drawing and 7r-rotation. of space (b), (d) under the action miffor its into takes fixes a vertical a picture image, where the left and right axis, is not changed. over/under of but the sign crossing are exchanged,
above
algebra plication
three
the
232
Monoidal
4.
categories
Int H
H
2-categories
and monoidal
H Int H
Int H
I
fH
fH
H Int
Hf T
Hf
fH
Int H
H
H
Hf
Int H
(b)
H
H
Int H
fH
Int H H
fH
H
H
11
f
fH
Int H
11
11
Int
H
H
70
f
Hf
07
fH
0
=
'Y
CInt H,Int
The
H
antipode
UH
another
The composition
fH
0
fH) W D fH)(f fH)_1 H
0
=
CIntHJntH(H f
0
H
0Hf)(Hf H(fH
f
0
H 0
Y)
-1
0
0
fH
Hf
H 0 Int H -+ Int constant
:
H
0
braiding Chit H,Int H : Int of Aute 1. The proportionality
IntH
IntH H -+
H -+ H 0 Int
Definition is
:
A0 B
4.2.8. -+
invertible,
or,
A 0 B is called
invertible,
or,
Let K be
K is called
an
lp
side-invertible,
equivalently,!
q
:
:
(4.2.4)
H,
(4.2.5)
IntH,
(4.2.6)
-+
IntH,
(4.2.7)
H is viewed
as an
4.3
(a) with
Hf
(0
object and let A, B E ObC. A pairing if the induced morphism p! : A -+ K 0 vB B -+ Av 0 K is invertible. A copairing q : K if the induced morphism. q! : K (9 B' --+ A is
invertible
side-invertible
equivalently,
H,
-4
is invertible.
at Figure Proof. The composition of the both sides of the identity are obtained IlInt H gives (4.2.4). The other identities similarly.
p
of the antipode
integral:
Y) UH f H)_1.
where the element
H
integrals
via
H 0
CIntHJntH(Hf CInt H,Int
expressed
4.10 [BKLTOO]). to
=
=
Int
H
(d)
4.2.7 (Proposition Proposition with an integral is proportional
Y
H
fH
Fig. 4.3.
0
H
Hf
(C)
-Y
Int
Hf
f
Int
Int H H
H
(a) H Int
Int H H
Hf
T
fH
H
'A (9 K
-+
B is invertible.
In order to prove that
Remark 4.2.9.
phism
use
or
Let p
:
copairings
the
A0 B q,
:
K be
morphism
qj
A
B
D
K
A
K2
'_(Pf
(Ki
=
isomorphisms.
ID
abelian
If
length(B),
f
then
category
A
:
f
is
an
is invertible, a monomor-
isomorphism. there
Assume that
where K is invertible.
c
0 B
-M
D0 A0 B
-M
D0
K)
(4.2.8)
=
(4..2.9)
Then p is side-invertible.
morphism
transposed morphism the
to
(4.2.8)
is
hence,
invertible,
the
partially
K, qj
D
A
(vD
B
(&
K,
1M vD 0
D (D A
P
vB
K
A is invertible.
Applying
0
-
vK2
to
(4.2.9)
10coe
we
get
-4 A 0 B 0 vB an
invertible
-M
K0
m9rphism
as
vB)
well:
A
4B;
P
are
K1, K2 such that
B O'C for invertible
K2
B is
-+
C
K
transposed
an
(AOK2 !M A(&BOC-L'-4KOC)
q2
B
P
Indeed,
=
pairing, :
of
observation.
B
D
are
a
D 0 A, q2
K,
K,
A
a
elementary
following epimorphism and length(A)
often
we
233
categories
in braided
hopf algebras
4.2
LB vB
]
K, 2
r
q2
=
C
K K (2) 1010001
4
vB
(A
locoe
-4 A
B
vB
XK2 1(Di(gcoe
14 K (D vB (9 K2 (9 vK2 10ev 0101
KOvB(&BOCOvK2
4
K0 C0
vK2).
234
Monoidal
4.
and monoidal
categories
2-categories
Thus,
(A iocoev4
pl has left
and
Dually, there
right
K
:
pairings
are
invertible
and is, therefore,
inverses
let q
-*
pi
:
A 0 B be
D0 A
'B)
where K is invertible.
B&
:
Assume that
CK2, such that K, and K2
are
and
D
K q
(D
Pi
K,
B
C
K q
A
(K
C
B P
A are
P2
K0
invertible.
copairing,
a
K1,
-+
20-4
A 0 B 0 'B
0 C
A
B0 C
A
K2)
K2
isomorphisms.
Example
4.2. 10.
Then q is side-invertible. Identities
H
Figure
at
4.3 show that
f
(IntH
)
H
H(&H),
)
H
H0
H),
)
Int
H),
Hf ) H#=(HOH m)H
Int
H)
=
H
Hp PH
are
=
=
(Int
f
H
(HOH'T)
H
fH
side-invertible.
Lemma4.2.11.
Let K be
be such that
pairing
the
0
an :
invertible
object,
H0 H
m
)
and let
a
Ht)
morphism
t
:
H
K is side-invertible.
-+
K
Then
Int H = K. H
Proof.
copairing r. : Int HH induced morphism i r. : XH 0 Int H -+ H is invertible. Since the
M= =
(HoIntH (HOIntH
M O'oInt
-0-0-H+ HOHOH H
)
H 0 H is side-invertible,
Hence, the composite
KOH)
KOXHOIntH
KOH)
the
is invertible.
a
split
Hopf algebras
--A-'I-+
K0 H
in braided
235
categories
particular,
In
L
is
4.2
epimorphism
(H
=
with
0 Int
H
(K MK 0 H
splitting
the
K06) K)
H).
H 0 Int
Wehave
H Int H
H
H
Int H
Int H
f.H
fH
fH
L t
t
t
K
K
K
Hence,
11K
tof
Thus,
s
K
Int H,
-*
=
H
(K
H (9 Int H
(K
HoIntH
Int H -+ K is
:
Since K and Int
deed, by tensoring
(Int H)
11N, where N is invertible.
Considering ]INV, and
phism. 1v 1
it Ob )
(1
1. In
!4'
N)
implies
Nv ON
Since N is invertible,
automorphism.
)
Int
Ht.f
=
deduce that
r
the claim
N splits we
the find
a
splitting
and
s are
to the case
idempotentp
(NI
that
equation
(D
r
1)
1K.
bt
isomorphisms. a
(N (1
b
) a
N
a'
11
Inb
1
c ,
1v
0
N 0 Nv
the left
Finally,
-!!f+
NI)
coeVN
=
coev
(1
1V 0
1)
121% Nv (2) 1v
.
1
hand side is 11
=
p.
N
'
)
an
1 (,
c
)
N) 1).
with p under the isomor-
which is identified N. The
--
with
K).
that
(1 coev) an
object
Nv
particular,
(1 coev,
is
we
transposed morphisms, Nv splits the idempotent pt, --
K)
K)
)
we reduce
the
Nv (9
coev
The
I
8
Int H
)
H are invertible,
with
6(&l
)
split epimorphism
a
r
(K
L
N 0 Nv
-!!f+
24 10 1V 1v 22
automorphism,
1) e,,
hence, p
1
=
1) p
2
P.
236
Lemma4.2.12.
and monoidal
categories
Monoidal
4.
Hopf algebra
For any
2-categories
we
IntH
H IntH
H IntH
Hf
CInt H,Int
fH
H
H
fH
CInt H,Int
H
'Hf
Hf
fH Int
Int H
H
fH
HT
ly
have
11
Int
11
braiding
where the
Proof. lowing
Due to
H
CInt HInt
(d)
equation
H
H
is viewed
Fig.
from
H
H IntH as an
invertible
4.3 the first
Int
H
constant.
equation
is
equivalent
to the fol-
one.
H
Int H Int
'Hf
Int H Int H H
H
Int
Ni Hf
Hf
Hf Hf
Hff
H Int
CInt H,Int
Y
H
H H
Hf
fH
fH H This
H Int
Int
equation
is
a
corollary
Int
H
Hf
H I
I
H
of the
fH
fH
Int H Int
H
following:
H
Int H
Y-I
HY
Int H
H
HT j H
one.
Int H Int
H
H Int H
fH
Int H
(d) from Fig. 4.3. (and its proof) is obtained from equation (c) in Fig. 4.3.
again equation
The second claim from the first
H
D
Hf
H
we use
H Int
fH
fH
To prove it
H
of the lemma
So it follows
by the
space rotation
particular
In the q
A) coevaluation)
1
:
-+
(resp. (resp.
B (D
evaluation
case
1,
=
pairing only
a
if and
p if it
A (9 B
:
1, then #H
If Int H
=
(H
(or
1
-+
be chosen
can
morphism. Then it can be complemented by p), which is called its side-inverse.
4.2.13.
Corollary
of K
is side-invertible
categories
in braided
Hopf algebras
4,2
m
(9 H
a
copairing evaluation
as an
coevaluation
fH
H
)
a
237
1)
)
has
q
a
side-inverse
UH 0Hf
Ho
1,
)
H&H-M
a
side-inverse
HOH),
H
HO
=
f
(1
H0
H
(f H
H 0
-1
Y)
H)
has
(H
Hf ) 1) HO=(H(&H T)H WfH)-l
0 H
has
H0 H
-'6-H-+ 1),
side-inverse
a
OH
(1
0
-124'Y
H&H
)
10,Y
HOH),
)
H
#H
=
(1
H0
H
H)
has
Wf H)_1 (H 0
Let
us
give
a
of
criterion
4.2.14. Proposition pairing. (a) Assume that
(Int H)
-+
(H
0 H
-M
H0 H
-E-"+ 1).
Hopf algebras
Self-dual
4.2.4
side-inverse
a
H, p"
c!
side-invertibility
Let H be
:
(Int H)' (Int H)
10 H
coev
a
(non-degeneracy)
Hopf algebra
and let
w :
Int H. H of
ol (Int H)
(Int H)
of
v(Int H)
0 H0 H
Hopf pairing.
H0 H
Suppose that there C, such that
0
a
exist
-4
1 be
(H
_-
H (9 1
locoev
H0
(Int H)v
(9 Int
H(&HoIntH
Hopf
morphisms /1'
0 H
12"o) (Int H)
0 1
--
Int
and if
a
H W01 )
loIntH-IntH)
H)
be chosen
can
for
is side-invertible
'Int
a
left
2-categories
right integral-functional morphisms of C
or
and the H
and monoidal
categories
Monoidal
4.
238
H (independently).
on
Then
H
(vInt
S
H0 H
H0 H
H
-MHOHOHI 4HO1!_-H),
(4.2.10)
(Int H) v
H
(H
S/I
(IntH)v
0
H0 H
H O'd "
are
H 0 H (9 H
-
10 H
_-
H)
(4.2.11)
invertible.
(b) If
w
is
(Int H)
H =, 10 H
(Int H)v
then
side-invertible,
0
(Int H)
Int H, the composition
--
0 H
H
lof
01 )
(Int H)
0 H0 H
-1-04 (Int H)
0 1
=
Int
H
or
H
,_
Ho 1
H0 -
(Int H)
10fH(D,
---+
can
be
H f--
chosenfor 1 (9 H
Hf coev
(91
(Int H)
H 0 H (9 Int H
H --+ Int H, and the
(Int H) i(D
or
(9
Hfol
(D
(Int H)
(Int H)
-%
10 Int H = Int H
composition
(9 H 0 H0 H
2-04 (Int H)
(9 1
fi-_
Int
H
w
Hopf algebras
4.2
H
10coe
H (9 1
-
-4 H 0
(Int H)
0
III
braided
be
Proof.
fH
chosenfor
f'is
(a) Since
0'= (H (Int H)
0
H
:
an
m)
(2) H
v(Int H)
lolntH-IntH
Int H.
integral-functional, ff
H
0 H
239
(Int H)
HOH(DIntH can
categories
)
it follows
IntH)
=
-10110 1 (Int H) H
(H
0 H
0 H0 H
by Example4.2.10
m)
H n-- 1 (9 H
1% (Int H) H
H
0 1
that
coev
(3)1
Int
-
H)
H
(4-2.12)
Int H
Int H
Therefore,
is side-invertible.
(v(Int H) is
0
H)
(9 H
where S' is defined
side-invertible,
In
H0 H
by (4.2. 10). Similarly, H
P"
=
(HOH, m)
JC4
H
(4.2.13)
1
IntH)
H
=
Int
Therefore,
is side-invertible.
H0
(4.2.14)
relating
these
(b) Follows
0
(Int H) v)
side-invertibility
implies
Both (4.2.13)
(H
12
where S" is defined
is also side-invertible,
and
H
and
pairings
w can
is
an
be used
of as
isomorphism.
from Theorem 4.2.5.
H0 H
by (4.2.11).
"J
)
(4.2.14)
1
Side-invertibility
of
(4.2.13)
w.
ev' H Hence, the canonical morphism. S" is an isomorphism. Similarly, .
S'
240
Assume that
w
is side-invertible.
2-categories
Let
denote
us
H
Int H
H
Int
HT
H
H
Int
H
Int H
H
sitH=
sit
SIH
st
H
H
H
H
Int H
morphisms
are
related
as
follows:
H
Int H
Int H
H
in
Hf
H
H
These
introduced
morphisms,
the
by
4.2.14,
Proposition
and monoidal
categories
Monoidal
4.
Int H
fH fH SIH
S/IH
H Int
Int H
H
H
Jnt Int
jH
I- n-t
----------------------------------
Int
Int
H
H
Int
fH
fH
H
fH
fH
V HV W
------------
H
diagram can be chosen for by Fig. 4.3(c). Similarly,
since the dashed part of the third Int
H by
Proposition
4.2.14(b) H
Int H
Int H
Int
H
HS"
H
S'
H
Hf
H
and
H
H
Int H
Hf
Hr j
(Int H)'
H
Int H
Int H
H
in braided
Hopf algebras
4.2
Int H
categories
Int
H
H
HS'
CInt H,Int
Hf
Hf
H
Int H
H
HS
H
Int H
H
H
H
Int
Int H
S11 H
CInt H,Int
fH
fH
H
SIH
CO
H In the last
two
equations
Lemma4.2.15.
(H Then
we
Remark 4.2.3
Suppose that
2-1)
0 H
have
H
H0 H
thefollowing H
w
is
and Lemma4.2.12
symmetric
w
)
1)
=
(H
are
thefollowing
in
0 H
-M
used. sense:
H0 H
relations
Int H
H
Hf fH' Hf Hf
Int
H
0
CInt H,Int
H
HS'I
0
S,H
I
H
H
H
Int H
H
Int H
C CInt H,Int
HS'
H
_fH fH
H
f
0
0
S/IH
J ^Y-
H
Proof.
The first
equation:
H
1).
241
4.
242
categories
Monoidal
H
2-categories
H
H
H
H
Int
and monoidal
Int
Int H
fH
fH
fH
ly
ly
W
W
H
H
H
H
Int H
H
Y
Int
H
Hf fH Hf Hf 0
CInt Hjnt
=
H
0
H
H
tioned
is similar.
equation
integrals are two-sided and/or Int H -_ 1, This holds for certain coends simplify. significantly braided Hopf algebras. Operators S' and S" are analogs
Remark 4.2.16
Whenthe
relations
amples
of
transform
4.3
The second
due to (4.2.5).
holds
on a
finite
abelian
categories
Abelian
the above -
our
men-
main
ex-
of the Fourier
group.
form
a
monoidal
2-category
proposed to construct TQFT's as double functors with values QV-Cat-modl in the 2-category V-Cat-modl. category of quintets version of the weak symmetric monoidal 2-catsemistrict is a The latter 2-category of its abelian bounded of categories [Lyu99]. Webegin with the description egory deWe k-linear of tensor monoidal. structure categories. product given by Deligne's the concrete example, in general and construct monoidal 2-categories fine semistrict In the introduction
we
in the double
which is obtained will
enter
In
our
monoidal
our
from
a
bounded abelian Then
construction.
construction
2-categories.
of
we
TQFT's
Nevertheless,
ribbon
add the next
we
actually
the strict
with
category
extra
structure,
that
namely, symmetry. consider weak symmetric
ingredient, need to
version
is sufficient
to check that
the
of equivalence machinery operates correctly. allows us to of bounded abelian categories the symmetric monoidal 2-categories because a lot of this is simpler, Technically perform the model strict constructions. That is why we the strict in identities case. become and isomorphisms, equivalences weak natural the one. artificial strict to the setup prefer The
of the strict
and weak versions
Abelian
4.3
Deligne's
4.3.1
product
tensor
categories
of abelian
form
a
monoidal
2-category
243
categories
categories are additive ones with good behavior of kernels and for instance in [Fre64]. Besides, we shall not need precise definition The categories of modules over the most general to categories case. equivalent finite-dimensional associative k-algebras are called bounded. They are automatically k -linear and abelian. Such bounded categories with a balanced monoidal strucbe used as input data for the construction of a TQFT. To introduce the exture will of terior 0 of bounded categories, it suffices tensor product to do it for categories Roughly,
abelian
cokernels,
see
For technical
modules. metric
reasons
we
of k-vector
category
tensor
deal with
shall
semistrict
a
version
of the sym-
spaces. of the
main properties objects we are dealing with. The 2-cate(resp. AbCat') is formed by small k-linearabelian 0-morphisms, or objects: essentially categories A with length, F : A -+ B, I -morphisms: k -linear left (resp. right) exact functors 2-morphisms: natural transformations (morphisms of functors). of a bicategory.) (See B6nabou [B6n67] for definition Monoidal 2-categories defined studied and are by Kapranov and Voevodsky The of monoidal notion more general a [KV94]. bicategory has been proposed by between these two theories and The Power Street [GPS95]. Gordon, relationship monoidal of AbCat and is clarified and Neuchl The Baez structure [BN96]. by AbCat' is given by Deligne's 0 -+ A 0 B. tensor product of categories : (A, B) The tensor product F 0 G of 1 -morphisms F : A the C and G : B -+ D satisfies condition that the diagram Let
us name the
gory AbCat
A
FxG
B
Ot ANB is commutative
only
up to
an
isomorphism.
up to
isomorphism. and is,
C
FOG
CZD
However, this requirement not sufficient
hence,
building
for
F0 G
determine the full
monoidal
structure.
In order
egories and right tor
exact
and let
functor
to construct
of finite
functors.
C be
F 0 C
mod-A due to the
F
)
the monoidal
dimensional
Let F
:
finite
:
mod-A 0 C
mod-B factorises
over
mod-A
dimensional
a
-+
consider
2-structure
modules
-+
associative
mod-B be
mod-B 0 C. The
via the
a
algebra.
associative
underlying
the
2-subcategory dimensional
finite
k-linear
right
of cat-
algebras func-
exact
Wehave to construct
composite
functor
mod-A (D C
the -+
mod-B 0 C -+ mod-B,
algebra homomorphism C -+ EndA M
EndB FM
for any A 0 C-module M. This determines (F Z C) (M) = FMas Also (F 0 C) (f ) = Ff on morphisms f E mod-A 0 C.
(4.3.1) a
B 0 C-module.
244
transformation
A natural
transformation
C0 F
the functor
C Ma: C MF
give
mod-A
a :
FZ C
:
-+
G0 C
structure
on
-+
mod-A'(9
such
FGM-+ GFMfor
:
mod-C 0 B
-+
ASG
C
tFZD
BOG )
mod-B (9 D
A 0 C-module M.
Clearly, for arbitrary right exact functors
isomorphism. does not exists. However, for following reason. Consider the exact sequence
an
for the
M(&AoA
(M,
where
M& A
a :
M)
-+
MoAoFA FMwith
which identifies
Using
a:
is
an
one :
M0 C
MOFAOGC
There
aA,B,C
phism
are more
mod-A 0
:
f4
GMo FA
GFa )
(B
one can
:
apply
4.3.2
Semistrict
case
of
work in
explicit
down first.
a
a
weak
Z
(B
Z
A (&
(B
0
C).
Z
G)M
If
G)(F the
0
C)M
associativity
comes
functor
from the isomor-
wants to avoid using Voevodsky [KV941 and is equivamonoidal 2-category
Kapranov a
D)(A
tOF,G
for instance,
that
the
one
and
2-categories
monoidal
to use the
In order
(F
C, which
of
Mwith
monoidal
to a sernistrict
be written
0,
A 0 C-module
0
[GPS951 implying 2-category.
FM-+
)
B)
0
the main results
Gordon, Power and Street lent
[KV94],
C) -4 mod-(A (A (& B) o C -+
0
a-'
algebras
of
such data,
elements
structure
Fa
under F
diagram
FM0 GC
1(&Pt M(& GCo FA
on a
Mvia the
-+
have it
MOAFA, where FA E A-bimod-B.
NF,G
defines
image
M(D FA
product
the tensor
sequences M0 A -+ M, -y
we
")M--+O,
A-module and its
)
functors
of A-modules
M&A
OFA-M(&Fm
c'
such exact
actions
aoA-Mom
F
mod-A 0 D
OF,G
an
defined.
are
2-category we also need for given of functors mod-D an isomorphism.
mod-B 0 C
ZF,G
the natural
mod-B (2) C, which is
-*
this
FOCt or
mod-B determines
-+
mod-A 0 C
:
GMfor any A 0 C-module M. Similarly, mod-C (9 B and the natural transformation
C Z G: mod-C (9 A
-+
the monoidal
mod-B and G : mod-C
-+
G : mod-A
F -+
map am : FM : mod-C (9 A
the linear
given by
To
Z -C
a
2-categories
and monoidal
categories
Monoidal
4.
The
6-category)
model situation
symmetric
definition
of
difficulty
here is that
a
is not so easy to write and to deal with a strictly
a
monoidal
2-category it should of it (a particular is why we prefer to
weak version down. That
symmetric
semistrict
monoidal
Abelian
4.3
2-category
equivalent
monoidal
metric
symmetric
strict
to the weak one
category
of vector
monoidal
category.
algebras
finite-dimensional The definition
of
a
categories
we
fonn
a
monoidal
key point
need. The
(which
is not
replace
is to
strict)
with
spaces Then we use the monoidal
category
monoidal
sernistrict
2-category
was
Definition
4.3.1
2-category
%equipped with
([GPS95,
KV94, BN961).
monoidal
A semistrict
Power
of
Coincidence
2-category
is
a
I E 9A (the
unit object); objects A, B E 9A an object A 0 B A' and any object 1-morphism F A
1.
An
2.
For any two
3.
For any
4.
the sym-
equivalent analogs of
given by Gordon,
[GPS95] and by Kapranov and Voevodsky in [KV94]. both definitions is elucidated by Baez and Neuchl [BN96]. in
AN B
an
and modules.
and Street
object
245
2-category
9A;
B E %a
1-morphism
A E 9A
I
F0 B
A' 0 B;
-+
-morphism G
For any I
in
B' and any object
B
a
-morphism A MG
ANB-+ ANY; 5. 6. Z
B E 9A and any 2-morphism A : F --+ G For any object A' A morphismANB: FOB-+ GOB: ANB-4 A'NB; A E 2( and any 2-morphism A : F -4 G B' For any object B morphism A 0 A: A Z F -4 A 0 G: A 0 B -+ A 0 B'; For any two I -morphisms F : A --+ A' and G : B -+ B' a 2-isomorphism
ANB-
FZBt (i) (ii)
thefollowing object A
a
2-
tFOB'
Ed
A'N B'
satisfied
are
and Z A 2-functors A 0 : 9A I Z Afor any object A, A A0 1 FZI F I Z Ffor any I-morphism F : A -+ B, I Z afor any 2-morphism a : F --+ G : A a 0 1 a B; The tensorproduct A 0 (B 9 C) is associative on objects: (A 0 B) 0 C. For any I -morphism F : C -+ C' and any algebras A, B we have A 0 (B 0 F) For any
=
E 9A
we
have
-
-
=
=
=
=
(iii)
conditions
A'
2-
ANB'
ZF,G41JI,
A'S B such that
AEG
a
=
=
(ANB) If
a:
A 19
(iv)
(v)
F
(a
ZFAZ (FOB) -+
0
G: C -+ Cis
B)
=
(A
0
a)
0
=
a
B,
For any 1-morphisms F : A have OANG,H = A 0 ZG,H,
(ANF)
2-morphism -+
(a
0
A)
A', G
(AZB) (B 0 a) 0 (A 0 B);
OB, FZ
=
then A Z 0 B B
=
-+
a
B' and H
:
=
C
(FNA) (A 0 B) -+
ZB. 9 a,
C' in 9A
we
OFBOH, OFGOC OFG0 C; B For any objects A Z 1B, and for % have B N we E 1A A, 1ASB F B' A B G -* : -+ : we A, 1AEG and any 1-morphisms haveolA,G OFJB 1FEB;
OFZB,H
=
=
=
=
=
246
(vi)
For any I
thefollowing
and monoidal
categories
Monoidal
4.
2-categories
-morphism F : A -+ A' and any 2-morphism two 2-morphisms are equal. AEG
A Z B'
A0 B
AZB
ASH'
O
FOB
A'N B
F,H
-+
A'
FOB'
A'M BI
WEDB
A'N B'
I-morphism G : B -+ B' and any 2-morphism are equal: two 2-morphisms thefollowing
(vii)
B'
A N B'
Op,G
FOB
=
-+
A'OG
Z--)H
A'
FZBI
AEG
G -+ H : B
F
For any
HOB
A' 9 B
AIR B ftv)mb,
ANB
HOB
-+
H
:
A
A'NB
FOB'
ZF,G
AEG
A'OG
OH,G
AEG
=
A'OG
HOB'
ANY
FOB
-morphisms F : A OF,HoG equals to thefollowing
For any I
(viii)
FOBt
A semistrict
4.3.3
A', G : pasting
B
4 NF,H
A'M
A'SH
2-morphism OHoG,F equals of the
2-category
C, H : C -+ D the 2-morphism
ASH
tFOC
A'RG
version
-+
AZC
ZFG
A'Z B the
-+
AEG
AMB
Similarly,
A Z B' ftiPOBI A' 0 B' FOB"
A'OB'
to
of
the
pasting
categories
AND
tFOD A'9D
0fNG,F
and
ZH,F-
of modules
of cateof the 2-category version examples of monoidal 2-categories Westart with a strict constructed by Day and Street [DS97] via enriched categories. (V, 0, 1, P), equivalent to k-vect. For instance, symmetric monoidal category V described of coordinatized vector V is the monoidal category by spaces k-vect, has internal it and is closed category Kapranov and Voevodsky in [KV94]. Then V a X' 0 Y for any pair of objects X, Y E V. homomorphism spaces Hom(X, Y) Following Kelly [Ke182] we define a small V-category C as Let
us
gories
construct
of modules.
in this Our
paragraph example is
a
semistrict
similar
=
=
1.
a
set
ObC of
objects,
such that;
to the
4.3
pair X, Y E ObC there object X E ObC there
for any for any
2.
3.
categories
Abelian
is
form
a
monoidal
247
2-category
object Hom(X, Y) E Ob V; morphism. 1x : 1 -+ Hom(X, X)
an
is
a
E V
(the
unit); for any
4.
triple
objects X, Y,
of
Hom(Y, Z)
Hom(X, Z)
-+
morphism cx,yz (the composition);
Z E C there is E V
a
Hom(X, Y)
0
such that
composition
the
1x is
a
left
c
is associative:
right
and
unit
cx,yw
for the
unit
left
A V-functor
property is similar). : A -4 B is defined
F
a map F: Ob A --+ Ob B; for any pair X, Y E Ob A a
1. 2.
Fx,y coherent
the units,
with
composition,
the
:
1FX
HomB(FX, GX)
&
1x
:
Ho MB
I
Fyz)
G
-+
E V
y
:
A
Fx,z
category
we
1
-+
(FX, FX),
and with
cx,yz [Ke1821. B is defined as a set of
given for all X
o
E
2 4 !Lom,3 (FX, FY) CFX,PY,GY
Kelly [Ke182]
to
monoidal
k -vect
F
HomI3 (GX, GY)
is referred
=
=
E V
Ob A, such that
0
mor-
the
11omB(FY, GY)
tCFX,FY,GY
arbitrary
for V'
:
Fx,
The reader an
o
0
llo_mB (FX, FY)
t
,\xoGx,y
Hom(X, Y))
holds:
the usual relation
(X, Y) HOMA
to
-+
Fx,x (Fx,y
Hom(FX, GX)
-+
1);
0
as
=
CFX,FYFZ o A transformation 1
(cx,yz
cx,z,w
c:
Hom(X, Y) ex'x'-y4
0
llomm,4 (X, Y)
i.e.,
=
morphism
i.e.,
A V-natural
phisms Ax analogue of
:
cyz,w)
(9
composition
(10 Hom(X, Y) IM Hom(X, X) (and the
(1
o
1LOmJt3 (FX, GY)
which apply general definitions, In particular, strict. necessarily and natural functors of k -linear categories, form small k -linear categories a 2-category for
more
V, which is
get the usual notions
-4
(4.3.2)
not
that (essentially) form a transformations and V-natural V-functors V-categories, Similarly, 2-category V-Cat. For any pair, A, B, of V-catThe objects of V-Cat are small V-categories. whose objects are V-functors egories there is a small category, V-Cat(A, B), transformations V-natural G F are -+ F A -+ B and the morphisms X : B is F F -+ : A -+ 1 -+ HoML3(FX, GX). The identity A morphism IF : F G A of The -4 : 1FX : 1 -4 HOMJ13(FX, FX). composition given by (11F) x and /-t : G -+ H is given by transformations.
Recall
k- Cat.
=
1
=
The
10 1
composition
Hom(FX, GX) in V-Cat is
0
Hom(GX, HX)
given by the functor
CFX,GX,HX
Hom(FX,HX).
248
B)
V-Cat(A, (a:
x
(F'-
=
where F
a
F
G
-
-
GF is the
=
F'.
Homc(G'FX,
(4.3.3) functor.
equivalent equivalence =
One can
f4"
G'
F"
G")
F"
G" )
a-G"
G"
G
)
F" -G"
and
of
by diagram (4.3.2). verify the associativity
The unit axiom
easily.
equivalence. :
A
-+
Weshall
B is exact
the functor
say that if F : A
a
-+
V-functors.
to
k-vect,
-+
B is,
etc.
defined
as
For any morphism in V choose and fix in the category V. Its V-category
associative the
1LomA(M, N)
Thus
its kernel.
A-modules
(M,
a
:
Coe
)
M(& A
=
that
transfer
the
V-category
in
is
if C is,
theory
of the
categories
of
Let A be
an
A-modules -+
M),
unital
associative is defined
M E Ob V,
a
as
the
E
V,
hom-objects =
Hom(M, N)
pair of A-modules (M, a) and (N, P) (recall
-01A
we can
V(1, Z)
way.
VA of right
Ker(Hom(a, N)-Hom(MOA, #)0(-01A): for any
-4
follows.
algebra
with
Z
C is abelian
V-category Many results
in this
results
V
Define the 2-category V-Cat, whose with length, objects are abelian V-categories 1-morphisms are right exact V-functors, 2-morphisms are V-natural transformations V-Cat-mod consisting and consider its full 2-subcategory
equipped
composition
for this
=
F
set of unital
GF"X)),
Honja (GF'X,
then V-Cat and V'- Cat If V is monoidally to V', equivalent for V equivalent In particular, to k-vect we have the 2-categories. V-Cat -+ k- Cat, C ObC, C1 where ObC 2-categories
functors of ordinary properties of ordinary corollaries are simple
modules,
(4.3.3)
1
G"FX)).
V(1, Hom', (X, Y)). Indeed,
the canonical V-functor
follows
C)
F'-G'-+
-0:
of V-fanctors
(11324
are
a
)
0 L-'
Remark 4.3.2.
Y)
composition
F"
!Lom,3 (F', F"X)
distributivity the identity
C (X,
a
(1 - 4
The is
')
F'.,3
G'
V-Cat(A,
--+
G Lef ej'f
-
a-G
G'
--+
G")
G'-+
(P.
0
C)
V-Cat(B,
F",fl:
F'-+
2-categories
and monoidal
categories
Monoidal
4.
(Hom(M, N)
=
Mv(&N
=
Mvol(&N
-+
that kernels
Hom(MOA,N)) are
MVcoeVA ON -+
chosen).
Here
MV(&Av(2A&N
Av(gMv(&N(oA-(MoA)vo(NOA)=Hom(M(&A,NOA)).
4.3
categories
Abelian
fonn
HOMA (M, N)) is identified (M, N) I-e-f V(1, HOMA M N Mor i.e. -+ E : V, such that f morphisms,
with
Note that
A-module
2
(M 0 A #
where a, To in
verify
associativity
tion to
VA is
(k -veCt) A is VA, to check
Clearly,
statements
The
we
a
=
an
to
to
also in
these
VA.
A-module
right
A
homomorphism of algebras
a
A
B
-+
are
-+
End M
def
a
exact
FM,N : HOMA(M, N)
associated
with =
Am E
VA, whose morphism. the sets of HOMA (M, N) VA
:
is
-
F
VA
:
V(1, HomB(FM, FN)) A
transformation A
VA _+ VB;
V-functors
_+
VB corresponds
a
V(1, HOMA(M, N))
'V(1,
transformation
follows:
transformations.
category
=
as
A in V;
algebras
right
are
V-natural
Note that
in V-Cat'
V-Cat-mod unital
associative
=
phism. Am
k-vect
=
V(1, -) valid
M E Ob V. A
in V and let
V (1, ffiomA (M, N)) sets are VA (M, N) A-module morphisms in V. To each V-functor unique functor F : VA _+ VB such that
A V-natural
for V
Applying they are
composition.
of the
algebra
2-subcategory
full
the above definition
Mv 0 M.
objects are I-morphisms the 2-morphisms
=
N),
-
is obvious.
equivalent
the
natural
N0 A
in mod-A. Therefore,
statements
statement
The
Remark 4.3.4.
(M 0 A 1-0-14
=
consider
unitality
and
M is
Hom(M, M) Define
V-category
a
Let A be
Lemma4.3.3. in
N)
the composithe usual category mod-A. Wehave to construct that the unit morphisms 1 -+ !Lomv (M, M) are in the kernel,
get valid
following
structure
--f-4
the set of
the actions.
are
that
see
M
)
249
2-category
monoidal
a
F
:
F
G
-+
:
-+
G
VA
:
VA
V(I, HOMB(FM, GM))
=
VB corresponds
_+
VB'
_+
HOMB (FM, FN).
=
which
to
a
unique
is the B-module
HOMB (FM, GM) given
mor-
for each
A-module ME VA.
4.3.4
Construction
We shall
of the 2-monoidal
make V-Cat-mod
into
a
structure
semistrict
monoidal
Kapranov and Voevodsky [KV941, Gordon, Power and explicit by Baez and Neuchl in [BN96, Lemma4]. Proposition
4.3.5.
V-Cat-mod
is
a
semistrict
monoidal
2-category in the sense of [GPS95], and made
Street
2-category.
250
Monoidal
4.
The monoidal
is built
structure
1) The algebra
phic
with
tensor product multiplication
the
-224
A0 B0 A0 B Lemma4.3.6.
objects
of two
2) The
data:
of V-Cat-mod.
A, B (algebras
V1
Note that
in
V)
in
is the
fL 4
A 0 A (9 B 0 B
A, B be algebras
Let
A 0 B-module in Mamounts
right
following
the
object
1 E V is the unit
2-categories
is isomor-
V.
to
with
and monoidal
categories
algebra A 0
B
A (D B.
V, and let M E Ob V. The
to any
structure of a algebra homomorphisms
ofthefollowing
in V
A (9 B
(i) (ii) (iii)
A
End M,
-+
KndB M, KndA M.
B
verification. Straightforward universal (the proof).
Proof. grams
Lemma4.3.7. -y
:
N0 B
apply V(1, -)
one can
M, N E VAOB Denote
Let
110-MAOB (M, N)
of
the action
.
N. Aen
-+
Or
B
by P
Hom.A (M, N)
-+
to the relevant
is the
:
M0 B
equaliser
dia-
-+
of
M, the
pair of morphisms 1Lom(,6,N)
RomA(M, N) RomA(M, N)
This holds
Proof.
Let F
3)
to construct
there
is
:
a
VA
'
VA
_+
FM0
A'(&
map on
it to the
Proposition Then
B)
be
a
F 0 B
V-functor
It makes FMinto
a
4.3.8.
an
B
-+
:
morphisms
(M
0
B, N)
V-functor, :
and let B be
VAOB__+ VA'(9B
.
an
algebra
in V. Wewant
For any module ME
VA&B
EndA M
En dA FM. Ki ,
A' (9 B-module with the action FM& EndA, FM-+ FM(D End FM-+ FM.
B
objects F 0 B : Ob VAOB-+ Ob VA'OB morphisms. Let F
:
VA
_+
VA'
be
a
V-functor
: !LomA (M, N) HOMAI(FM, FN) (M, N) -+ llommx oB (FM, FN) HOMAOB
Fm,N
M,N
L.A
algebra morphism
an
to extend
OM
in k-vect.
P':
Hence,
HOMA(MOB,N)
HomA(MOB,NOB) Hom(MOB,2+) .
-01B
gives
-+
rise
to a functor
denoted F 1Z B.
is constructed.
and let induces
and the
Wehave
M, N E VAOB. morphism (F 0 collection of such
a
lre;M
4.3
Proof.
original by #':
Denote the
the induced
action
FM(& B) with
(resp.
an
of B by
action
FM(9 B
A-module
:
M,
FN
X-module)
(resp.
B
y
-+
251
2-category
inu-noidal
M(9 B
FM, -y'
-+
a
N0 B
:
N and
-+
Equip
FN.
M0 B
structure
-(234MOAOB22f*M(&B,
M(&B(&A
-(2-34M
FM(& B (o A'
0
A'O B
!%
B,
FM(D
of A'-modules
OM 111 as an object in k-vect The homomorphism of A-modules 0 : V V. We use the equivalence M0 B -+ Mrepresented as : 1 -+ HomA (M 0 B, M) composed with FM(&B,M induces a homomorphism. F0 : F(M 0 B) -+ FMof X-modules. Using the basis where a, a'
F(M
bi
E
0
B)
is
the actions.
FM0 B due to the fact
V(1, B)
finds
one
F# Compare
Then there
are
-+
the
isomorphism
an
B
that
-_
11
=
that
=
(F(M
B)
0
0
-4
two commutative
following
4 FM).
diagrams
ILO-m('3'
ILornA (M, N)
6
FM0 B
Ho MA (M 0
t
FMOB,N
HOMAI(F(M
FM,N
B, N)
&
B), FN)
t Hom(OM1,FN) iLomA,(FM,FN) 11omA(M,N)
-01B
4
-
ILomA, (FM
Hom(,8',FN)
H-O-M(11L))---+
1LomA(MOB,NOB)
o
ffomA(M(8)B,N)
tFMOB,NOB HOMA1(FM,FN) with
coinciding equaliser
t
-01B
left
MO-MA1(F(MOB),FN)
t
HOM(O-14N)
HOMA1(FMOB,FN(&B)
and
right
columns.
tFMO.B,N
4
Since ri
:
Hom(O-1,FN) M
HOMA1(FMOB,FN)
H-OMA(&B(M, N)
-+
(M, N) HOMA
with this map and endof the two upper rows, any path starting the composition in the lower right comer gives the same morphism. Therefore,
is the
ing
.
MO-MAI (F(MOB),F(NOB))
FM,N
B, FN)
Fm,N 0 ri equaliser
factors
through
of the two lower
4) Let G :VB
_+
r,
VB'
module ME A 0 B there
:
HOMAIOB(FM, FN)
rows.
be is
an
Functoriality a
V-functor
-+
!LomA, (FM, FN)
-
the
of F 0 B is clear.
and let A be
algebra morphism
an
algebra
in V. For any
Monoidal
4.
252
a' It makes GMinto
5)
For a
A (9 Y-module.
an
transformation
The natural
AZ B
A
the action
:
3)
to
F
:
F 0 B
of the
we
get
a
VA _4 VA' we con: VA(9B _+ VA'(&B
:
This
can
EndA,
EndA M
FM0
ILomA, (FM, GM)
GM
jL0M(3,Gm
HOMAI(FM, GM)
(1 -
HoMA,
Hom,, (FM 0 B,
RomA, (FM, GM) 14
HOM A
jLom(F
are
AM:
1
This defines
6) a
For
a
a
V-natural
to
5). 7)
for
a
of B.
actions
transformation
V-natural
:
VA
V-natural
A0 A
:
VA,
G : VB
transformation
ZFG
_+
FOB
1
right and
the functors
(A'
Z
ones.
G) (F
Amfactorises
B)
are
B, GM0 B) 0
B,
GM)),
equaliser
via the
as
F
-+
G VB
AZ G
VB be
a
right
_+
VB'
we
construct
VAOB _+ VAOB' exact
V-functors.
Similar
Welook
AEG
)VA(DB'
1
FOB'
AV VA'OB'
F and G, not only for arbitrary assumption (F 9 B') (A C9 G) right isomorphic. necessarily
without not
:
'
_+
A Z G and F 0 B exist
However, 9
-+
11
VA'OB
exact
A
A0 F
VAOB
Note that
0
A Z B.
transformation
transformation
Let F
(FM
HOMAI(FM
Lemma4.3.7
By
I
GM))
MO-MAI 013 (FM, GM) -+ liomA, (FM, GM).
V-natural
given
(FM, GM),
equality
the
to
and -/
1
tcPM,PM,GM CFM,GM,GM
ILornA, (FM, GM) 0 KndA,
0
as a
tFm,m(&,Xm
t
where
be written
EndA M0
XmoGm,m
(1
of A-mod-
morphisms
the
algebra V(1, B).
t
which transforms
AZ G
functor
G ED B
B
1 (9
GM.
G
-+
-+
A commutes with
transformation
with particular, commutative diagram in
EndB,
Similarly
transformation
V-natural
given
a
V-natural
follows.
as
ules,
M Gm,m,
Ep dB
-+
VA(DB'.
VAOB _+ struct
A
:
2-categories
and monoidal
categories
the
for
exactness
Let
(M,
a :
M0 A
-4
M)
MoAoA the exact sequence in
gives
f'onu
Ab' Ii4i!
4.3
be
an
A-module.
a(&A-Mgm
S
a
FMA : A 0 S
=
for
a
-+
(N, 0
B-module
`
ASA'
Q
N& B
io
=
=
N)
-+
AVA'
=
is
FM-+
equipped
Q -E'-3+
BVB'
GB E
diagram
commutative
M(9 A (2) S (9 B (9
t
(234)
,30AOS OQ
in
are
is
GN-+ 0,
equipped
M(& S (9 B (2) (23) #(&SOQ
ce(91-10a0i
with
row
the left
following
of the
t
Q
_M(&S(D'O
(23)
t
GPOS
GaOS-GM(Do,
GM&A(gS there
Therefore,
M(9 S 0 B (D
is
an
exact
Q ED M& S (&
)
sequence in
A (o
Q
GFa
GM&S
GFM
VA'OB'
[001o(23)-10iP]E)[a(8)1
Mos(&Q
GFaoG(OO
FGM. sequence determines these versions of) (permuted sequences
GFM-+ 0.
exact
Furthermore, M(3 B (9 S (9
are
related
by
a
chain map
MOS(&Q
Q ED M(9 A (9 S (9 Q
t(23)
(34)E)t(34) M(D B (&
action
M(D s(&Q
GPOA01
A similar
Similarly,
VB'
exact.
-MOA(&S(9V)
t
(234)
S.
action
'
Q
M(&A( DS(DQ
the left
with
EndA,
.
VA'OB
(4.3.4)
0,
the exact sequence in
obtain
N0
BQBI
)
EndA (AA)
-4
we
Fa
GMB: B 0 Q -+ Q. (M, a, 0) E VAOB The columns and the lower
Now, let
VA
c)M-*O
MOA
FA E
=
NOBOQ where the bimodule
Then the exact sequence in
MOS
S coming from A :
253
2-category
VA'
MoAoS where the bimodule
monoidal
a
Q (9 S o) M(9 A (9
It induces
a
GFMsuch that
M(&
Q (9 S
unique isomorphism
OFG(M)
between the cokernels
Q (& S
FGMand
254
Monoidal
4.
2-categories
and monoidal
categories
F(M
M(gs(&Q
Q)
(9
FGM
(23)1 G(M 0 S)
MOQ(&S Clearly,
it
the V-natural
gives
Now we have all
structure
't1Z.F,1G
GFa
NFGthat
morphisms,
and it remains
0
)
looking verify the
for.
we are
to
2-category. algebra A E V
semistrict
(4.3.5)
(M)
GFM
transformation
0
)
axioms of
a
monoidal
A 0 V-Cat-mod we have 2-functors : (i) For any Z A: V-Cat-mod -+ V-Cat-mod. V-Cat-mod, the conditions: (ii) The unit object 1 viewed as an algebra satisfies A (9 1 A 1 (9 A for any algebra A; F M1 F F : VA _+ VB; 1 Z F for any V-functor 10 a for any V-natural transformation a N1 a a : F -+ G : VA _4 VB. -
-
=
=
=
=
=
=
product is associative : VC -+ VC' and
The tensor
(iii)
For any V-functor
F
(B
AZ since
(F
F)
(A
=
morphism A Endc M. Similarly,
give
to
Ao B
Z
-+
A) A Z (B Z a) a N (A 0 B).
a
Z B. If
Z
=
a
(A
:
0
F
-+
B)
(9
-+
on
any
=
:
-+
SF,G 0 C. The resolution VB' OCbecomes
=
Mo B (& C (& GBo C
B)
=
is
a
natural
(4.3.4)
N a) Z B, and
(A
written
'Y01Oj3(&1-GMB0MCO(
transformation
:
4-+
VA
Denote
by J
=
7
get the expression
o
P (9
1
:
Z
A)
MB
'
-+
VA and G
Wehave to prove that H = GZ C
M0 GB0 C
M(D B (9 C -+ Mthe action.
4
HM-+ 0.
From definition
(4.3.5)
for ZFH via
F(M
M(& FA (& GB(& C
(432)t M(&GB(&C(DFA
Clearly,
(a
then
for the functor
G,3o-i(&1o(2
we
0 C.
=
VC
'
VBOC-+
B)
=
VC
(iv) Using the notations from 7) let us assume that F VB -+ VB are right exact V-functors. Let C be an algebra.
OF,GOC
0
indBoc Mis equivalent to giving a morphism (F 0 B) , AZF)C9BandFN(AOB)
A C9
G
(A
VAOBOO__ VA(DB(DC'
Z F:
N a, A N (a 0
B)
objects: A 0 (B 0 C) algebras A, B we have
G(JOFA) -
& GB0
G(M (9
FA
C)
FG6
)
GF
2)
FGM
)
0
)
0
tZ.F,H GFM
diagram projects onto Diagram (4.3.5) using the action -Y : M(9 ZFH OFGNC as stated. Similarly, OAZG,H A 1Z NG,H. Consider now given right exact V-functors F : VA _+ VA' H: VC -+ VC' and denote S transformation FA, R,= HC. For any algebra B the V-natural OFEB,H is found from the commutative diagram this
C -+ M. Therefore,
=
=
,
=
4.3
aflar,
,
oncs lorm
caL
Fa0l
M&SoR
MoSoBoR
a
monoidal
255
2-category
FHM
FMoR
tO.FSB,H
(234)t -
2F,BZH
transformation
and the V-natural
HMOS
HM 0 So B HM(DSOB H,801o(23
B MOROSOB
HFa
by
is determined
)
HFM
the commutative
dia-
gram
(432)t o S MOBOROS
The first nal
in these
rows
arrows
(v) Let IdAdenote B Id A(&B =AOId
M0 R M(&R&S
diagrams
)
HFa
equal composite morphisms.
OFEB,H functor
identity
the
HM(gS
H'Yol
determine
Therefore,
also coincide.
tOP,BMH
OS
+
FHM
FMoR
FM(9 B (D R
MoSoBoR
OFBOHVA _+ VA. Clearly,
Id
HFM The
Id
diago-
AMB
.
The V-natural commutative
OIdAG
transformation
(23)
t
MoGB(DA
a
:
VB'
VB
_4
0--i>
GM
is found
from the
diagram
c2lt>
M(D A (& GB
as
G
for
G,30
M(9 GB
rl-GMOA
G
Ga
to'dA,G t>GM
identity morphism 1AZG- Indeed, the above diagram is the image under G of diagram, which states that actions a and P on Mcommute. Similarly, NFjdB the
1FOBA' and,0 : G -+ H: VB (vi) Let F :VA -+ V. transformations the following two natural are equal.
vA(DB
ANG )
#A&P
VA(&B'
VA(&B
) AOH
FOB
F,H
FOB'
=
VA'OB Let S
=
V
A'ZH
FA, Q
=
VA' B'
GB, R
=
)VAOB'
P,G
FOB'
A'OG
0
VA'(DB'
VA'(&B
HBand ME
Wewant to show that
AOG
z
FOB
'4
VB'.
VAOB There .
is
a
diagram
(4.3.6)
256
4.
Monoidal
and monoidal
categories
2-categories FGj8oF(a0Q)
M(&S(DQ
F GFao
M(&Q(Ds
G(,3&S)
GFMk_
--->
MOSOOB
Fom
IPFM +
F(a(g)
FH)3o
M(2) S (& R
MOOBOS
FGM
FHM
5
k" '
HFaoH(OOS)
Mo R (& S
HFM
and the lower is similar. where the upper face is Diagram (4.3.5), front and the back walls commute. Hence, the right wall commutes.
(4.3.6) is proven. (vii) Let G : VB _+ VB'ando: transformations two natural following
The left,
Thereby
the equa-
tion
HUB )
vA(9B
F
-+
H
:
VA
_+
VA'.
Similar
the
(vi)
to
equal:
are
VAOB HUB, VA'(&B
VAOB
FOB
A'UG
ZP,G
AUG
=
A'ZG
ZH,G
AUG
HZB'
VA(DB' Let F
(viii) V-functors.
:
FOV VA'(DB' VA
_+
VA'
,
G : VB
Wewant to check that
FOBt as
a,
/-t
o
A'OG
A for A E
MFG
=
VA(&B There .
GFM& C -+ GFM. Pasting
(4.3.7)
VC and
-+
G,801
MOGBOFAOHB
(23)t
(23)t )
) VA'OB'
FOB'+ H:
VC _+ VD be right
ANH
vA(&C
tFZC
exact
vAOD
t
4 NF,H
VA'(D (A'
N H) and /-t
=
are
actions
GM& C
fits
into
Fa(&l ) FM(DGBOHC MOFAOGBOHC
0
G)
the commutative
FGa01
GMOFAOHC
(A
(4.3.7)
FZD
VA'OD
ASH
G,8(91 H-y(Dl 11 MOGB(&HCOFA GM(&HCOFA
(34)t
1 V)ZB'
pasting
NFG
VA'(DB
OF,HoGLet (M,
the
AUG
VAOB
denoted also
V,
40B'
*
NFGequals
--+
GMand J
diagram FHGa
HGM(DFA
C> FHGM
tIN )FGMOHC
ZF,G(M)(Dlt GFM(DHC
HFGM
H6
til HGFM
to
Abelian
4.3
and it is determined
by
HG,80 il,
2-category
monoidal
a
FHGq
HGM 0 FA
(F
f>
t
(23)
257
0
D)(A
9
HG)M
tOF,HG(M)
M0 FA o HGB
Look at these
two
whose walls
FMo HGB
diagrams
HGF,8
(A'S HG) (F
rectangles;
the vertical
Z
B)M
parallelepiped maps are 3 identity
top and the bottom of
at the
as
made of 6
are
a
and 3 maps of the form 10 Hx ,and 10 Hx (9 1 in the left
half
right
maps in the
f6mi
The value Of NFHGon Mis found from
it.
M(& HGB& FA
diagram,
categories
half, where x: GB(9 C -+ GBis the action and Hx: GB(& HC-+ HGB. The left, front and back walls are commutative, thereby proving that the right one is as That is' 'p
well.
o
Weconclude
4.3.5
Braided
Wewill we
deal with
in the
proof
Proposition it is
over,
sylleptic Proof.
Clearly,
the full
definition
4.3.9.
strictly
V-Cat-mod
pair
of
RA,B
:
For any
-
a
is
that
2-category.
M)
_4
4
one.
The
2-category, since meaning is specified
monoidal
semistrict
in the most strict
More2-category. (in particular,
sense
I.
---:
A, B
algebras
VA(9B
braided
a
monoidal
semistrict
symmetric
is, braided
-
=
braided
a
RA,B RB,A
MoA(&B -4
RA,B RB,A
monoidal
semistrict
strictly proposition.
symmetric, and
a
of
of
case
following
of the
is
2-structures
particular
a
[DS97])
(M,a:
ZFHG-
=
monoidal
give
not
A
that V-Cat-mod
E V there
is
an,
isomorphism
categories
of
VB(DA, (M,M(&B (2)A Mop) M(&A(2)B
1-
The squares
VA(&B
FEB
VB(&A Also the
BZF
AEG
RA,Bt
VBOA'
VBOA
VA'(&B
vA(&B
VA(8)B'
tRA,B' GOA
VB(&A
prisms FOB )
VAOB 4AZB RA,B
VA(2)B
tRA',B
RA,-
commute.
VA'(&B
t
VB(&A
G.ZV
BOF
)
t
RA'
4BOA VBOA' BOd
,
B
RA,B
t
VB(&A
AEF )
#AZA VA(&B' AW FOA
t
RA BI
) VB'(DA
4,XSA GO
,
2'
M).
258
are
4.
Monoidal
and monoidal
2-categories
Therefore, RA,B together pseudonatural transformation. of V-functors: equalities,
commutative.
1, RA,G
---
I is
:
Wehave
transformations
RFB
RA,B(&C
(VAG)BOC
vB(8)A(&C
RAOB,C
(V A(8)B(&C
VAOCOBRA,CZB VCOAOB
verify
from the left
modifications
RA,B gives
To check that
need to
identity
with
a
They give identity
tive
categories
a
following
the
braiding
hand side to the
right 2-category
in the monoidal
hand side. V-Cat-mod
we
axioms:
S 1) For any V-functors cube
F
VA
:
_+
VA',
FOB'
RA,B
FOB
there
t
is
a
commuta-
VA(&B
VA'(&B'
RA1,B
RAI,B'
BOF:
vBOA
RA,B1
VB'
G : VB
VAOB
VA(&B'
vB(9)C(9)A
SGJ,
VB'(&A The walls
are
for any M E for
M1 is determined
The top
commutative.
VA(&B.
=
a
2-morphism.
Weuse the notations
RA,BM
by
VB(&A'
).
B'123F
similar
=
(M,
is determined
of 7).
M0 B 0 A
The bottom
-23)
+MOAOB-4M)
diagram
M1 0 Q (9 S
GM' (& S
GFO'C>
1 GFM
tOG,.F
(23)t M10S(&Q Pasting the above diagram 1,
we obtain
S2) square
by Diagram (4.3.5) 2-morphism. written
and
FM'o Q
Diagram (4.3.5)
NG,F (MI) ONF,G(M)
We have to check that
and
(M)
FGM'
taking
into
account
that
(23)2
of the cube. 1FGM. This is the commutativity 1. In other words, that the following ZFRB,C =
=
commutes:
VA(&BOC AORB,C,
-17-AOC(DB V
t
tFOCEB
FSBOC
VA'(DB,&C
A'ORB,
S,
-I;rA'(&C(DB "
Denote D
algebra
B 0 C. This
=
have
RB'CD
takes
the form
acts
the natural
D with
=
M& FA (&
categories
Abelian
4.3
in ME
VAOBOCviax:
FM0
DFx
(23)t
our
259
2-category
monoidal
a
of C 0 B. In
action
DFc,014
form
-+ M.We MOD case
Diagram (4.3.5)
FM
)
OF,RB,C M(2) FA
Mo D (2) FA
Fa
FM
)
right vertical arrow is the identity map since this diagram diagram, expressing the fact that Mis an A 0 D-module. L we check that S3) Similarly ORA,B,G The
F-image of
is the
a
`
Other or
follow
numerous
axioms of
symmetric category
in
strict
a
2-category
monoidal
RBA o RA,B
Moreover,
from the above.
semistrict
braided
a
are
obvious
1, hence V-Cat-mod
=
is
a
sense.
symmetric group SN in the tensor VA. Namely, for any permutation power 0N __+ which equals VP-1 : VA ANN ANN, _+ is a functor R, : a E SN there 1 A&N the of A(&N -+ AON is the automorphism algebra VAON, where P, : R,(,, and given by the action of the permutation a-' E SN. Clearly, R, o R, In
particular, ANN
there
is
ON
VA
=
of the
action
strict
a
A
of the category
=
=
R(i,i+,)
A(&i-1
=
Now we
algebras into
can
is built
RA,A
from k-vect
data in
the
2-category
in the
of modules
dimensional
way as V-Cat-mod is built This makes it a monoidal 2-category
same
is not strict.
aA,B,C,D)-
the monoidal
finite
over
from V,
comparison with V-Cat-mod (some
tagon 2-morphism. To define
0 A
that
that k-vect
account
non-trivial
0
notice
in AbCat'
2-structure
are
(resp.
trivial,
still
V-Cat) limit
with like
we
taking more
the pen-
recall
of its
k-
that
subcate-
from Ob AbCat' (resp. V-Cat) is an inductive mod-A (resp. VA) For categories gories equivalent to mod-A (resp. VA) [Del9l]. the tensor product exists and is again a category of that type [Del9 1]. In general case Deligne shows the existence of the tensor product of inductive limits from AbCat' by proving that the inductive limit of tensor products satisfies the properties of the definition [Del9l]. Applying this procedure to V-Cat' and V-Cat-mod we get a in structure that is, a monoidal 2-category weak version of a monoidal 2-category, the Since limit. 2-category V-Cat'. It depends on the choice of the above inductive in 2-category structure to V-Cae we also get a monoidal AbCat' is equivalent any category
AbCat'. Since V-Cat-mod are
symmetric In order
remark that the 2-functor
in
a
weak sense,
to define
t-+
COP.
2-categories instance, [Lyu991.
symmetric, see,
symmetric isomorphic
for
the
monoidal
the
AbCat is
C
strictly
is
to
AbCat,
2-structure
and the
AbCat' in
AbCat,
isomorphism.
and V-Cae
simply given by
we
is
5. Coends and construction
of
Hopf algebras
previous chapter we described properties of Hopf algebras in braided tensor class of examples of such Now we are going to construct an important categories. Hopf algebras. They are built as special inductive limits, namely, coends. of a large class of coends in abelian tensor categories Webegin with a discussion that are determined by an expression with operations 0, C9, Y, etc. Wecompute several coends and establish canonical isomorphisms among them. We review how to construct in any rigid, abelian braided category a natural In the
Hopf algebra using happens precisely dimensional
as
well
=
f
XEC
as
over
of F
co-actions
of F
as an
on
inductive
coend exists
if this
to the
category
in C. This
of finite
k-algebra. The structure are obtained Hopf-pairing, to special tangles due to the
associative
C and
a
natural
associated
transformations
as an
object,
is, equivalent
that
dimensional
finite
a
from natural
property
X 0 Xv
when C is bounded,
modules
morphisms very explicitly universal
F
limit.
special Hopf pairing, w : F (9 F -+ 1 for such a Hopf algebra F. Whenthis pairing is non-degenerate the bounded braided category C is called modular. From the modularity of integrals of F is isomorphic to we deduce that the object the unit object. Furthermore, the integrals for F are two-sided. Another fact closely related to one of the basic topological equivalences arises for the natural transforfunctor mation of the identity in Homc(F, 1). of C corresponding to the integral We use the theory of squared Hopf algebras to show that its image is of the form 1 ED ED 1 for all objects in C. Weconstruct
a
.
...
5.1 In set
The coend
an
to
rem
extended two
1.8.6]
purposes as an
this
TQFT formalism is mapped to
that the can
object
a
circles,
object representing
be written
as
F
of C Z C if and
=
cylinder, object
an
f
only
this
XEC
viewed
as a
in C & C. It
cylinder
is
a
cobordism is shown in
special
coend, which for
X 0 X1. Furthermore,
if C is bounded.
That is
section.
T. Kerler, V.V. Lyubashenko: LNM 1765, pp. 261 - 282, 2001 © Springer-Verlag Berlin Heidelberg 2001
from the empty [Lyu99, Theo-
why
this we
coend will
study
our
exist
coends in
5.
262
Coends and construction
General
5.1.1
Let C be
P
x
coends
k-linear
a
Hopf algebras
of
abelian
POP -+ C is defined
with
category in [Mac881
as an
B(X,Y)
B(X,X) f
where
X
:
Y
B(X, X)
:
The coend C of
a
bifunctor
B
of C, which is the inductive
limit
diagram
of the
ix
length. object
runs
MorP.
over
B(YY), C is
That is,
equipped with a morphism diagram
C E C for each X E Ob P, the square in the
Y)
B (X,
B (Y,
Y)
tiy
B(X,f)t B (X,
X)
C
(5.1.1)
-
D is
: any f The last condition
for
commutative
objects.
such
jX gles
B (X,
:
X)
X
reads: exists
we can
(D-
B (X,
f:X-+YEMor
exterior
unique h
a
essentially
C
:
universal
commutes -+
between
for
making
D
system
a
the trian-
Let
us
case
=
pX
say that the sequence in the
Y)
B(X,f)-B(
the
particular
:
Mac Lane
uniquely
f
diagram (5. )
[Mac88]
through
object
of
e.
generally,
More
Let p : P -+ V be V C9 V and let B :
=
C
0
it exists from
functor
a
P
x
POP
an
-+
C,
1.
XET'
PX Z (PX)V. 1)
for
the definition
of
we
as j immediately :
B
a
j )
bifunctor
dinatural
:
"
find
D.
D to
and its i
an
transfor-
the value
transformation
) h
dinatural
as a
with
transformation
B
C
(5.1.2)
expressed
be also
dinatural
transformation
the coend
definition
can
C, where C is the constant
any dinatural
From this
-
B(X, X) 2%C -+
category (pY) v. The coend is denoted
The coend of B is the universal
particular,
Ind
Z
*9
B
as an
case.
C
P, let
The commutative i
cocompletion
(D
the coend exists
F
mation
X).
P.
consider small
Y)
B (X,
XEObP
small
essentially
XEP
P
So in this
is exact.
B (X,
f
The coend C is denoted
If P is small,
ties.)
the
if
commutative.
Notation.
for
-+
D, then there
-+
Y E Mor P, and C is
:
B
object
C. (See proper-
"
)
C. In
D factorizes
The coend
5.1
A be
-+
a
Then there
exist.
X
f
and
natural
is
B(X, X)
Let 7 that the coends
assume
XEC
f
1YX,X:
IX.7.1).
Proposition
(Mac Lane [Mac88] transformation, a unique morphism
5.1.1
Proposition COP x C
B'
B
:
of B and B'
XEC
f
-+
263
B(X,X)
such that B (X,
^tx,x
X)
B(X, X)
---+
I
ixI XEC
f
X)
f
X
XEC
^tX1_X4
B,
(X, X)
any X E Ob C.
commutesfor
Multiple bini
B (X,
if X
coends
be
can
We will
theorem".
computed consecutively
have to
use
as
them,
stated
dealing
when
following
in the
with
"Fu-
with
surfaces
many
holes.
COP x C
f f
X X
such
functor,
The double coend
PEP(fXEC B(P,
f
thatfor
Then it determines
exists.
Q, X, X).
P, X,
X))
a
f(PX)EP
P
functor x
C
B (P,
P
x
(P, Q)
POP -+ C,
P, X, X) and the iterated
simultaneously.
exist
x
x
They
are
isomorphic
and
0 satisfies
isomorphism
the
a
Q, X, X)
B (P,
coend
A be
-+
B (P,
IX.8). Let B : PIP Proposition any pair P, Q E P the coend
(Mac Lane [Mac88]
5.1.2
Proposition
(PX)EPXC
B (P,
B (P,
P, X, X)
P, X, X)
t0 f 5.1.2
A
XEC
particular
dimensional.
finite
Weneed the notion
Deligne
and Milne
functor and
f
T :
:
X
A -+
PEP
k-linear Let F
of
[DM82]).
a
:
C
such that
categories, -+
tensor
A be
a
product
The easiest
k-linear
B (P,
X))
P, X,
of
an
k)
the
k-spaces
Homc(X, Y)
functor.
object
way to introduce
k-vect -+ A, such that T(X, Y, where k is the field viewed
x
(fXEC
coend
Let C and A be additive are
f
P, X, X) 2 4
B (P,
=
as a
with
a
vector
it is to choose
X, T(f,
A)
one-dimensional
=
Af
space (e.g. k-bilinear
a
:
X
vector
-+
Y
space.
264
Coends and construction
5.
If V is any finite ei : k -+ V, pi
dimensional :
V
-+
k,
of
Hopf algebras
vector
it has
space,
basis,
a
is,
that
a
system of maps
1 < i < n, such that
pj
e.,
o
Eei
Jij,
--
opi
(5.1.3)
11V.
=
i
morphisms X T(X, k) T(X, V) and T(X, V) the thus same relations, obey making T(X, V) into a direct T(X, k) that T exist and they are all isosuch fanctors sum (DnX. It follows immediately define Now the of tensor we an product morphic. object X with a vector space V Hence, there =
are
=
X, which
as
X0 V
=
T(X, V),
f
0 g
T(f, g).
=
just another notation for the functor T. this to V us apply Homc(X, Z), with X, Z E Ob C. Let ei : k and Homc(X, Z) Homc(X, Z) -+ k be its basis. For any linear map g : k pi denote Homc(X, Z) g (1) : X -+ Z E C, for any morphism f : X -+ Z denote the linear map 1 k -+ : Homc(XI, Z) f f So it is
Let
=
=
Lemma5.1.3.
Let n
evx
=
E j=1
Then the
following
(FX
0
Hom(X, Z) MFX 0
diagram
commutativefor
is
any
FXok
In
particular,
evX does not
Proof. By linearity Homc(X, Z), and
depend
on
Ae morphisms evX
(see diagram (5. 1. 1)).
f
:
X
-+
evx
:
X
FZ). -+
Z
FZ
of basis.
the choice
that
assume
transformation Let
morphism f
-f%
f
=
ii
:
X
-+
Z, f
=
ei
:
k
(5.1.3).
Lemma5.1.4.
Proof.
FX
tFf
Hom(X, Z)
we can
use
=
FX
I(Dft FX 0
k
Y and g
:
Y
-+
:
FX 0
Z be
Hom(X, Z)
morphisms
--+
FZ, define
in C. Then in the
a
dinatural
diagram
The coend
5.1
IOHom(f,Z)
Hom(Y, Z)
FX 0
FX &
)
265
Hom(X, Z)
1
FX
FX (& k
FfOlt
Ff(&l
Ff
all
0_
'my
Hom(Y, Z) in FX 0 k and
paths starting
evx
FY
FYOk-
FY 0
t
ending
F7
.
in FZ
equal. Hence,
are
the exterior
com-
mutes.
The system (FZ, evx) is 5.1.5. Proposition the (X, Y) -+ FX 0 Hom(Y, Z). bifunctor of
Proof.
Let ix
:
functor
the constant
n
Hom(X, Z)
FX (9
=
P be another
X(:c
f
the coend
FX 0
dinatural
Hom(X, Z)
transformation
to
P. Set
(FZ
FZ 0 k
=
10=14
Hom(Z, Z) -1 4
FZ 0
P).
Wehave to check that
ix Set Y
=
Z, g
Z E C. In the
FX 0
(FX
=
0
Hom(X, Z) -!14
following
1OHom(f,Z
Ff(&1t
(FX
0 k
This
implies
(5.1.4).
f
:
X
Ff
t
ell
Hom(X, Z)
+
ix
I
FX (9 =
(5.1.4)
iz
from FX (D k and
10=44
FX F
Ff
Hom(Z, Z)
paths starting
P).
FX 0
-
FZOk
all
)
diagram
Hom(Z, Z)
Ff(&1
r'
lemma and take any
11Z in the diagram of the previous
=
FXOk
FZ 0
FZ
ending
Hom(X, Z) --!L+
(FX
0 k
10=f
in P
are
equal.
In
P
particular,
P)
FX 0
Hom(X, Z) 114
FZ
P).
266
Coends and construction
5.
5.1.3
Coends for bounded
categories
A set S C ObC p-generates
5.1.6.
Definition
hX
epimorphism S E S, f : S
an
ENEA --r>
:
X E C there
f by S also
Wedenote Let B
B'
:
S
:
C
SIP
x
Proposition B' exist
the full
COP -+ A be
-+
A its restriction
5.1.7.
If
a
an
full
Without
bifunctor
S as its
objects.
of
set
in each variable.
exact
S p-generates
and the canonical
XES
B'(X, X)
A.
(B (X, X) of dinatural
generality
lack of
Dinat(B)
Let
Denote
-+
C, then the coends of B and
map XEC
f
B (X,
.
denote the category of dinatural i -+ V are such f
ix
-*
D
-f-+
X)
r
is full
:
(B)
Dinat
image of ip from the diagram
Dinat
-+
and faithful
in the
the
since
B(P, P)
:
B (P,
M)
r
4'
S Using a resolution S, P E S we find a unique following diagram commutative B (S,
M)
B(1,p)I P)
admits
S)
an
of im :
P
i
:
B
D,
VX
A that
E
=
inverse.
B(M, M)
:
---t>
M, P
-+
D is
E S. This
M)
)
D
bijective
is
P
-!-+
arrow
im
M)
B (P,
on
objects.
M -+ 0 of :
B (M,
an
M)
B (P,
P)
S
Let V
arbitrary -+
B(M, M)
3!
im
t
B (S,
D'
-*
B(1,p)J
B (S, B(1,3)
(B')
image
B (M,
P)
the functor
with
D
tim
B (P,
given.
transformations :
D for p
-+
B(1,p)f Now we show that
that S is additive.
Y) for any X in C. Weshall prove that the categories Dinat(B) and Dinat(B) are isomorphic. Namely, the
transformations
The functorr
contained
we can assume
morphisms
Its
restriction-to-S-functor
follows
X-
'>
S.
subcategory
in A simultaneously
-
as
of C having
subcategory to
hx
ENEI Si
k -bilinear
a
f factorizes
isomorphism.
Proof. D E
--2-+
S
:
C iffor any X E ObC there exists I is finite, Si E S andfor any
X such that g, such that
exists
x
f is
Hopf algebras
of
D
:
B
object
D be
ME C
D, which makes the
0
5.1
the upper row is exact and the left column as well. In particular, 0 -+ P -+ P -+ 0, which implies ip = ip, P E S we use the resolution the that show to Wehave system im : B(M, M) -+ D is a dinatural
Indeed,
Note that
mation.
P, Q E S as
morphism f
any
M -+ N E C lifts
:
9
PI corresponding to f is a diagram with the
transforof
covers
Jq N
corresponding
from the relation
follows
B(g,l)
B (P,
B (Q,
B(g,l)
N)
B (Q,
=
B(M,NfW9B(N,N)
a0v
Q)
N)
t
=;B(1'f)
=B(l,f)
B(1,g)
to g. In-
exterior
commutative
Q)
B (P,
M
Q
f
M
deed, there
morphism
for
in
P
The relation
to a
267
The coend
Q
B(M,M)r
B(P,M)
Therefore,
D
P
B(P,P) i E
Dinat(B). P
Moreover,
set
f
=
I
M-+ Mand choose two different
:
q
r, M of M. The above diagram shows that Q the two constructed morphisms im : B(M, M) -+ D coincide. Hence, i depends and Dinat(B') are isomorphic. Finally, only on V and the categories Dinat(B) and the are initial tile discussed coends (if they exist) objects of these categories, follows. proposition
M and
resolutions
P
Corollary
5.1.8.
If the category
exact
in each variable
k-bilinear
mined by the
exact
b
(aj)jEJ
is
a
bifunctor
projective B
:
C
x
generator P, the coend C of a CIP -4 A exists and is deter-
sequence
EDjEjB(P, P) where
C has
a
basis
of
B(a,,l)-B
the vector
("L+ B(P, P) space A
=
-+
Ende P.
C -* 0,
268
5.
Corollary
Coends and construction
Hopf algebras
of
If C is semisimple if and only if J isfinite.
5.1.9.
is bounded
Note that
above
in the
with
the list
In this
case
Now we
apply
C Z COP. It follows
the
general
that
the universal
theory
then it
=
equivalent
mod-A via
to
the fanctor
to the
bifunctor
universal
0
:
C
x
COP
coend XEC
f
XOX
in C 0 COP.
Corollary
Remark 5.1.10. variable B: C
JSj}jEi of simple objects, EDjEjSj z Sv.
Homc(P, M).
M -+
exists
C is
assumption
F
well.
as
x
5.1.8
Indeed,
COP
F
C ED COP
rigid
a
bifunctors
A, where F
B(X, X)
monoidal
=
are
right to
in
each
of the
form
exact
one
Therefore
exact.
XEC
XNX.
k-linear
abelian
right
is
f
F
that
isomorphic
is
XEC
f Let C be
for
holds
any such bifunctor
category
length
with
such that
the
coend XEC
exists
in C 1Z C. Then C is
mod for
some
finite
equivalent
dimensional
1.7.5]. Wecall such C bounded. Corollary 5.1.8 we can write
of
GjEJP The exact
isomorphism perfect,
k is
fanctor as
0
-_
the functor
of tensor
(C
x
In an
PV
0
(as
unital a
certain
k-linear
category) to the category Ak-algebra A [Lyu99, Proposition F is identified
sense
with
A*. In terms
sequence
ajgl-lga
product
C0 C
a
XVzX
associative
exact
C
XEC
f
X z XV
F
---A
(9
C0
P0
,
:
C
C - L*
Pv x
C
C),
-+
-+
F
-+
C
0.
decomposes
where & is left
up to
exact.
an
Since
C is exact
5.13 (vi)]. by [Del9 1, Proposition If exactness of (5) is known (say, C is semisimple), k we can consider non-perfect Now we prove that F and Inv(as well. 0 -) obey functor analogs of relations between coev and ev, see Definition 4.1.3. to They might be called side-inverse each other. Instead of equations we get new isomorphisms for z and n, important 6)
:
-+
what follows. Lemma5.1.11 Denote by 1 N Inv, Inv 01 : C 0 C -+ C the left ([Lyu99]). with identity the functor Inv : C -+ k-vect functors obtained by multiplying in X isomorphisms Homc(1, X), 1 is the unit object. Aere arefunctorial
exact
X
(10 (Inv +-2
e)) (F
o
Z
X)
(10 (Inv
=
f f
0))
o
(1ZInV)(F12'(&X2")=(1ZInV)
YEC
0)
o
+-2 -
Z
1) (X
0
0 1)
(Xi,
(Inv
Proof. Since 10 finite C (essentially For an injective
zx
Y0
X
we
1)
1Z
1)f
YEC
f
Yz
f
(YV
(D
Y (9
Hom(Y, X)
X)
nx
=
:
ev
fZEC
(the morphism A-modules).
commute with
ev
Remark 5.1.12. side-inverse
boundedness. Remark5.1-13.
[Lyu99].
to
(X
vZ)
X0
and
Y)
ED Yv
0 Z
f
=
n are
Hom(l, Yv
Y (9
ev) yof
that
f
of left
to
X)
f (y) for-A-modules),
-+
ZEC
(X
Hom(Z, X)
5.1.5
(9
X
0
ZEC
X reduces
& Z -+
over
YEC
(Inv 01)
=
the coend
as
X reduces to
Proposition isomorphisms
It is shown in z
0
Hom(Z, X)
:
-+
")X.
isomorphisms.
YEC
Hom(l,
One deduces that Theorem 1.8.6
f
0 1)
(Inv
of
Y(&Hom(Y, X)
:
X,
(XOY)ZYV
YEC
(the morphism
X)
(9
XzYzYV (-
the first
gives
(YV
YEC
zX and nX
define
i
YEC
these functors
This
limit).
Z
(Inv
=
exact,
are
inductive
(0)
o
F1, 12)
0
object
((Inv
=
(D 0 1
e,
(19 Inv)
:
F)
-
YzYVzX
YEC
((Inv
269
The coend
5.1
f
0
OZ
z
-+
zX and nX exact functors
VZ)
0 z
ev)X
f (z) in the case isomorphisms. (see the proof of
are
[Lyu99]). It
is
Inv(-
explained (9
A side-inverse F is
one
-)
[Lyu99, equivalent unique up
Theorem 1.8.6]
in
is
is
of the
main
to
to
the existence an
that
the existence
of
of F in C 9 C, thus
a
to
isomorphism.
examples
of
squared Hopf algebras,
see
270
Coends and constmction
5.
5.2 Braided We assume be
abelian
an
In such a
that
k
End 1 is
=
finite
with
category
that is, C is
set,
of
everywhere
field. In this section C will a perfect objects and finite dimensional k-vector condition: isomorphism. classes in C form
length
of
Homc(A, B). One more technical
spaces a
algebra
function
as
Hopf algebras
of
essentially there
a case
is obtained
functor
o
the universal
coend F
of C and
C. The
-+
f SBV
B'
:
ED
words the coend F
AoBv
f OBV-A0
to
can
t
14
a
(5.2.1)
iY
L(=-C
Lv,
L 0
morphism f
be defined
via
EDLOLv
is
similarly. a cylinder, Hopf algebra in the
a
discuss
Wewill
tegrals logical special
EDiL
B, and similarly
-+
exact
F
--
F
corresponds
(D F
=
cobordism.
sequence
(5.2.2)
0.
-+
Hopf operations
in details
to
a
1 -holed
in this
section,
A 1 -holed
torus.
[BKLTOO, Yet97],
category
and
so
is F in
particular,
in
the in-
for F. Relations
later with elementary topoamong them will be identified We define modular categories for which the as the bounded ones,
moves.
Hopf pairing w : F 0 F -+ 1 is non-degenerate. between integrals and w. From the modularity we deduce is isomorphic Weprove that the integrals to the unit object.
Choose
a
full
p-generates F
=
of
subcategory
monoidal
C) lim
Also
we
that the
discuss
object
for F
are
relations
of
integrals
two-sided.
properties
General
consists
an
be defined
d.
tion
A
:
LEC
As F represents
5.2.1
here to
BCDBv
f
A
f:A-+BEC
torus
of the coend F
I
Av is the transposed
-+
for F. In the other
can
restriction-to-the-diagonal specializes
of the
t
AZAv
F
object
as an
X 0 Xv E C Z C. The
object satisfying
ASf
ft
Xv 0 X
XEC
f
coend F
a
f
X (9 Xv
by application general definition
AZBv
where
f
=
from the latter
C0 C
:
a
d [Lyu95a]
cocompletion
former
small.
exists
and Ob Co is
(P
:
D
identity
-+
C),
a
:
f
-+
Co As
an
C C such that
and two
source(f
),
=
Co is equivalent
limit
inductive
where Ob D
morphisms sref
set.
Ob Co U Mor Co and
morphisms
tgtf
:
f
to
the coend F has
-+
target(f)
C (variant: a
presenta-
morphisms
of D
Braided
5.2
given
for
is
one
f
each
an
E Mor
identity,
The functor!P
4i(M)
=
!P(f)
=
x (& YV
for
!P(tgtf)
=
f
ft
(resp. -OX) has
a
F (9 F
02 A f)
02(srcf,
02(tgtf
7
N) N)
Note also D
x
D
,Px4i
Similarly,
) one
Here
we
inductive
C)
D2
:
and
(M, target (f ))
(f, N)
(source(f
), N)
(f, N)
(target(f
), N)
,
)
N) N)
)DXD
P2 is
,P
x
43
C
x
Mv,& N (2) Nv
=
1 (9 10
=
1 (9
=
f
ft
f
(9 1
M,N
for
E
ObCO
f6r(f:X-+Y)EMorCo
(f
X
:
-+
Y)
E
MorCo
M(9 Mv 0 X 0 Xv
-+
YV
M(9 MV0 X 0 YV
-+
M0
MV0
Nv
-+
X (&
XV (2) N 0 Nv
Yv
0 N (9
X (9 Yv 0 N 0
the diagram 02
C
C
- 4
-24 C,
M0 Mv (9 X (2) Yv
(9 1 (D 1: X 0
0 10 10 1:
C
for
that
can
C with the
Nv
-+
Y0
be embedded into
same
colimit
a
F (2) F
Yv
0 N (9
Nv
diagram
bigger
=
Y0
IiM( p(2)
:
D2
!P (2) _+
presents =
functions
Braided discuss
(see [Lyu95b,
C Ob'D
of D x D
subcategory
M0 MV(&xOYV
F(D'
5.2.2
a
(M, f
M(9
x
Co
liM(4i2
limit
(M, tgtf
=
tgtf)
E Mor
commutes with
functor
inductive
(M, source(f
4'2(f,N)=XoYvoNoNv 10 10 1 (9 ft IN(Mi srcf) 452 (Mi
Y)
-4
(M, f
02:D2c,
=
this
as an
N E Ob Co. The functor
Co, M,
=
X
:
sref
(tgtf
02(M, N)
(f
ObCOc ObD
YV
morphisms
identity
of
(srcf E Mor
Y (&
of F. Here A is
(M,
f
pair.
ObCOx ObCOU ObCOx MorCO U MorCO x ObCO,
=
Mor A consists
in D at least
the
X (9 Xv
right adjoint,
present
to
us
presentation
ObD2
for
-+ -+
:
allows
to the
Yv
x o YV
(D i
Since XO-
X&
:
271
follows:
for ME
1 (2)
similar
as
M0 MV
=
This
Co. For each pair of composable morphisms composition equals to the other morphism of
D -+ C looks
:
4i(srcf)
limits.
the
so
algebra
function
lim(4i,, as a
the structure
LM94, Maj93]).
:
E),,
-+
C)
=
lim(4j(n)
:
Dn
-4
C).
Hopf algebra of the coend F
as a
Hopf algebra
in the category
C).
Coends and construction
5.
272
The
Comultiplication.
comultiplication
Indeed,
the
and ix
:
hand side
Xv & X
-+
(X the
coalgebra
right
0
are
the
equation
transformation
1)
)
dinatural
a
with
'X
FOF).
values
in F 0 F,
transformation.
equation
the 5
F
X0XV0X&XV
dinatural
(X
=
ev
XV
0
,
1).
with
transformation
multiplication
To construct
form.
following as in diagram
fh
by
determined
values
in 1. The
verified.
easily
in the
formation
by
gives
+
dinatural
a
-1:14
Xv
hand side
axioms
Multiplication. mations
gives
F is the universal
in F is determined
The counit
Indeed,
XocoevoXv
X 010xv
==
right
uniquely
in F is
A)FOF)
(XoXv-!: 4F (XOX"
Hopf algebras
of
for
Assume
we
F
we use
found
a
42
D2
again dinatural m and a
functor
transfor-
natural
trans-
C
M
D Then the system
(!P2 (U) forms
a
cone,
!P2(U)
771U) !pM(U)
'P2(h)t
MV)!PM(V))
to construct
the
Concretely,
a
multiplication we
h
morphism
induces
system from (5.2.3)
define
.1
(U)
lim!p
-Pm(h)t
!P2(V) for any
(5.2.3)
UEObV2
diagram
because the
is commutative
IiM49
!PM (U)
m:
:
on
F. -+
I
U -+ V E D2.
morphism. D2
ZM(V)
m:
D on
liM42
By the definition
-+
objects
(P
Of HM02 the
lim.0.
Weuse this
M0
Y)
as
m(M,N)=M&N
where
M,
N Ob Co,
gt(m' f)
(M
M(f, N)
(f
(f
:
X
-+
Y)
0
f
:
M0 X
0 N: X & N E Mor
C0, and
on
Y0
N)
morphisms
we
set
morphism.
5.2
=
srcmof
M0
f
-4
M0 X
M(M, tgtf)
=
tgtMof
M0
f
-4
M0 Y
N)
=
srcf
ON
f
o N
X0 N
N)
=
t9tf
ON
f
&N
Y0 N
m(tgtf 7 n-
:
42
-+
0
o
m,
the
fn=
same
that
fn-(M, f):
M(D
Mv 0 X (9 Yv
ffi-(f,
X0
YV 0 N 0 Nv
N):
MONO (MON)v
(M
M0 X 0 X 0 N0
-+
(Y
Y)v N)v
0
0
expression
AoCoDvoBv
(AOBVOCODv
A 0 C0
morphism. of functors and the existence of mfollows. graphical notation for Fn is given below. Accordingly, the diagram below. F 0 F -+ F satisfies
Then fn is
(B
0
a
the
The m:
273
is,
ONONv MOMv
fn(M,W):
by
I
algebra
function
M(M, srcf)
m(srcf, Define
Braided
Lv
L
multiplication
Mv
M
L 0
Lv
(9
(M
o
Mv)
F&F
3t
LOct
and
MF
LOMO (LOM)v Mv
M
L
Lv
(5.2.4) The unit
is
given by the morphism 77: 1
follows
Associativity
1)3
MXIt
10 1v
following
from the
A
=
'j53
F.
of natural
transformations: 0,3
A
tOX1 1XMt
0
C2
E)2
A
C3
t1X0
j52
C2
Mt
Mt C
D to an
equation
( 123456)1256344
where the braid
L4
identity
C3
V
which reduces
---
u
E
in
o C
D
B6 o
(65432)
(432)
o
(654)
+
B6 is the positive
lifting
of
a
permutation
a
E
S6.
D)v).
Coends and construction
5.
274
Wedefine
Antipode.
morphism
a
Hopf algebras
of
-y
:
F
following
F via the
-+
commutative
dia-
gram
I(gU2
M(2) MV -C4
MV(2) M=4 MVo MvV
t
im
F
using
transformation.
F
)
composition of the morphisms verification A straightforward
that the
the fact
natural
IiMV
3-y
upper-right
in the
path is a diantipode
shows that -yF is the
of F.
diagram corresponding
The
to the
antipode
7F
:
F
-+
F is, hence,
given by
F
^YF
(5.2.5)
D F
There is
6X
a
natural
X
:
Lemma5.2.1
of F in
coaction
===
X o XV 0 X
X0 1
([Lyu95b]).
X E C
objects
The
pairing
w :
F0 F
(5.2.6)
FOX.
-+
1
,
F
F
(5.2.7)
is
Hopfpairing.
a
It
satisfies Ann w 'Ne:-f
where the
of
w,
Ker(I
which w :
The
Annieft
Hopf ideals -+
antipode
symmetric
Example 5.2.2. of isomorphism.
E)iEJXJ
0
XjV-
with
1)
wo
by Proposition
object
case
-y is
w
Annr'gh'w
=
and Ann right
of a rigid Fv) respectively.
in the
F
w
Annleft
=
respect
wo
w are
F coincide
to
(10
w
E C
the
left and right
with
in the
F 0 F
Ker(W following -+
annihilators
F
-+
vF)
and
sense
(5.2.8)
1
4.2.2. Assume that
classes
of
the
simple
category
objects
C is
semisimple with a represented by JXJ}jEJ.
finite
number
Then F
Biaided
5.2
Example
modules
H. Then F
=
multiplication coend F -
by
H*
X
f
the left
a
from the usual
Drawing
pairing
c
(FOF
by
and
w :
right
the
described,
It is
one.
e.g.,
1)
instead
of
(FOF
The
[Lyu95c].
The
with
the
single
lines
W)
FOF find
we
of H
antipode.
1).
that
F
F
F
action.
satisfies
1
=
di-
Hopf algebra
the two actions
composed
translations
-+
in
275
of finite
coadjoint
H 0 H-mod is H* with
"')
F
F
=
F0 F
FOF
double lines
in F is the
and the H-action
X 0 X1 E C 0 C
The
H-mod is the category ribbon quasitriangular
=
finite
coalgebra
translations
Lemma5.2.4.
Proof.
over
as a
in F differs
=
the category C dimensional
Assume that
5.2.3.
mensional
algebra
ftuiction
F 2 U
since the
morphism.
transposed
to
c
is
ru
U2
(5.2.8).
by equation Therefore,
the
functor. the
identity
Hom(F, 1) equipped with the 1), 0,,o E Hom(F, 1) is -
:
Hom(F, 1) O:F--+
-
:
End Id
between these spaces.
-4
1
---+
End Id
Consider Id
fanctor
:
C
-+
k-algebra
as
well.
the commutative
-
C. The space of coin-
product 0*0
convolution a
hold for F.
of Lemma4.2.15
and the conclusion
hypothesis
Endomorphisms of the identity k-algebra of endomorphisms of 1
F
F
F
F
F
F
variants
equals
This
c.
There
=
(F
are
'a
,
linear
End Id,
-4 x
=
(X 224
Hom(F, 1),
FOX-"--X
a F--+
a
=
(F
1OX
F
F
maps
X),
1)
276
Coends and construction
5.
Proposition that
([Lyu95b]).
5.2.5
inverse
are
Assume
implies
that
now
The above maps
rigidity
and
-
Inv F
Hom(1,
an
F)
1 is
-+
isomorphism
is
Due to Lemma5.2.1
,r(r,)x 0
:
Inv F,
-+
a
algebra
are
is
Non-degeneracy
0
'0)1).
algebra isomorphism.
an
:
Inv F
End Id,
-+
-+
0(a)
(F
=--
a
=
morphism fl,
such that
1),
FOF
10 F
which
are
called
monodromy
inverse
to
each other is defined
via the
F
S?r
f2Fr,X
=
F (9 X
:
Fo X
-+
(5.2.9)
form
w
The our
case
of
is
category Ann w 0.
non-degenerate:
implies
Boundedness
Side-invertibility it for
ribbon
A bounded abelian
5.2.7. the
X
categories
Modular
(PM)
tangle
X
F
Definition
W
Hom(F, 1)
---
The maps -r
isomorphisms,
Here the
5.2.3
of
spaces
01 (X 1OX %FOX-224FOX -L-* 1OX X),
=
End Id
this
([Lyu95b]).
5.2.6
algebra isomorphisms
non-degenerate.
of vector
rz:l-+F --+(F IOF- FFOF Corollary
are
-
relations.
F 0 F
w :
that there =
Hopf algebras
each other
to
Reduces to standard
Proof.
of
and
that F is an object non-degeneracy of
degenerate
form
C is called
modular,
only thing. [Lyu95b].
of
if
=
w was
of C w
(and
is the
considered
not
a
cocompletion
same
in
Weshall
not need
purposes. The
Lemma5.2.8.
Proof. Set t given by Figure
=
(F
Hopf algebra f 10V-1
)
F
5. 1. Therefore,
F in
6
)
1).
a
C satisfies
modular category
Then the
it is side-invertible
pairing and
0
=
we can
t
Int
o m:
apply
F
1.
F (9 F1 is
Lemma4.2. 11.
5.2
Braided
function
F
F
V-1
V-1
algebra
277
V-1
V-1 V-1
Fig. 5.1. Non-degeneracy
Proposition category.
theform
(a) (b)
there
(criterion
5.2.9
exists
a
f
-+
I =
(F
-
left integral-fiinctional; exists a morphism /2 functional; (d) there exists a morphism p" is
fF
F
'u'01)
F 0 F
w
)
1)
(5.2.10)
F
of C such that (5.2. 10)
I
F
of C such that
12L
F 0 1
is
a
integral-
right
1)
F OF
(5.2.11)
a
Assume that or
Ff
Now let
f
10 F
1
--
there
Proof. be
(F
left integral-functional; exists a morphism /-t" functional. is
ribbon
a
f (e)
bounded abelian
I is
there
(c)
a
equivalent:
are
thus C is modular; non-degenerate, F such that 1 of C morphism M' : -+
F&F
w :
conditions
o m
Let C be
modularity).
of
thefollowing
Then
of t
fF,
prove (b)-(e) prove that (b)
we
set
p"
F
=
1
-+
F
of C such that (5.2.11)
Then Int F
(a) holds.
we us
:
1
by
Lemma5.2.8.
4.2.14(b). using Proposition is equivalent to (e). Assuming
(1 -L+
F
F
JUL
F).
that
is
a
right
integral-
Taking p', IP (b) holds,
so
to
that
Then F
F
F
V2
fF f
F
0
Ff Of
F
G
5.
278
Coends and construction
Hopf algebras
of
is a right Similarly, tion
Thus, (e) holds. integral-functional by Lemma 5.2.4 and by (4.2.6). (e) implies (b). Conditions (b) and (e) together imply (a) by Proposi4.2.14(a). (c) is equivalent to (d) and together they imply (a). Similarly,
Proposition
5.2.10
unimodular,
that
The
Proof the
following
The Hopf algebra F in a modular category ([Lyu95b]). on F are two-sided, is, integrals-functionals fF F f: the
"
integral's property implications:
F
(4.2.3)
and the
F
F
non-degeneracy
F
of the form
F
F
w
F
F
F
F
F
F
F
By that
notations,
we
F
F
F
Corollary
=
(F
Ff
5.2.11.
of the
F1
F
integral
Ff
1
1).
The
integral-elements
)
F
have
F
the definition
fF
gives
F
F
In other
C is F
_"
Since
F r.
-+
: 4 0, in F
1 there
fF are
can
exists
F F1
a
constant
be rescaled
two-sided,
f
to F
give F
=
f
E
r,
:
k, such
r.
1.
1
F.
13raided
.2
The
Proof.
in F1
integral-elements FV
f
Hopf algebra Fv
The
Lemma5.2.12. 7
fF
0
Ae
integralsfor
F
f F.
take
we can
Fv.
F.
are
under the action
invariant
composed
integral-element
with
279
of the antipode:
f F'
=
F
integral-element.
right
coincides
to
since
FV
t
Ff =f
isomorphic
is
fF, f O'Y1
The left
Proof. a
=
t=
fF
=
two-sided
are
algebra
function
there
Therefore,
with
is
a
antipode
the
proportional
Hence, it is
constant
f,which
o
fF
:
1
f
F
)
F
o
-
F is
-+
by Corollary
1, such that
E k
c
'Y
F
to
5.2.11
Cf F.
F
From the is
proofs of
isomorphism
an
Lemmas 5.2.8
and 4.2.11
number),
(an invertible
know that
we
where t
f 1OV-1
(F
=
1 )
t
)
1
F
Therefore,
(1 is also
an
f
F
)
ly
F
t
F
)
Wehave t
isomorphism.
f
C(i t
o
:
F
F
1
t
F
)
as
)
1)
(5.2.12)
proof
graphical
following
the
shows.
XV
x
xv
x
NII V-1
V-1
F
Thus, the left with
right
the
required
hand side,
equations
Notation.
that
1
4.2.5,
thus,
a
that
7
Denote p
Notice rem so
hand side of
we can
=
is
find
that
we
F
f
o
(5.2.12) =
f
fF
F
1 (9 1
f
c
:
to
follow
1 -+ F the
00 -2-+
(1
p in such
===
a
1 &1
)
we
integral-element
F 0 F
1.
Comparing
have proven
this
one
of the
by
Theo-
from it.
1
number in k 1. Weassume that this
rescale
t
F
1
1. Hence,
=
The other
.
F =
equal
1 is
an
in F.
isomorphism
number has
a
square root in k,
way that
% F0
F
1)
=
11.
(5.2.13)
.5.
280
Coends and construction
pt
We choose
isomorphism 0 allows
The last
Fv
fF
=:
Ff
=
equation can that (5.2.13)
_=
0
(F
on
functor IrX
X K
-r(It)
:
and
u :
modular.
KX
C
O'x
Id
are
natural
act: -T
:
Inv F
-+
T(O)y In
particular,
following
F via the
Fv
.
equations:
'0)1) '0)1).
1.
=
=
X
-r(p)
the
natural
an
epimorphism
uniquely
to a functor
transformation
action
of F
FOY -24 End Id in the =
and K
:
monomorphism.
a
C
of the
-*
identity
morphismr(p)x
C, such that
7r
:
The map Id -+ K
transformations.
The natural
Remark 5.2.13.
extends
0 1
graphically.
Consider
X into
ObC extends
K
Fv
For any X E ObC decompose the
E EndId.
ObC
An
transformation
C is
Assume that
Fl.
implies
A(p) Coupon
on
'4') 1)
Fv
)
,
=(F 10F%FOF =(F I&F-f-(84FOF
be checked
Notice
5.2.4
Fv Ou
0 F (& F
integral-functional
two-sided
a
A
integral-functional
1 to be the two-sided
-+
1 (9 F
-
to fix
us
Hopf algebras
Hopf algebras
of
F
:
F'
:
of
(Y
can
=--
on an
object
FOY '60y) sense
I&Y
Y
that
10 Y
be drawn
Y of C
y-+
-00
F0 Y
act
Y).
as
X
X
X A
(5.2.14) F
fF X Lemma5.2.14. n
>, 0.
For any X E ObC the
X
X
X
object
KX is
isomorphic
to
1'
for
some
Braided
5.2
F
-+
Id endomorphism an epimorphism Decompose
Consider
Proof. 1.
the
-+
Id
7r
and
it into
'r
Id
K
-
algebra
function
281
corresponding to the integral a monomorphism a
c-L4
Id
A
.
Then F
A
of
By definition
f
(F
")I)=
)F
integral
left-right
XV
x
=
xv
x
xv
X
XV
x
atx
WX
'T TK
x
xv
KX
F
atx
epi,
are
Hence, the natural
coaction
JKX: Theorem 2.7.1
coaction
and
(X
0
(5.2.6),
Y, 6x
makes
of Theorem 2.7.1
in notations
of loc.
cit.
on
)
1
--14
X0 Y
-+
F).
F 0 KX.
2.7.2 from [Lyu991 product C 0 C. Namely,
:
F
KX equals
10 KX
Corollary 0 Y
ev
V
KX
F- 7 1
F
0 KX
of F
(5.2.6)
that
state
the category
CXC
the functor
F 0 X0
Y),
where
Jx
FC of -+
FC,
is the natural
FC into C 0 C.
KX belongs
The F-comodule
proof
(KX
in C is the tensor
F-comodules -+
=
1v
1
implies
equation
this
iKX
(X, Y)
1v
1
1
F Since 7rx and
KXV
KX
ortx
to
the
[Lyu99]
it follows
KX as
an
image ( (C that
x
On the other hand, with
by
the
10 KX. Indeed,
isomorphic
of C 0 C is
object
1).
it is identified
to
the
following
kernel
4112'2" (3 Ker(F12' (9 52" (9 2112'2" Ker(F121 772" -
=
=
-
11 0 KX2.
KX :
F12
:
F12' -+
(9
F12'
KX2,, (9
-+
F2"2"')
F12'
(9
72"
(9 KX
2...
(D
KX24)
282
5.
Coends and construction
Therefore,
KX belongs
equivalent
to k-vect
intersection
By is equivalent
to a
of
Hopf algebras
the essential result
of
intersection
Deligne [Del9l,
to 10 1 and the
C s -E n.E z c, where I
Proposition
lemma follows.
=
(1)
5.14] this essential
is
chapter
6.1 of this
In Sect.
Then
TQFT's.
proceed
we
For this lar category. coends of expressions
give
definition
precise
the
book, namely,
involving
tensor
products,
sequence of lemmas transformation a natural
a
we
extended
an
of
large
a
the
a
a
a
of such modu-
given
holes,
with
as
F and the func-
Hopf algebra
show that
TQFT and
class
TQFT from surfaces representing of
functors,
some
of
the existence
the construction
with
define
we
In
of invariants.
tor
we
of this
the main result
state
TQFT-Double Functors
of
6. Construction
cobordism
between two
functors. corresponding class of transformation to an equivalence To achieve this we first associate a natural wider it that show a on Then ambient we equivalence under depends isotopy. tangles the category of cobordisms as moves, which defines class, stable under topological of of the 3.0.6). a quotient tangles (Theorem category determines
surfaces
In the next three
phism patible
coloring.
with
surfaces
setting
In this
CZa
to a functor
_+
check that the
2-morphism First
check
Then
functor
from the
transformations
vertical
with
compatibility
we
compowithout
these results
reformulate
with
a
is
braiding.
This finishes
only
transformations
to natural
cobordisms up to
an
isomorphism,
which
of the double
the construction
is
pseudo-
(TQFT).
Weend up the of
map from the 2-moris com-
constructed
of natural
Weobtain
6.6. The map which takes we prove in Sect. with horizontal is compatible compositions
constructed
so
set
incoming and b outgoing holes corresponds with the vertical comagain the compatibility which with the horizontal the compatibility composition,
surface
a
we
holes.
colored
CZb.
More involved
position.
we
to the
compositions.
both
with for
sitions
sections
set of cobordisms
between the
which show the
with remarks
chapter
necessity
of
our
conditions
modularity.
6.1
Main result
Definition
plicative
6.1.1. on
Theorem 6.1.2.
pseudofunctor
A
objects,
TQFTis that
a
double
For any modular
multiplicative Ve
on
=
(Vc,
pseudofunctor is equivalent
S')
is, V (Lj,,+,,,
bounded abelian
P)
to
&b
-+
V (U,"
S')
:
category
(TQFT)
objects a,
V
:
eWb -+ QAbCat,
T. Kerler, V.V. Lyubashenko: LNM 1765, pp. 283 - 311, 2001 © Springer-Verlag Berlin Heidelberg 2001
QAbCat multi-
C there
0 V (U exists
,,
S').
a
double
284
TQFT-Double Functors
of
Construction
6.
of symmetric monoidal isomorphism P is obtained from the structure The isomorphism coends. canonical AbCat and isomorphisms of 2-category of C. a uses also the braiding of In the model situation, given a modular category C from V-Cat-mod, one can model the pseudofunctor achievefor where the
Ve
that,8,,,o
=
1,
Colorations,
6.2
with
a
&ib
:
-+
respect
to
QV-Cat-mod vertical
TQFT functor
not
and
in
to
chapter.
Liftings of
tangle
functors.
Our
[Res8 8], [Tur8 81 and [RT90]
which the strands
are
colored
do
need to be irreducible. To
a
sociate To
of this
is the construction
the way the tangle invariants analogous strategy difference is that the objects with The defined. are is
compositions.
subject
is the
Transformations,
Natural
step in building
The first
0)
pseudofunctor
of this
The construction
a,
V is strict
that
so
(Ve,
=
a
surface
Zg,a/b
fanctors
two
of
diagram
a
First
of all,
g,alb
tangle,
transformation
natural
of genus g with and
a
Yg,a/b,
representing
which a
to each surface
Zg,alb
are
cobordism.
between such functors there
as
outgoing
equivalent
forms
between surfaces
correspond
standard
n
to Theorem C.2.1
-Fg,a/b
these :
planar graphs represent
CNa g
(Cop)Nb
-+
k-vect,
functors
we as-
of each other. we
graphs
Gg,alb
According
holes
follows.
n
Gg,alb
and b
incoming
associate
a
Y99,alb(XI
Z
Xa ID Yb 0
Z
...
=
CD YO
...
(Xi
Invc
Y'g,alb
Tg,alb(xl
0
Yl,...,YbEC
=f
fy
(XJ
5.7].
Lef
...
ZXa
F09
o
that
implies
Tg,alb
X10
-+
F09
0
a+1
OM) (Xi
the functor
isomorphic
to
In this
Z
...
section
we
N Yb
...
0
T9,alb
Yb-
is exact.
Proposiby [Del9l, finite an essentially
is exact
C, since it is the functor
Oy2VNy2Z*
...
*'Zyb
YbVO. *OY2VOY20" *
the above functor
Weconclude
is exact.
V
f)
Invc
:
Invc
--+
an
as
(Xa(l) (9 (Xi 0
defined
'E
92,a/b-
Specifically,
This
(9
Xa
braided
FO92
0
tensor,
(&
((fa(l) 0 f) (M I IXS'},
Xj -
*
-+
strictly
YV). 1
I
rigid
J
and
hj
a
:
Yj
fa(a)) 0110 (htO(b) IY}) Invc ((f, 0 0
...
YI
-+
J
0
(9
...
fa)
we
(D
Yj a##
and
=
(6.2.2)
length arbitrary
with
category
XJ
4. 1. Furthermore,
in Section
Xf'
look for
we
YV b
permutation of labels is given by U(M) property, namely, family obeys a basic naturality :
natural
V
of C. The
morphisms fj
a
it suffices
(9Xa(a)&FO91(&Y,3(b)O-OYVI)) N
...
...
balanced,
abelian,
and coend F E C,
' gi,alb
:
dib(a, b),
Egi,alb -+ zg2,a/b, _+ 'Fg2,a/bClearly,
M:
Jrgi,alb
:
ME
given cobordism,
a
holes,
labeled
with
between those functors:
I
Here C is
for
wish to construct
surfaces
(M I 1XI M)
=
YJ Z
Y2 0
thefunctor
YbV (9
[De19 1 ], hence,
give a natural transformation family of morphisms:
Invc
0
0
Y2,...,YbEC
OXa(&FO-'Of NXaNFO'Nf
to
(1.2.2).
YJV)
(9
is exact.
transformation
objects
Y2V)
0
G)
over
YJV)
0
...
a, b > 0
If
the coend
is
YbV 0
...
0 is also exact
Yg,alb
Ybv
(6.2.1)
YbV 0
0
bounded category.
a
of k
between connected
of
0
Y2,...,YbEC
((0
The functor that
Xa
Therefore,
limit.
&
I
Xa
0
...
(D commutes with
Hence,
inductive
XIS
0
0
Let C be
Perfectness
Proof.
F O-q
0
Oa _+ Ob
:
!2--'YbEC
Lemma6.2.1.
tion
(Xi
Invc
Xa
0
Xa)
0
...
285
Liftings
and
Transformations,
Natural
Colorations,
6.2
are
as
defined
have:
h,6(1)))
010
in
that for any sequence
f)
(htb b 0
(M I JX}' fyk}) ...
(9
htl)). 1
(6.2.3)
286
6.
of TO ,Fr-Double
Construction
Functors
tangle, T E 7-glplan;* (a, b), which R2 a given coloring a morphism. in C, represents which is a composite of the natural commutativity, and rigidity morassociativity, of that C. attaches in C to an object phisms By a coloring we mean an assignment with (4.1.3). every piece of strand between extrema in T in a way that is compatible The coloring of a non-closed strand is determined by the color of one of its ends. Obof a split tangle is given exactly by a coloring of the strands serve that the coloring the since and have at we emerging top-line, only topthrough-ribbons. To represent from rg,plan;* of Fig. 4. 1. The a tangle we use the conventions R2 The transformations
ribbon
twist
constructed
are
M. The first
from
a
with
step is to associate
denoted
P)
V
T goes to the ribbon twist To a coupon with n
the to
respective natural
j-th strand. cointegral
v.
A(Al
object,
transformation
Eventually,
penetrating
vertically (9
...
(&
A of the
of the coend F
as
X
identity
we
(A,
Ende
E
Aj's
C. Here
on
0
...
defined
in Section
5.2.2.
X
morphism. of that belongs coloring of the
the
associate
0
An),
is the
choose A to be the transformation
will
we
strands
An)
obtained
Wecan write
X
X
F
where
fF
ix
X
Using the rule for the
coaction
r F we
get the expression
for
a
F X tensor
product
XOY
X
on a
=
XOY
VI F
coupon with -two entries
F
Y
I
X Y
X
from the
X
X Y
and
Trans oimations,
Nwturai
Colorations,
6.2
287
Liftings
Y
.
X
=
17
fF X Y
X Y
Similarly,
we
for coupons with more entries. of the maps in (6.2.2) we shall always color the first a exthe objects in the order X,(1), Xa(a), and the last b external
expand expressions
For the construction
ternal
strands
with
with
strands
external
tical
Y
objects,
the
strands
Y,6(1).
f XC, }
Hence,
ordered
the first
to
of
union
ver-
object
the
associate
we
XC, (1)
0
0
...
(6.2.4)
Xa (a)
Yv 0 Also, (9 Y' object IY,3}' ,8 (b) the that the with so same we object, Pk, pair associated TIP. The to in also extends object along top-strands uniquely coloring the obtain To indicated in such a pair is, thus, Pk1 (9 Pk, as object assigned (6.2.2). tensor of all strands, we form the ordered to the collection product of all of the objects that appear in (6.2.2). at any JIJ, IJ1} at the top-line In general a pair of through strands can originate In order to keep the notation and end at any other such pair at the bottom line. simple let us assume that the through strands are attached exactly to the first mpairs both at the top- and the bottom-line, i.e., the j-th pair starts at JIf, IjOJ and ends at for the general case The construction m, and -r E S,,,. 1, I I, (j), 1,0(j) }, for j the same, only that the order, in which the objects of the pairs occur in is literally at the a tensor product, is permuted. Wedenote the coloring of the through-strands j-th pair by P,(j) so that the object for the first 2m internal ribbons at the top is and to the second set of strands color
=
[P-1-1 Thus, the object
L,,,,
where
where
P70
=
3
Pj
=
balancing isomorphism.,
.
Wedenote the
as
the
PIV(1)
(9
in
object
(6.2.5)
P1_(1)
corresponding
Pv 0
Pj
if the
[P#]
denote
0
coloring
but without
:
V
P" (.)
2m strands
0
..
=
(P,#)v
[P]#
--+
is
(9
and
(6.2.5)
P,
at the bottom
through-strand
top-pairs respective permutations. of the
to the
(D
...
a
associated
'
,
.
pt#p[P] that
=
straight through-strand isomorphisms, P(Pj) : Pj
for
The
=
at the
Lj
We also
if it is twisted.
.
the
of the k-th
strands
the internal
(P,#)
PO 3
=--
=
Pjv,
0
PY for 3
Lj
a
V
twisted
define
thus,
of T
0
(g
(PI),
one.
a
canonical
(6.2.6)
by Qj, with j
2(g,
0
Pv
Pj
[P#].
set of
L,
=
=
(PI)
(D
...
[P]#
is
and
straight
-
m)
=
1,
strands
gi
in
[Q],
-
M, so
defined
288
6.
Construction
TQFT-Double
of
Functors
simplicity,
For the sake of notational
we
of T occupy the next 92 mpositions the top-line in the same order in which
also
(before
-
assume
that
splittings
the
splitting
the
ribbons
of the closed
ones)
at
they are top-line pairs is given by BJ, with j 1, m, so that [B] g2 is as in (6.2.5) the object for the respective obtain set of strands, we as a collective bottom for the we object pairs [B]#, i.e., replace Bj1 0 Bj by Bj (9 Bjv if the from the is the last n pairs a twisted one. Finally, through-strand resulting splitting of internal strands at the top-line the of closed with colors are ribbons, splittings n. 1,... CjJ With colorings given in this way we may now associate to a tangle a morphism between the objects at the top- and bottom-line by composing those associated to the singularities in the diagram. Composing the result with the isomorphisms coloring
attached
at the
at the bottom-line.
=
If the
-
.
.
,
.
=
0,#p,
ot#B]
and
defined
as
(6.2.6),
in
we,
T1(T'PJJXJJP],[Q],[B],[C]):
thus,
JX,,J
(9
find
the
[P,]
0
-+
JX}
morphism:
[Q]
[B]
0
[C]
[P#]
0
[B#]
0
0
JYj ,6}v
0
(&
I
JY}v.
(6.2.7)
For the colorings colorings. JX} and fY} (6.2.3). Using the naturality properties of the elementary morphisms we can deduce some relations for the colorings of the internal strands by "pushing" an arbitrary morphism between two colors through the diagram. For top-strands (with colors Qj or Cj) we verify what we called in Sect. 5.1.1 For example, if f : Q, is any morphism between colors then the dinaturality. following diagram commutes:
The system is natural with respect this property is expressed precisely
(2)
...
...
OIQV 10fo (2)
Here
we
relation
we
special
other find
objects that through-strands
of
....
E
)
we
0
:...
when
case,
for T1 (P)
P
straight -+
Q,
we
choose
Endc (Pv
not
let
PY 0 Pj 3
us
changed by consider
dots.
instead
0...
..
...
(D
Py
only one through strand arbitrary morphisms f : P
have
0
P)
pairs of through-strands. equation
P (9 Pv the
are
)(6.2.8)
.... ...
corresponding general situation,
For the
of the
maps
TIM(91 for
o
...
have to deal with
If
QV,
....
G..41
the
case
strands.
relation
...OPOWD...
Q,
(2)
Q1V (2)
T,(...,Pj the
by
. .t
indicated
in the
when
i.e.,
01V
to the
3
0
with
Pj
0...
color
and g
,
P in T and :
P
P,
-+
no
we
the relation 0
f)
=
(910 MINI
In the twisted
case
(6.2.9) we
find
for
TI(P)
:
PV (9
6.2
Colorations,
(910 f
T1 (P)
and
NaturaiTransformations,
Y"
91)
(9
T1 (P)
'289
Liftings
(6.2.10)
-
transNote, that if a system of morphisms, & Pv 0 P -+ Z, defines a dinatural also is the twisted for o o the then case) formation, composite & TI(P) (or &v TI(P) a
Now, the coend
ip
PI 0 P
:
F, of this
-+
the tensor
product
composite morphism
of T1
!P*
(6.2. 10). provides us
by (6.2.9) given in (5.2.2)
transformation
dinatural
of the
Let
IX,,}
:
JX} which is
now
dinaturality example,
dinatural
as
in all
(6.2.8)
in
is
Q is the color
if
[A]
a
transformation,
universal
Av,
=
(D
A,
(9
(9
...
Av,
& Ak
-+
:
with
(TP JJXJ,[P],[Q],[B],[C],JYJ)
with
denote for
us
F. Wemay then consider the natural [A] (9 1X} i[P#] 0 i[B#] 0 JY}v. This yields the
iA, by i[A]
(6.2.7)
in
as
type.
or
0
[P,]
0
FO'
0
colorings.
internal
[B] (D [C] 0 JY1'3}V FO(92-7n) 0 JY}V' (6.2.11)
[Q]
0
As
0
we
the existence
equivalent a given pair, to
of
we
discussed
of
have the
5.1.3,
in Section
to the coend.
lifting following a
For
for
factorization
T1 *:
Tf
QV (& Q (2)
(D
...
OFO
...
...
...
...
(6.2.12) for
Q.
a
in all
procedure
to
*
T1**(T'PJJXJ,JYJ) JX,J
E
Tf*(T'PJJXJ,9,...,*,JYJ):
FO'
(9
(9
FO(91 -+
natural
which is still
In summary,
JX"}
JX,}
(9
diagram is uniquely
following
[P" Q, B, C] Ofyfl I'
FO(91
(9
0
JX}
to the
respect
have the
-')
(9
Xj
FO(.92
-M) 0 FO' 0
FOm(9 FO(92-M) and
JY}V'
(6.2.13)
diagram:
JXJ
(D
[PO, B*]
0
JY}v
t1X1O'[P#,B#](8)fY1V
t
+92-m+n)
(D
Yj.
commutative
Jyj '3}V
JY, Jv
(&
ly '3}V
T'**
fX}
0
F092
(2)
JYJV (6.2.14)
given system of maps IP as in (6.2.7) the existence of such a of T1 and that !P** to the naturality is equivalent properties determined by this diagram for a choice of sufficiently large objects as
that for
with
we
0
fX- 1(9i[P-Q,B,C]
It is clear
a
that
The dots
natural
morphism that no longer depends on The lift is, of course, still are not changed. replace as usual colors and of the other colors Applying this can, therefore, again be lifted. find eliminated a morphism, TV until all colorings are we is
where T-1*
(6.2.8),
map as in
all
for
a
colors
colors.
Finally, going from
notice
that
the upper left
T1* may be included to the lower right
in the corner.
diagram (6.2.14)
as
the
arrow
290
6.
Topological
6.3
of
Construction
TQFr-Double
Functors
Invariance
verify that the maps constructed in the previous sections are intransformations moves. Thus, the natural depending topological class of in on a fact, on a diagram of a tangle equivalence of diagrams. The depend, moves TI, TD*, and TS*. An equivalence relation is generated by the topological So Theorem 3.0.6. construct cobordism. we shall class a by equivalence represents which only depend on the represe 'nted cobordism. in (6.2.2), the transformations braided tensor category. The following is immediate from the axioms of a rigid, In this
section
variant
under the
we
Tf*, and T1**
The maps T1, THO, for all colorings.
Lemma6.3.1.
through Proof.
For
given coloring
a
follows
TH-invariance
duced,
so
that
the
equation
follows
T13-invariance
follows
braided
a
rigid
=
1
identity c- Oc=1=cOcin Definition the hexagon equation
4.1.4
of
follows
TV* and T1** up to
Lemma6.3.2.
between
properties
from the
and
ev
follows
are
of the
coev
in Definition
4.1.3
braiding.
from the definition
constructed
directly
from
of
a
monoidal
category.
T1, they also depend
on
the
isotopies.
Thefollowing
equation
holds: F
F
(6.3.1)
F
Proof. with
Let
the
X 0 Xv
a
category.
T18- and T19-invariance
tangle only
TV
=
1
from relations
follows
T15-invariance
Since
moves
category.
TI4-invariance of
=
from
1
=
from the
T12-invariance
isotopy
ribbon V_ U2, in a strict equation U-2 -1 2 case the compensating morphisms u 0 are introU2 UO U-2 0 U-2 holds U2 0 anyway. -1 -1 0 0 _1
from the
In the non-strict
category.
under the
invariant
for T1 is standard:
proof
the
are
us use
the definition
integral-functional -+ F equals
A
:
F
related transformation of the coupon as the natural F -+ 1. The left hand side composed with ix
6.3
xv
X
x
Using the definition XV
x
M
fF
F x XV
ix
41
by Fig. 4.3(c) Lemma6.3.3. up to
a
sign)
and
(5.2.13).
Let C be
of
The a
required
equ,
modular category.
the two-sided
integrals
suc,
XV
Pbpological
Invariance
291
6.
292
Construction
under the
invariant
TQFT-Double Functors
of
TI8,
TH-TI5,
moves
TI9,
TD2, TD3*-TD5
TDI,
TS1 *-TS3*
and TS4.
Proof.
Wehave
suffices
TD1
already
under TI
moves
follows'from
TD3 *- and TD4*-invariance
follows
Lemma5.2.14.
from the fact that Int F
fF =fF
TD5 * -invariance
follows
from the fact
TS 1 *-invariance
follows
from
TS2*-invariance
follows
(5.2.14). from normalization equation
TS3*-invariance
follows
from Lemma6.3.2.
Finally, there
is
TS4-invariance
braiding
a
c
That is, for
(10
Y
Indeed,
01 )X
(10
c
y
Wedefine
:
C
both sides
10h,
(5.2.13).
objects
=
of the
h
(1
1 -+ X
:
h0l
(2) y
(10
c-1
y
X0 Y
:
(Inv X)
to
-+
0 Y
have
)XOY
equal
equation
=
we
Y E C
X,
c-1
C-1
Y0
)
X).
to
Y (2) 1
10h
Y0
X).E3
follows:
as
(M I f X}, JYJ)
the results
X)
X)
y 0
the map in (6.2.2)
(Lemma 5.2.8).
1
(Lemma 5.2.12).
morphisms
of these two
morphism y 0
)
c
y 0 1
Summarizing
=
equation
restrictions
arbitrary
an
of
f)
o
is proven as follows. For arbitrary -+ Y (2) X and an inverse braiding
(2) Y
by naturality
that 7
X (9 Y
Y 0 X. Wehave to show that coincide.
in Lemma6.3. 1. It
following.
and TD2-invariance
-
the invariance
proven
the
to notice
now
of this
(V (T I JX},
Invc
=
section,
and
f Y}, jT)).
justifying
(6.3.2)
the notation
in
(6.3.2),
we
have:
Proposition
6.3.4.
with
category
the cobordism
In this
section
it is
a
will
we
functor
corresponds we
ME
--+
In order
to
of the holes, we choose
that
V respects
compositions
db
a modular constructedfor Then it depends only on
b).
Surfaces
:
is
integrals.
true
Colored
-4
k-vect
for
of cobordisms
fixed
a
coloration
over
surfaces,
of the holes.
composition in the double category picture. the colorings of the holes by lifting V to a fanctor
This
Following of the form
Fun(Co', CNb). show that
we may use
to
(M X}, f Y})
using the
over
verify
'P
map f)
above,
the vertical
to
eliminate
Gob(a, b)
as
d7b(a,
Compositions
6.4
i.e.,
Suppose the
coend F
represent
V (Mi) V (M2)
=
V (MI
the fact that V does not
the cobordism.
o
M2)
compatible colorations particular split tangle contains assume that the tangle
depend
We, therefore,
for
on
the
Compositions
6.4
no
through strands,
which
can
be
arranged by applications
splitting
that the
over
of
T1SP are
Colored
Surfaces
293
of the TS3 Moves. More-
not as in the
previous sections right of the last b external strands. If we have two such tangles T?P with j the cobordisms Mj, then Tl'.P2 1, 2, representing TI'P o (T2P U 111,p) ribbon whose closure represents strands is a split M2 o Mi. Here 11,p are parallel that extend the pairs of strands, obtained from the splitting ribbons from the top-line of the diagram of T1'P to the top-line for T2P. For each of the tangles with colorings the morphisins we can construct we assume
over,
the left
to
ribbons
but to the =
Tf2
:
=
VaIOOO 0 [Q2] T11
and Here
we
IX,,,}
:
already
have
[B*]2
T'lo2
T11
=
JXaloa2 1
(9
(9
[C21
IYl}v
(2)
(TV2
0
In this
ly)3,o)32}V
(9 (9
colorings
chosen
of the internal
[Q1]. o
[B21
[Q1]
(2)
To make the colors
condition
(9
[Bi]
0
[Cl]
--+
of the external
strands
JXJ
-4
(9
JX}
strands
[B2*] [B*]
that
are
compatible we have form the composite
also
case we can
to
fy)3,}V
0
JY}V. compatible. impose the
[Bi, C1])
[Q2, B2) C21
(&
ly,810132 IV
[Bi, Ci]
(2)
-4
JX}
(2)
[B,*]
(2)
JY}v.
morphism. TV (TI.P2) associated to the composby the source objects. Since the assignment the prescriptions of elementary is local, for constructmorphisms to singularities of the diagrams ing TV (T,, ,P2) and the composite T/1o2 are the same in the interior is for compatible colorations. At the boundary between the diagrams the coloring and the morphisms of both parts are composed. continued, consistently Wenow wish to compare this to the split tangles with colors given
ite of the
Thus,
we
find:
For the
T11** T/ 2
tangles
IX,,}
:
:
JXcil
oP2)
T1 (Ti
Lemma6.4.1.
(9 oa2
TsP
we
F091,1
10
T11
02
obtain
(9
F 092,1
with
JY,,, IV 0
(9
FO92,2
92,2
=
F(391 (2) F
2
g1,1 (9
O'n2
morphisms
Foni
_+
(9
F091
2
(&
JYJV
IYO1002 IV Xa
which satisfy the relations (6.2.14) with respect ing this and Lemma6.4. 1, we can compute for composite tangle
JX}
(D
JY/ '311 V,
morphisms Tf, and T/2. morphism obtained from
to the
the
F 092,2
Usthe
294
6.
Construction
of TQFT-Double Functors
(IXI 0'[B*j (M O'tB#]
lf*(Tlo2)'P
fYJ)11(T1o2) fY}) T11 (Tf2
'9P
0 0
111
0
i(Q1]
i[Q1] ({(IXal} T 1** (IT/2 **(fXCt10C12} (S)
F0(9',2+n1))
T11** (Tf2** Thus,
we
liftings
have found
to coends are
The the
If
following
(T,'OP2)
we
apply
this
is
product
tensor
given integrals:
construction
0 IL 092,2 to
To this
apply
we can
V(TjP)
=
the result
of colored
with
colors
of the holes that
ZaX '
_+
map that
ZJXbJ b
these conventions
Proposition
6.4.3.
assigns
consists
we
are
find
The
Lifting
V(M)
In the formulation
of this
to each
of coends.
i[B,,C,])
-
Since
any
F(D(91,2+n1)).
0
by composing
to derive
section Its
are an
the
&
we
T1**
11091,2
find
same
the
:
bc.i
concisely surfaces,
(T,'OP2) 0
with
P(9ni.
immediately
composition
law
M : Za
if
introduce
the
we
Z{xl,
as
in C. The set of
object
by the cylindrical following:
mapsfrom (62.2)
to
objects
hole in Z
cobordisms
of all
more
connected
'
6.5
R** (T $P)
0
of )(T 2 P)'
in order
Gobc.l.
surfaces
gether
a
Of Y,310,62}V
V(Mj).
Wecan formulate
category
IV) 10 i[B1,C1]
0,32
in Lemma6.4.2,
)(T"P) I
Invc-functor,
the
fy,31
P(&n2 0IY131002 IV
the formula
(Tsp for
(2)
0
(6.7.5)
in
as
02
[Bi, B2])
0
product
the
to
S
M) T11
i[B1,C1]
0
2,C21 0
( T P)
0
(T12
(D'[Q2,B2,C2]
/**
now
of
0 F (8)92,1
MeI0012}
=
(fxC1100L2}
lp**(Tlo2'sp)='
morphism.
Pall
0'[Q2,B
of Tl*(TI'012) a lifting unique we conclude:
Lemma6.4.2.
i[B1,01])
(2)
fyf3l}V)TI2}
0
0'[Bl*j
[Bi, C1])
0
fy)31}V
0
(IXI
` _-
-+
Zb in
pieces
dib
in
db
to-
morphisms
such that
in W coincide.
the
With
define afunctor. --+
k-vect
Color-Independent
Natural
Transformation
of
6.4.3 the definition of the functor still depends on Proposition reflect the coloring of the holes. This formula also does not intrinsically in the colors. property that the resulting maps are natural In this section the dependence on the colorings we eliminate of the holes by V as a functor reformulating Gbb(a, b) --* Fun(Co', COb) The maps f)(M), a
particular
.
6.5
which
that
associated
are
holds
compositions
in the category
of functors
Specifically,
isomorphic
transformations
compatibility
be-
with
the
Coa
:
Eg,a/b
1+ 1-cobordism
to the
c0b,
__
functor. from
Mold
a
of
product
a
the
into
category
of functors
category
(CZb)
The
transformations.
bifunctor
natural
a
natural
as
lifting a dinatural, color-dependent Only now we lift it to an object
(6.2. 1) associated
V(Eq,alb)
*z--
following
to the
Wecan define
coend-object.
the
from
)7g,a/b is
to the
is similar on
and natural
the functor
appear
295
well.
as
morphism
to a
thus,
to a I + 1-cobordism.
The idea of the construction transformation
Transformation
to
above, will,
have constructed
we
tween functors
vertical
)
LiMag
OPP
X
CM-4
Fun
(CEa'
&
(Y1
0
Yb)
0
...
A0
X
Zb)
0
...
T Y'
Z g,alb
-+
(6.5.1) where the functor
jr(YA
'
JXJ
with
xI z
-
g,alb
is defined
...
defined
as
Wecan define
First,
set
1179,
/
a
(for
:
C
a
Z
Xa
in
(6.2.4).
0
J-1b
the formula
(JXJ
0
Fog
coend
as
follows.
Invc
+
candidate
for
a
0
IY}V)
Zi 0
Z
...
(6.5.2)
Zb
category):
strict
CNa
b
The functor
J7bo
a
by
a
C
)
JFb'
C
COIFOCO(b-l)
LL-4 C g CO(b-1)
'1700 C)
cs(co
)
Inve
CZ(b+1)
by letting
inductively
is defined
F0
0
-OF09
-1
[ COb
be the
g
CUb.
identity 0
CZ(b-1)
(6.5.3) C, and
on
gCOb
CN(b+l). (6.5.4)
Here F is the coend from
Co'.
Note that
(6.5.4)
in
tors
for
-+
Y'
thought Y. Composing
of
Z
=
F
can
be
we
0 Z as
of it here
think
be defined
can
g,a/b
F with
the functor
Y' E Y
(5.2. 1) and
functor.F(Y'Z)
the
k-vect
:
natural
as a
in the
-+
C02
.
transformations
these individual
functor
same
F
k -vect
way if
we
-+
replace
Now the transformations
between constant
transformations
we
define
a
func-
dinatural
transformation
IY For b > 1 it is often
tor,
which
we
denote
composite:
'97ggo lb
CNa
:
more
by
J7(YY)
-Fg,alb-
,q,alb
useful
Y7g7a/b*
to
use
Similar
a-1
C
-
0
isomorphic
another
to
(6.5.3)
FO-q
it
)
C
(6.5.5) version
is defined
by
of the func-
the
following
C0b.
(6.5.6)
296
6.
Construction
TQFr-Double
Functors
graph
the
It represents
of
G' g,alb
If b
=
ZO,2/1,
pants
Another
'Fg7a/i
I the functor
which
often
Yb)
9,a/b
-+
In
analogy
we
(Xi to
the top
as
'
)
'-
b
0
(6.5.5) Y row
c 71 / b b
=
Tg7l
0
0
/
0
7a / 1
J
'r' 1
version
Xa
Xa
0
F091
define
Y2 0
0
...
that makes the
0
Ybv
0...
0
transformations
the
0,
oY 1
pair (9
of
F09.
(6.5.7)
*
-
0
-
e
F1 1/0 Invc. functor also a system =
Y2V)
0
Icl
TOO'(k
Y2
0
Z
...
g,alb
Yb.
'r'
-
g,a/b,
the evaluation Yb. Clearly, evy, following diagram commutative.
(6.5.8) where
may be used to
Too, (f')
eVg,a/b
g,alb
g,alb
IYt
ti;l ng,a/b
Xg,alb X and
1
by setting
for the latter
19
...
we can =
to
jr(y)
For fixed
are
follows:
X1 0
0...
abbreviated
define
/
construct
we can
fanctors
y700, (Y2,
examples
-
extend the notation
-97g,alb,
of colored
Prominent
yields the functor.T' 0,2/1 e, and also F1 9,1/1 writing (6.5.6) is, thus, the following factorization:
way of
As for
simpler.
=
J Ta Wewill
looks
(k) k, T" (X) g,alb
(6.5.9)
c7g7a/b
may be identified
with
the coend
f
Y'
F(Y) (X). g,alb
J7(-,-) : Cop X C X C1 -+ C2 and T : C, -+ C2 T(YY) -+ T, such that for any object, X E C1, we Y have that.F(X) T(YY) (X), then.F is also the coend of 37(yY). In is the coend f order to understand this assertion let us consider a general functor more concretely G: Ci -+ C2 and a system of natural transformations Gy : Y(YY) G, which is dinatural in Y, meaning, for which In
general,
and natural
if there
are
transformations
Iy
Gv o.F(f,v) for all
V, WE C and f
functors
:
:
GwoT(wf) VW.
:
F(WV)
)G
F
The coend
ural
transformations.
that
Gy
hx
:
T(X) by
exist
h
=
is characterized This
9(X),
-+
means
consequence In our situation
this
Now the functors
categories are all to the expressions
Y
F(X) to view
*
9 such
)
else but the system of morphisms, = hX o IyX for all Y E C, which a
is then
transformation
natural
a
for each X. X
fixed
the above coend in Y, for
as a
coloring of on N-products Zx 0 Y, and the permutations and, thus, commute with the coend. Applying this
-+
we see
that
themselves.
f
and
9797a/b
g,alb
(6.5.6),
and
fy
g7.,(k)
=
Y1 (X),
follows.
as
(6.5.8),
we
have
g,a/b
(6.5.5)
and
system of transformation,
as
IT,
and
Iy,
transformations
is in fact
summarize this
us
(6.5.3),
k
coend in
their
Let
(6.5.2),
_=Y Y(Y)
corresponding
such dinat-
'.
thefiinctorsfrom
For
297
Transfonnation
Natural
object of the category of unique, natural h : T
these form
that
(k) (X),
for TI,
a
h is
us
functors,
exact
is
fixed
with
(9,
-97g,,alb where the
the initial
of
and, hence, also for the functors Lemma6.5.1.
as
that there
fact
allows
but still
coend of functors
Color-Independent
nothing C, such that GyX
X E
assumption. The of the universality
the
to
Y, and this
for
Iy
o
V(M)
Lifting
6.5
given
are
in
(6.5.9). allows
Lemma6.5.1
it
to
a
Wesuppose a system of maps With from (6.2.3). the relation
fies
T Yp,yp
Xcl
-
gi,a/b
*
color-dependent
a
by the functor Cob3 (N) natural color-independent,
is associated
cobordism,
lift
to
us
(,)z
ZXa(a)
...
I'll
k -vect
-+
Col
V(MJJXJ, JY})
given
as
in
as
(6.2.2)
(JXJ(9F091 OJY,,Iv)Y,(,)N
Invc
i-+
is
6.4.3
-Proposition
in
of functors
transformation
to
a
follows:
and satis-
OY,6(b)
...
and
(Y,Y) OT
0-1
92,a
Xa (1) this
yields
0
by
o a
b
-
0
...
action
Xa(a) on
the vector-put
V(Y) composite ity properties consequence
:
needed for
this
lift
a
leaves
V(M)
a
(9
Fog'
1
Gy
lifting us
:
0
with 0
x
to a
0
Ygl,alb
IY}V)Y3(1)
F(\YY)
0
92,a/b
a) Ygi,alb
Iy
thi-functor natural
(8)
9
...
0
Y)3(b)
transformation
natural
Jr(Y,9,YP) gi,a/b
transformation
The
(JXJ
Invc
-+
x
o
a.
V(Y)
as
has the dinatural-
in Lemma6.5.1.
As
a
transformation -
)
-Fg2,a/b
o
a.
as the system of natural maps (6.2.2). exactly the same information reflects exactly the role of a 2-morMoreover, this form of the TQFT-functor Summar1 -arrows. and vertical in a square of horizontal phism. as a transformation
It contains
ily,
we
have the
following.
298
6.
Construction
Lemma6.5.2.
Gob(a, b)
Bor
C
category
a
the squares
to
TQFr-Double
of
in
Sa which is
strictlyfunctorial
in vertical
direction.
cobordism,
P(l)
If
this
way
be obtained
(T')
V'
JXJ
:
0
t t'//'V
.
to
(M)
circles,
a
cOb
oftertical
is
a
natural
V(M)
transformations
I
and the 2-arrows
-arrows
and the 2-arrow
right diagram
in the
in the left
diagram
transformation.
between two functors
given
in
morphisms
from
F091
(9
YV
0
13(b)
*
*
-
JX}
-4
that
in
squares
CNb
cZa
is the union of
one
1, the natural
=
can
F__V+
)3
under compositions
and the
the
map from
a
-Fgl,a/b)
CZa
Sb
diagrams Sa
In the
is
have
we
Sb
)
t 1 11M t
.
above
as
AbCat,
Y-,I,a/b
Sa
Functors
V
0
Y,8(2)
0
FO92
YV 0 b
(2)
YV, 2
(6.5.10)
in the same way as V from (6.7.5) a given tangle is. Only E 7'gl (a, b) represents Tg'l' (a, b), such that T I(TI) M, where I acts on a representing tangle by adjoining another strand to the right. with f) (T). From this Thus, the morphism V (TI) 0 Y11 is immediately identified if V(M) is conjugated we easily see that by the isomorphism n : T -+ 'T' we obtain exactly VI (M). Hence, VI defines in this sense an equivalent functor. The most basic example for V' is the cobordism, B : Z0,2/1 de_+ 'E0,2/1 constructed
are
here TI
picted
is
from
tangle
a
in
=
Figure 1.8, between
in
source-holes of two external
e
i.e.,
VI
Cob3 (1)
:
morphism, Let
mations extrema
First, b + 2
=
for
describe to
connect
Fun
of the strand
or
to functors
C0 C
--+
k-vect,
V (n)
:
k-vect
-4
C0
consider
target-holes
in
(k -vect, C)
the
case
source
arrow
tangle Thus,
T'
we
at the
consists
have
(6.5.11)
U,
C, strictly
preserves
the braided
way. the construction
be
as
where
a
to itself.
The
of
a
transfor-
vertical
assignment
arrow
of local
follows: i-+
Invc (X
k
-+
F
we
of natural
strand
X0 Y
C,
and target
where
surface
target
will
:
the vertical
under-crossing.
examples also 2-category,
V (U)
we
or
cobordism.
a source
case
0
in this
selected
for
over-
In this
S2. The associated
E
a
V'(B)
-*
enlarged
the
simple
a
if it is constructed
us
is allowed
conn
of pants.
generator
with
strands
pairs
two
is the non-trivial
=
f
have cobordant
surface
respectively,
(9
Y),
(6.5.12)
Xv Z X. surfaces
with
b > 0 and
and the vertical
arrow
for the
holes
source
complicated straightforward. Wefurther
imum have
Zgi,alb
:
Coa -Fgl,alb+2
C
-
explain
Wehave to
simple
a
In the first
strands.
in the second
Z92,a/b,
to
Z 92,a/b+2
of
consists
target
parallel
case
how
min-
case we
cobordism.
a
we can
associate
tizv(U)
F92,a/b
a
a
v
COb
_
IIIV(Mn)
tiov(n)
'Fg2,a/b+2
CZa
(U))
-
-Tgl,a/b+2
(6.5.13)
CZb+2
Ygi+l,albi V'(M')
The transformation
via Theorem C.2.1.
obtained
external
ribbon
returning, by interpreting The resulting an internal top-ribbon. the
obtained as
TI), thus, represents V (M). Note that the
a
topological
The
M:
handle of that
-5'g1+1,a1b
6.6 Horizontal
of
section
from Sect.
composition the composite functors composition of surfaces the
of natural
cylindrical For this
tangle (topologically
identical
to
= Z and we set V'(M') 92,a/b, since the moves in the category
source
surface,
is
quite obvious: the cylinder in Mu, in Mas another simply reinterpreted
V(Mn)
is similar.
Compositions
and the linear
The results
defined,
T
92,a/biS
-+
T' that represents
a
surface.
The construction
previous
tangle _+
Ygl+l,alb
in
of Mis also
interpretation
and ends in the
starts
functors,
cobordism
map Mu -+ Mis well also moves in the category of T.
are
surface
Ygl,,Ib
isomorphism
natural
Ob io
In the
and
Ob
-
Webegin with
of Tu
cOa
cobordisms.
to these
tion
b vertical
ClZb+2
/-/ -/V(MU)
that
at the
arrow
set of
1"gi,a/b+2
-4
are obtained using somegeneralizations but are, in principle, in (6.5.12),
More
transformations
natural
Mu
a
M1 from
cobordism
of the form Mn
299
-ions
of the functors
the vertical
that
assume
maximum following
or
a
identity.
is the
combinations
times
Coinposit
izcjnial
IL
have replaced the vector spaces associated to surfaces by to cobordisms by natural transformations. maps associated of cobordisms over a 6.4 allow us assign to the composition we
of the natural and ask how over
transformations,
holes.
are
related
Furthermore,
associated
boundary components. horizontal composition
However,
transformations.
they
to the we
not
expect
to find
glued the
also form
associated
have to discuss
to two cobordisms
we can
we can
functor
the
over
same
to the
composi-
the vertical
kind of strict
one for the following, as for the vertical simple reason. In order to have functoriality of cobordisms we admitted in the class of objects only one a simple presentation surface for every isomorphism class. Thus, the objects on the lowest level can also
300
Construction
6.
be identified
TQFr-Double
of
integers,
with
we
find
a
strict
on
Co.
Still
[92, b1c]
equation
Co
([92, b1c], [gi, a/b]) with
for b >, 1
Now the functors
condition
associativity
we
associated
to
this
surface
a
"0)
a/b])
defines
were
find
a
Let
us
ofunctor, so
into
a
the category
pseudofunctor. entirely by the we have an assignment
from
we can
provided
functors,
we
(2) of a* makes T* into
surfaces
F' as coends of g,alb given The (6.5.7). composition of two functors
the functors
we can use
and the factorization
[gl, a/b]).
a pseudonly, we assume b > 1 in (6.5.8) the expressions
we want to deal with connected
Since
o
a
of
transformation
too.
that
define
([92, b1c]
of cobordisms
pseudofunctor that corresponds to the choice natural a begin by constructing isomorphism. that
suitable
one
standard
a
determined
direction genus and number of holes. Hence, in horizontal : Co -+ Fun : [g, a/b] '-+ Yg,alb- On the category at most
cannot
we
[g, a/b]
Z*
Y*
therefore
that
the canonical
as:
([92, b1c]) *Z* ([gi,
'*
:
obvious
an
obvious
(2)
may then be rewritten
Together
it is also
-
In a more formal language, as in (1. 1.5). a isomorphism., whose 2-morphisms are all identities, and the latter 2-category,
an
as a
+ 92 + b
way and inverts if we assign to
pseudojuinctor,
a
by
enumerated
are
1, a/c] for b this way by Co.
in
C0, but
b,
a -+
:
[91
=
defined
--
goes the other
which
construct
we can
we consider
(2)
Z.,
[gi, a/b]
o
category
Cob2
functor
a
functor
and choose
surface
a*
have
[g, a/b]
morphisms,
and the
the genus g, with composition Let us denote the combinatorial
Clearly,
Functors
can
be written
as
F00
Assuming Hence,
a
mined
by
-F 071/c
) 7C00 91,a/b
0
g2,blc
that C is
Y
17
-
7b/l
0
jr*,(Yb,---,y2)
...
7
Yb)
.
can
:
YbV
(iyb
The
composite
tion
from the fanctor
in
the
071/b
g7
0
C Y00 91,1/1
0
-
01) /l
F0
we
be
given
X0
as
(6.6.3)
in
Y2V 0
0
to
(9
...
Figure
in
Y2V 0
1.3
Yb
Yb)
)7%1/1.
decomposition
W,
X+Y,1/1* is deter-
(6.6.2)
the coend of the functor
YbV 0
depicted
Y2
(6.6.1)
*
)7bcf1,1/1'
071/b
g7
iy,)a(Y2,.
0
isomorphism in (6.6.2). If we apply this to
0
X
of the braids
product isomorphism.
'77(y 2,
.
0,11b
Now the
a"
07b/1
of fanctors,
composition
0
of the above
g7
:
) 707b/1
0
0 Y1 also have that 171 X'1/1 Y'1/1 Y' to composite g2+gj+b-1,a1c of the middle part of the product:
category,
isomorphism.
natural
ab The
Ycoo g2,111
transformation
natural a
strict
a
0
gives rise
_+
then
This
(6.6. 1)
Yb-
Y2
can
we
YbV 0 yields
to the natural
Y2(6.6.4)
natural
then be lifted obtain
the
braid
Y2V 0
Yb
a
(6.6.3)
transformato
define
isomorphism:
the
a-F
Q92, b1c], [gi, a/b])
=
FO71/,
o
.
,'2'1/1(a-F)
T
97c2,b/c 92 .00
obeys
form it
Due to its
0
301
b
-T9010,a/b
associativity
the basic
Compositions
Horizontal
6.6
g700
(6.6.5)
92+91+b-l,a/c-
and, thus, makes T,, into
condition
a
pseudofunctor. (6.6.5)
erly
we
could
in order
relevant
of the braid
instead
Note that
when
which
pseudo composition with the compatibility
law for
we
a
verify
from Section
cobordisms
eWb -+ QV-Cat-mod for particular, and M2 : Z9I,b1c --+ 2 In
the
becomes
Our choice
T110.
for the three-dimensional
-product
2.6. 1.
Let C be
Lemma6.6.1.
o,
isomorphism. in permuted the objects prop-
to define
have chosen any transformation obtain
to
used in (6.6.4)
we
a
pseudofunctor cobordisms c ,-composable in the
double
any two
Z9',b1c
there
is
a
sense
M,
2
:
-57o,alb
- 'A,a/b
-+
1
1
expressed by thefollow-
prism,
commutative
triple (V, ay, 1) B.2.1. of Definition
Then the
modular bounded category.
a
is
ing equation
c0b
c0b V
Y
c0a
0C
cOa
0C
C
)3
') g
+ 9 2'
ZZlo
C
+b-la/.a
ON
I cOb C V
c0a Proof
(V, in the
triple
The considered
sense
11)
a
:
is
dib
by
construction V' )
Chapter 3. Wehave to prove that The map V" sends [g, a/b] to from
--Fg,alb(Xl
0
f Since
tion are
...
composition
of two
triples:
QV-Cat-mod is the
the second
equivalence of double categories triple is a double pseudofunctor.
Xa)
Y
Invc (Xi
we
consider
homogeneous obvious.
0
the
CZC
'Fgt,+g2t+b-1,a/c
7-gl
triple
B.2. 1. The first
of Sect.
cZa
COC
'Fgt,+g2t+b-1,a/c
Let
us
Xa
0
only
also a, prove
0
(v).
0
surfaces, >, 1. The properties
connected c
F09
YbV 0
0
we assume
(i)-(iv)
Y1V) Oyl
0
b > 1, and to
and
(vi)
...
0
keep
of Definition
Ybnota-
B.2.1
302
Construction
6.
TQFT-Double Functors
of
that
Wehave shown in Lemma6.2.1 coends
application
of
Mac Lane's
over
a
(.7g,,alb
'T-7g2",blc
0
Inve (Xi 0
Xa
0
Invc (Yj
0
which
0
...
0
F091
Invc (Xi 0
Y2
0 0
0
...
Inve (Xi
0
(&
...
(9
YbV 0
V
Fog2
0
Zcv
Xa
0
FO-Q1
0
Yb
0
Yb
0
FOA 0 Zcv
0
(9
F(8'9"'
X,, 0
2
F09'
2
(9 0
0
V
OYD0
...
(9
Y>J,Z
Y
are
Yb
Y>1,Z
f
and, hence, comTherefore, repeated
limits).
Xa))
0
...
Y,Z
f -;,-4
coend.
(X1
7
,
inductive
isomorphic to multiple coends by In particular, the composition double coend. Eliminating 5.1.5 we Yj via Proposition And that is precisely the isomorphism. a:
is the
single
Y is exact
the functor
finite
for coends [Mac88].
theorem
"Fubini"
of two such functors reduce it to
C
(essentially T gives repeated coends,
mutes with
...
0
z1v)
0
Y2
0
0
z1v)
0
...
...
Y2V 0 Y2 (9
ZV
0
...
(2)
V
(9
...
ZV)
0
(D
...
zi z
...
Z,
...
(9
zi 0
0
YbV(2) ...
0
0
zC
0
ZC
Yb
(9
Z,.
lemma is reduced to commutativity of the diagram in signs are omitted. The operation MlAhM2 is described in Fig. 2.3. The lower square of the diabetween MlAhM2 and M, c M2; Y is the braid gram expresses the relationship used in the definition of the last operation Vertical (see equation (2.6.3)). compositions are denoted a-97 in the prism to prove. Thus, we have only to prove commutativity of the upper square. The proof goes as follows: the isomorphisms we present # and 0-1 in this diavia The the is tangles. isomorphism tangle gram The statement
of the
6. 1, where most (9
Fig.
U
B+
lifted
to the
Invc (U
coends, 0
Yj')
see
0
Invc (Yj -+
Another
tangle
Sect.
6.2. This
0
Invc (U
W) 0
Yjv
tangle
is
a
graphical
way to write
the
mapping
-+
0
Yj
0
W)
Inv(loev
o114)
InVC (U
0
W)
6.6
Compositions
Horizontal
303
N
ti N
Cq
IV
N
CV
6e"
N
ej
4
Fig.
6.1.
Coherence of F with horizontal
composition
304
6.
Construction
of
TQFT-Double Functors
B-
=
0
represents
Notice
a
morphism Invc (U
Invc (U
0
product
in the order
that their
V)
B+ o B
holds
in
7-gl
due to TSl*
and TS3*
endomorphisms
of the
of Sect.
moves
functor
identity
composed
(6.3.1) from Lemma 6.3.2 U, V E Ob C
tion
map in
7-gl.
Indeed,
It
is also
an
equation
This
follows
identity
is the
-
V).
0
0
between two
0 F 0
with
2.4.3.
Idc. F
from equa1. We deduce that for
-+
arbitrary
(U Therefore,
fl-1
is
from
pseudofunctor 2-pseudofunctor
Topological modularity 6.7
sary.
assumption Anyway,
On the other
logical
that it is
(V, a-7, 1) V-Cat-mod
C
=
required
-+
V(SI) by
b
:
imply
hand, the braided
structure
composite
Take the
moves
considerations
monoidal
U&
V)
the
I[.
=
product # o M, 0 M2 o exactly M1Ah M2
which is
2.3.
double
The
UOFOV
of fl-'. Hence, tangle presentation by the tangle B+ o (Mi U M2) o B-,
Proof (Theorem 6.1.2). the
B-)
B- is the
represented
Fig.
0 V
in the
-+
the
properties
is bounded abelian our
an
choice
monoidal
extra
of
Proposition
of the
B.2.3
with
AbCat.
of integrals
is natural,
and
although
not neces-
of the target double category for TQFT. of C is deduced from the topostructure
in Lemma1.3. 1. Also the
of C is
sense
QV-Cat-mod from Lemma6.6.1
assumption,
rigidity
(existence
but existence
of
duals)
of the ribbon
for the structure
Topological
6.7
imply
moves
the
modularity
305
in the category a Hopf algebra (Lemma 1.4.6). The one-holed torus has to be represented by a Hopf algebra [Ker97, BKLTOO, Yet97] in C. Our choice F for this Hopf algebra is deduced from F, assigned to a sphere with two outgoing holes, i.e., a cylinder.
is
a
corollary
-
of cobordisms
-
important
The
of the double F 0 F
imply relations relations
of this
implies
exactly
that
that
topological
moves
is,
w :
moves
elementary cobordisms. These integrals for the braided Hopf
the
between
the relations
topological
modular,
F is
how the
also
We demonstrate maps representing
between linear out to be
turn
under all
is that invariance
section
V under construction
non-degenerate.
1 is
-+
result
functor
F.
algebra
Weshall
separately
consider
First,
moves.
deal with
we
the
moves
splitting
involving
ribbons
and the other
the second type.
the isotopies, a moment, the only moves besides TD1 and TD2. If we consider are, thus, splitting-ribbons, from the corresponding of morphisms that are obtained the equations equations into the relations then TDI and TD2 translate for arbitrary of tangles colorings, for all X and Y. Here x, respectively, x 0 Y and vx x c jx,y ( x 0 Y) and vx is the ribbon twist. c )x,y cyxcx,y, if we use the isomorphism from These conditions more concisely can be restated Corollary 5.2.5 between the coinvariance of the coend and the set of natural transforx we use in the construction mations of the identity on C. For the transformation A from the following of TV we obtain corresponding diagrams:
Disregarding
which
the TS4-Move for
involve
do not
=
=
=
xv
xvo x
0 X
ixt rigidity,
(D X
(6.7.1)
evx
A
F
Now, using
)xv
the condition
TD1
for
can
be restated
that
by requiring
the
morphism.
Y)
f(x,
:
xV&x(&YV0Y
xV(& (X)(&xV(&( (X'YV) is
equal
in both
coend.
to
)
we
x)]
0
may,
Using the pairing
w :
all
(9 evy for
F0 F
-+
1,
we
AOF
paring
(6.7.2)
ulo the kernel
condition
with of
(4.2.3), w.
More
for the invariance we see
precisely,
is dinatural
when lifted
to the
find
4FOFOF
(6.7.2)
k9w
1
F
equivalent
f (X, Y)
the condition
t
,X(&Ft
evxoevx(&evy
X and Y. Since
consider
equivalently,
FOF
as an
xV0x0xV0x0YV(&Y
XV0X(&Xv0X0yv0y
[evx o(Xv
arguments,
Xvgcoevx OXOYVOY
that the
arbitrary integral-functional
under TDI for
a right image of the difference
A is
colors. on
ComF mod-
of the two maps
306
(A
Construction
6.
1)
(2)
A
o
77
-
TQFr-Double
of
A
o
F
:
Functors
F is contained
-+
ker(w).
in
Let
call
us
element
an
In the modular case, when w Coinve (F) with this property an w-cointegral. of course, that A is a right integralis non-degenerate and A 54 0, this implies, in the precise sense of Theorem 4.2.5. functional The condition in this language, if we use the for TD2 car). also be reformulated which is either induced by the coproduct of F or * on Coinve (F), multiplication of natural transformations. In summary, we have the following the multiplication
/\
E
statement.
Lemma6.7.1.
sponds
A E
to
X
AX used
-+
5.2.4
in Section
corre-
Then the maps T, T *, and Tf
5.2.5.
under
invariant
are
Suppose the transformation Coinv(F) as in Proposition
X, Y of C
TDI, that is, for all objects
1) the Coupon-Crossing
(X&Y
r-
X0Y
2
-
x0y)
)
(X0Y
=
"X (91
)
x0y),
that is, Diagram (6.7.2) if and only if A is an w-cointegral, commutes; all X E Ob C. : X -+ X for 2) the Coupon-Twist TD2, if and only if v o =
Observe that
X
if
automatically
TDI is
X
:
through 10)
X factors
-+
fulfilled
and the
one
V,
which about
is a
satisfied
sign,
Next
we
by
and its
Section
square is
already
guish
the
that
also
implied
by
split
from other
tori
tori
of TI, T1*,
in the
boundary
or
on
splitting
actually
Tf**
of M,
to the
sition
of cobordisms
usual
TQFTcomposition
construct
Tf**
over
we
with
the
with
connected
surface,
a
law to
did
we can
to this
nothing at most
tangle split
the
represents
to distin-
expect that
to M.
corresponds
in the
may also be viewed as a componamely S' x S1. Since we assume the
an
arc,
(see introduction),
situation
we have
single
Given such p
we
-
arc:
]_)
=
/,t:
1
get the map for the cobordism.
JY}, p)
to
components of the split ribbons with a morphism associD'), M : 0 -+ S' x S' (or more precisely p : D' -+ S' x S'
V
P(TJJX},
apply
of the
in the tensor
ated to the cobordism
i.e.,
composition
decides
moves.
the
these maps are functions of M*P but are ambiguous with respect The regluing of a full torus in the surgery operation, which
language of tangles
only
coupon-twist
the
the other
maps that do not depend explained, the split tangle TIP
we already anymore. manifold M'P. Since in the construction
for
to
(6-7.3)
wish to construct
As
by naturality
1,
=
Recall
4.1.
ED I then the condition
...
for TD2 reduces
=
F.
-+
with
(6.7.4) the
reglued
tori
Tf**(T"PIIXI,IYI)O(IX,}OF0910M'D(91-'+')OIY,3,})
JX,,J
0
F091 0
IY,}v
--+
JX}
0
F092
(&
fYJV.
(6.7.5)
"'fbpolup;al
6.7
As
in the notation
suggested splitting
the
it
representing
was
into
constraint with
arc
again
that
so
on
/.t
we
Lemma6.7.2.
the
independent
The
a
p
o
=
following
P(T I JX 1, JY}, p)
invariance E
/.t
invariant
are
is
i.e.,
(6.7.6)
1 -+ F.
/-t:
element
7-invariant
compose
we
where (5.2.5), easily deformed
in
or
tangle
resulting
of ti,
when
2.4.3
in TD5* in Section
diagram antipode 7. we infer -/-invariance
have the
For
morphisms
for M, which is of
choice
P is actually
immediately
be derived
can
the
7
Conversely,
suitable
a
need to prove that
the braided
used to define
an arc
have to make
still
we
307
modularity
the
tangle.
of the
One necessary the
we
the moves, and
with
consistent
imply
maves
property.
Invc (F)
(such that (6.7.6)
under the
moves
holds)
TD3 *, TD4 *, and
TD5*
Invariof (6.7.6). from the above derivation The argument for TD5* follows from the fact that /-t is in the invariance TD3 * follows specialr-move
Proof
under the
ance
Specifically,
is natural
c x,y
of F, and that
we
in both
ojx,y (A Since
we use
the
with
the
objects
and the
identity
X
if either
or
Y is 1.
have:
invariance
jA for
using CF,F(M
0
same
move TD4*
(D
Y)
=A 0 Y.
every
P)
=
splitting,
we can
consistency
deduce
Y 0 /L
tangle can be related by the moves TD3, indeed only depends on T. implies that TD4, and TD5. Lemma6.7.2, therefore, of the internal let us discuss the implications surgery moves TS 1 Following, and TS2* for the special elements A E Coinve (F) and /-t E Invc (F) For a color TS2* gives rise to the X the (reflected) tangle on the left side of the cancellation Recall
that
of the
splittings
two
same
-
X & X'
X 0 X1
morphism
1-v4 1,
lifted
to
A
:
F
-+
1. Since the ribbon
of contribution the multiplicative of a closed ribbon, actually a splittiag under TS2*, thus, k. Invariance this isolated subdiagram to V is A o p E Endc (1) imposes that this number is 1. for TS 1 *, and strands in the picture If Y is the collective object of the vertical to the left tangle X is the color of the split annulus, then the morphism associated in TS2* is
=
of TS 1 *-move is to a
Y
given by
Yoxoxv
Y 0 X & X1
Y. It lifts
morphism
-M
YOF
f
and the TS I *-move same
morphism
can
XEC
Yoxoxv requires be written
it to be as
fC-2
01
equal
to
XEC
YOX(&Xv
y.
Since
the
=
source
Y(DF
-M
of P is 1, the
Y
308
6.
(8)
Y
4
Construction
TQFT-Double Functors
of
f
F0 Y
XEC
X(&XV(DY XEC
fl(&C2 and the
graphical
presentation
Xv
X0
of the
=
F0 Y
01 -M Y,
(6.7.7)
becomes
equation
Y
0 Y
Y
Y
T Y It lifts
to
F
Y
equivalent
as an
equation
(F 20 4 Let
us
summarize
findings
our
1) the Modification (6-7.8) holds;
TS1 *1 that
2) the Cancellation 6.7.3
remarks
relate
as
F0 F
w
)
TS2 *,
are
in order
is,
A.
=
(6.7.8)
JY}, It) is constructed as P is invariant under
(1
y,
equals
1)
F
conditions
to how the
above
Then
morphism (6.7.7)
if and only if as
1)
follows.
Suppose the morphism P(TIIX}, p E Invc (F) and A E Coinvc (F).
Lemma6.7.3.
from elements
Several
Y
=
if and only
if
11.
in Lemmas 6.7.1,
6.7.2,
and
to each other:
Rem.1 If A and p fulfill the condition condition on the normalization
for
TS 1 *, then the
on
y,
provided
one
the
for TS2* is
pairing
of p with
simply itself
a
is
non-zero.
Rem.2 If tt is an integral, Rem.3 The -y-invariance Rem.4
pairing is never zero by Theorem 4.2.5. since -Y is (6.7.8) imply that A is also 7-invariant, symmetric with respect to w; see (5.2.8). of an element A E Coinvc (F) implies that it is a left W-coin7-invariance Note, that we have made no tegral if and only if it is a right w-cointegral. the latter
of p and
in Lemma6.7.1 in regard to the sidedness of the precise specification the diagram of TS 1 * and its reflection. gral. The two choices distinguish Rem.5 In terms of the natural
variance
is
equivalent
transformation
associated
to
A the condition
of
inte-
-Y-in-
to X
This
(4.1.4).
allows
us
to
slide
a
coupon
through
a
maximum
or
a
minimum, cf.
Topological
6.7
Rem.6 If A is fact
related to p by (6.7.8) w-cointegral w is a Hopf pairing implies that p
that
all
those
do not involve
that
moves
imply
enough
to construct
of the
a
in the
in the obvious,
sense
diagram and, they
of the
A and p: namely, of Lemma6.7.3. For
example,
This in the
of the Reshetikhin-
and in the construction
invariant
Hennings
on
3-manifolds.
for closed
invariants
boundary
the
constraint
dual to each other
integrals,
have to be two-sided
construction
TS 1 *, then the
required by an w-integral
is
309
modularity
the
sense.
hence, the boundary of the 3-manifold is, thus,
imply
as
an
analogous Thus,
moves
necessary assumption. under the o-move invariant require this constraints are fulfilled, all that TS3*. Weshall show next that, given previous A and that the fact also and to of C, P boundary-move is equivalent to modularity the following In order to make the arguments concise let us introduce are integrals. natural (co)pairings
for closed
Turaev invariant
This
changes
m
#:FOF In the
case
when
Molds
modulaCity
we
where A and p
are
left
or
right integrals,
these
Corollary 4.2.13. For instance, Ffl and Corollary 4.2.13 gives
Ot =OF,#
=
(6.7.9)
FoF
pairings
when /.t
from
This follows
a
also
Pt:1-L+F
and
1
F
is not
V is
that
=
f
are
F ,
A
side-invertible. =
Ff
(F"to')FOFOF'O"014FOFOF-L(24F)=(Aop).IIF '8
with
invertible
equation
constant
A
o
[t
_=
F
f
0
fF
in the
right
hand side.
we
have
(6-7.10)
Explicitly,
this
reads: F F F
F
Aotfl
I
F
=
AOP1
=
I
'Y
I
F F F
310
6.
Construction
TQFT-Double Functors
of
F
Fu
(6.7.11)
F Combined with
the
impose
to
in order
A
equation
o
it
=
6.7.4. Let C Proposition sponding braided function
be
1, this
is
precisely
under the mirror
invariance
assure
bounded ribbon
a
Hopf algebra.
the condition
version
and let
category,
we
of the TS3*
F be the
under the
Assume invariance
have to
move.
corre-
following
moves:
for
TS3 *:
some
p
:
F, A
1 -+
:
F
1
-+
(6.7. 11) holds,
or,
equivalently,
(6.7. 10)
holds; TS2*:
A
o
TS1 *:
A
=
Then
Proof
w
A=
1;
(F 124
w
F&F
1).
)
is side-invertible.
Indeed,
from
it follows
(4.2.12)
(F
0 F
10 F
'6101)
#
=
that
S'01)
w
F0 F
)
1),
where
S' compare with
=
(F
c!
1004
F0 F 0 F
F0 1
F),
(4.2. 10).
Therefore, "'01
(F where
a
=
(1
'Ot
FoF
)
FOFOF-1 4
10'Y
(XF -1 4 Since the
objects
FOF F
F and YF of C have the
10s,
'01 )
F0
XF)
same
F)
=
length,
=
F).
1,
Equivalently,
1[vF. the
epimorphism wl
is invert-
ible. Remark 6 7.5.
Assume that
(that is, w-cointegral are two-sided integrals for hence, integral-functional, 5.2. 10. by Proposition is
an
hypotheses
A 6.7.4 hold. If, in addition, Proposition or (6.7.2) commutes), then A and P F. Indeed, non-degeneracy of Wimplies that A is a right The integrals are two-sided p is a left integral-element. of
the TD1 Move holds,
6.7
hiOVeS
IMPly the modularity
311
A and it, unique two-sided integrals, wish to use exactly the coend F (and that is associated to the punctured not a quotient as in [Lyu95b]) as the object torus, that and is an essential from Proposition 6.7.4 that modularity we find as umption, the functor V. there is (up to a sign of it) only one way to construct In Section
in
a
4.2.3
we
modular BTC with
asserted
a
that
there
coend F. Thus, if
are
we
of
7. Generalization
alent
functor
defining enhanced double categories a7bn which are equivcalled arc-diagrams, These 1-arrows, Cob but have larger sets of 1-arrows.
chapter
In this
modular
a
to
we are
coends.
of certain
encode the structure
The
reason
to introduce
them is to define
hor-
possible require braidings. is The construction only canonical isomorphisms. of double the of modification tangles described in Section 7.3. It requires a category in the enhanced cobordism-category. the modifications to reflect look like in familWedescribe how the braided Hopf algebra F and its integrals monoidal of the case a semisimple category we consider iar examples. Specifically, of modules of a linear ribbon Hopf algebras as the example of the category as well in the original sense. compositions
izontal to
define
a
theory
suggests
much
structure
functors.
for honest
allows
that
more
bundles
be also be
with
combinatorial
complex-analytic
this in
one
which vectors
on
that do not
TQFT functor
The additional
minimal
Hence, it will
we
introduce
Indeed,
here
on
the context
the surfaces
is the
of conformal
field
structures, namely spaces of complex curves A detailed connections are defined. exposition on
involved
with flat
approach
to
modular functors
is
given by Kirillov
and Bakalov
[BK01].
7.1
Enhanced cobordism
categories
1-morphisms of the double category Cob were chosen as triples with marked boundary. Replacof homeomorphism of surfaces ing this double category with an equivalent one, we get other classes of horizontal is to take for such class a set of graphs which 1-morphisms. One of the possibilities CIZa _+ CZb. Many of these functors are functors encode certain combinatorially the graphs called arc-diagrams definitions. but have distinct Similarly, isomorphic, but not as plane in a certain category, with genus g and fixed a/ b are all isomorphic comgraphs. Enlarging the set of 1-morphisms we make the third (isomorphism) simpler. Namely, it does not include an ponent of the TQFT double pseudofunctor braiding isomorphism. The braiding is used only to establish equivalence explicit looks So the enhanced picture double categories. of the enhanced and the previous preferable. aesthetically The horizontal
[g, a/b]
-
classes
T. Kerler, V.V. Lyubashenko: LNM 1765, pp. 313 - 334, 2001 © Springer-Verlag Berlin Heidelberg 2001
314
Generalization
7.
7.1.1
Graphs
with
of
a
modular functor
and crossed
nested
arcs
and fixed genus given a and b (the number of incoming and outgoing tentacles) of combinatorial the set is a finite set nature set of 1-arrows g the corresponding of isomorphism classes of arc-diagrams. for [g, a/b] consists of base-line with arcs, which are halfAn arc-diagram each other, circles that start and end in points on the base line. Arcs do not intersect will do have an arc a commonendpoint. nor Further, diagram always have a 1 they which and b tentacles connected 1 tentacles are quarter-circles outgoing incoming in one endpoint. to the base-line should be divided into three intervals, such that the incoming (outThe base-line interval without the left attached tentacles to are crossings, the arcs are (right) going) A generic example is given in the following attached to the middle interval. figure. For
-
-
-
r'1_
rrr
1 a
=
4
g
=
b
5
=
(7.1.1)
5
1-arrows is a category of combinatorial The objects are posnature. 1 integers a. The composition of two graphs consists of gluing together b of the second 1 incoming tentacles outgoing tentacles of the first graphs and b graph, so that new arcs are formed. Let Arcn be that category.
Set of these
itive
-
-
7.1.2
Enhanced cobordisms
choosing one standard surface for each (g, a/b) we choose a pseudofuncSurf G -+ ZG in the following way. Wetake a closed --neighborhood oriented G of the graph G in the plane of drawing, so that the 2-dimensional manifold d with boundary is retractable to G. Then we glue together two copies of the thickened ubjoku graph 6 U -0 along the part of the boundary o9G Ljai,k 1 i" V I homeomorphic Any two choices of ZG for a given graph are almost canonically in the sense that any two homeomorphisms that fix G combinatorially are isotopic. Clearly, there is an isomorphism Instead tor
of
Arcn
-+
,
-
ZG Oh ZH canonical
Further,
equivalence
up to
an
ZGoH
isotopy.
we construct
classes
--
-
of
a
double category Obbl out of Arcn. Its 2-morphisms described by squares of the form
pairs (M,,O),
G a
al
b
M H
b,
are
where
S,, and P
E
a
Arcn,
from
Mis
a
E
Sb
315
TQFT
graphs (1-morphisms) isomorphism of the boundboundary holes of EG and ZH
G and H are
permutations,
are
V)
and
(a cobordism),
3-manifold
of enhanced
of die construcdon
Skate,',,
7.3
is
an
joining the to (1.2. 1). 0, similarly permutations according dibn the is The double category framing extension of GObn. It is defined simhas and Sect. (M,'O, n), n E Z, as 1.6.2, equivalence classes of triples ilarly to 1.6.3-1.6.6. those of Sections The 2-arrows. operations repeat literally ZH U -ZG
ary of Mwith to
U cylinders,
a
of
7.2 Formulation
and
TQFT as
a
in the extended
functor
double
case
In this
section
dibn
defined
we
state
using
of Theorem 6.1.2,
extension
an
the enhanced category
previous section. This will result in a double TQFT functor, for which the isomorphisms JG,H : 77G 0, YH -+ -77GoH are constructed canonically, Here we use the coherence proven in Theorem C.2. 1. without the use of braiding. in the
For any modular bounded abelian
Theorem 7.2.1.
multiplicative
pseudofunctor
VC
objects
on
=
(VC j, p) ,
category
C there exists
a
double
(TQFT)
bn
:
(7.2.1)
QAbCat,
_+
the structure of a symmetric isomorphisms J and P are obtainedfrom 2-category of AbCat and canonical isomorphisms of coends. In the model situation, given a modular category C from V-Cat-mod, one can strict: make the model doublefiinctor where the monoidal
VC RemarkZ2.2.
Strictness
it does not matter
modular
abelian
VC is strict
V:
7.3
J and C with
of
oEb-7b n
a
_+
0 an
pseudofunctor b pseudofunctor
Now we discuss
is
which is
for
or
one, an
not.
applications.
replacing equivalent one,
d4bn
a
TQFT
replacing
the structure
and
of Theorem 6.1.2 -+
For
Nevertheless,
we can
is obtained
and the double
a
of
achieve
as a com-
pseudofunctor
QAbCat.
Sketch of the construction
which
identities
equivalent
(7.2.2)
QV-Cat-mod.
of AbCat with
2-category (7.2. 1).
double
_+
VC is irrelevant
are
The double
Remark Z2.3.
position
category in
bn
:
of the functor
whether
monoidal
symmetric that
(VC, 1, 1)
=
precisely a positive
the structure
of the double
the enhanced
integer
of enhanced
a, to
TQFT VC (from Theorem 7.2. 1),
pseudofunctor
TQFT. We force
VC
category Coa
A vertical
the
.
to map
an
arrow
object a
E
,In &b _-1 n S,, of Cbb of
,
316
Generalization
7.
of
modular functor
a
permutation functor R,,, : CZa _+ CZa. Now we are going to assign graphs from Arcn, which are horizontal arrows of b n. is graph G E Arcn is read from left to right. More familiar interpretation
goes to the
functors A
to
obtained
if
incoming
(-
For each
'
+
2
endpoints
by
the base line
rotate
1 + -, where e > 0 is very small. Thus, the 2 do maxima. Each arc in the middle not contain top one) tentacle bottom (right or one) contains one maximum.
(left or outgoing
and each
interval
the
we
tentacles
-
graph, denoted 0, belonging to the baseline,
-)-rotated
of arcs,
k-vect-+C
functor
assigned to G. This tangle G under
is
planar
Z
C,
C Z C -4C.
6)
the
to
the
Coend
The composite responding to
assign to the local maxima and following functors:
we
nothing
is
thick
else but the functor,
P7T
the trifunctor
cor-
V-Cat from
-+
Theorem C.2. L
Vc (G)
The value of the functor
represented
be
in
a
_+
COb on
an
object X,
Z
0
...
Xa
can
Z1'--Zg,Y2--'YbEC
f
G(Xi,
G(Xi)
Xa; Z1,
1
...
Xa; Z1,
Zg; Yb,
-,
-,
X1 where the list g,
CZa
:
coend form:
g),
j
determined
built
are
elementary
from
Considering rigid
a
monoidal.
general,
in
Example
7.3. 1.
Vc (G) (Xi
0
...
0
case
category of
a
a
Zi,
(9
...
permutation if
C from AbCat
modular
may indicate
graph (7.
X4)
0
ZiV2g
0
YbV 0'
Yb,
0
V
0 Y2
of the sequence (1, j < k. To the first
1, 2, 2,
.
.
occurrence
:--
[
1.
the
we can assume
parentheses
1) corresponds
that
C from V-Cat-mod
category
for the tensor
the functor,
it is strict
it cannot
product.
representable
isomorphic
as
Z1'---'Z5,Y2,---'Y5EC
X1
(2)
...
(9
X4
0
&Z1(9Z20Z3(DZ3 VOZ2V(&Z4(DZ4VOZ1VOZ5(9Z5VOY5V& "(DYVEY20" 2 There is another
...
of
so we
To the
Y2 Z
Zjv. The position of the pair (j, j) is endpoints of j-th arc in G. Indeed, there are canonical to Vc (G) (X), depending on Zj and Yk dinaturally. They isomorphisms, listed in Appendix C. 1.
modular
In the
made strict
is
0
than k in the list, appears earlier Zj, to the second corresponds
j corresponds by the position mappings of the integrand of
i2,)
0
Y2; G)
Xa
(ii,
of indices
such that
Y2; G)
Zg; Yb,...,
-
form of the functor
VC(G):
5.
be
V (G) (Xi,
f
-
X")
-,
-
of enhanced
Sketch of the construction
7.3
=
Yl,...,YbEC
Zj EC
f
Homc 1,
G(Xl,...,
.,Zg;
X"; Z1,
f
Homc (Yi,
4
ev
-
)
Yb-
)
Yl,...,YbEC
Prop. 5.1.5
0
-
0
-
VC' (G)
f
Y1 0 Y2 Z'
Yj')
V and V is given by
0 between the functors
isomorphism.
Y2; G) 0
Yb, 0
The canonical
317
TQFT
Zj EC
!9 (X1
[fZjEC
2,...,YbEC
f
Zg; Yb
Xa; Z1
9(X11
...
)
Xa; Z1
i
...
i
Y2;G))OYNY20
....
Zg; Yb)
-
Y2,...,Yb;ZjEC
f
9(X1,
The V form is useful
-
Xa; Z1,
-,
-
defining
for
the
-
G)]
Y2;
-,
OY20* ''Nyb
Y2; G)NY2 Z
Zg; Yb,...,
-,
-
-
0 Yb
...
Z Yb
...
-
TQFT on 2-morphisms. V(G)
G
CZa
b
a
V(G)
'r-11M
a
tl4lv
R,,
cNa
b
H
(M)
Rp
V'(H)
+ a
Ob
4OG
+
COb
40H
V(H)
X1,
For fixed
V, (M) -4
f
comes
:
Inv
.
.
.
f (f
Xa the morphism
,
X10
Inv
X,,110
from the natural
V'(M I X, Y)
:
Inv(f which is determined
V'(M I X, Y)
itself
in
/3V(1))
ZVW V(b)O,
(&Xa(DG(Z,
Wv)OYbvO OYjV)OYjN
OX,,,-1aOH(W,
...
...
0 Yb
0 Yb
Xj, Yj morphism.
X1 0
...
(9
Xa
(9
G(Z, Zv)
0
V
Y (b)
W
Xa-11
by
0 Y1 0
z
(f
Inv
...
...
a
(9
-
*
(2)
Xa-la
0
tangle T, representing
is defined
below
as
H(W, Wv)
0
the cobordism
)3(l))
0 Y
0
rbv
0
...
M. The
0
V
YjV),
morphism
318
7.
of
GenevAizabon
a
modular functor
V'(M I X, Y) similarly
to
=
Invc
(P(T I
X, Y, /_t))
(6.3.2).
tangles we are dealing with here. The objects G' of G' is the arcthe morphisms of Arcn. Precisely, 7-glArc;* are bijection of added the middle number infinite between G with arcs an diagram auxiliary An and the of tentacles. of interior arcs arc right group outgoing auxiliary group and auxiliary is not placed under any other arc. Interior arcs are distinguished by the For example, to G from (7. 1. 1) r6le (and by thickness in our graphical notations). corresponds Let
us
recall
what kind
G'
of
with
in
(r - r)
=
interior,
Thus,
an
object
an integer equivalence
which
attached.
are
auxiliary to an
7-glArc;*
of
g >
untwisted,
g
=
of
of
a
arc-diagram A morphism
of the above type and T : G' -+ H' is an
of coupons and top and through strands, and the baseline, such that all tentacles
number of them,
finite
unknotted
an
arcs.
tangles, consisting of arcs, to endpoints
except
C)
auxiliary
consisting
is
r--
5
0, the number of interior
class
arcs,
r-)
and unlinked
complemented
are
annulus
with
a
coupon
with
on
a
top strand
it.
(7.3.1)
that (G', T, H') are related plane diagram of the tangle T, isotop crossing signs and strand twists, and H. The equivalence be the arc-diagram Werequire
new moves
TD3* *-TD5
*
TD5 *, TS I *-TS3 *, in which the
iliary
Let
eliminate relation
us
in the
picture
all
*, TS I
pair
*
*-TS3
all coupons, The result must
curves.
generated by
is
the
moves
TI 1 -T15,
and TS4.
* *
are
K7 /
of intervals
draw the union of Gand the
plane ignoring
closed
TS 1 * *-TS3
T18, T19, TD1, TD2, TD3* *-TD5 The
follows.
as
the
3
of the
versions
Kt 3
is
moves
TD3
with
an aux-
replaced
arc.
The map
V(T I X, Y, p)
:
f f
z
X1
0
X._11
'
''
0
0
Xa
(2)
G(Z, Zv)
X.-Ia
0
(2)
YVb) N
H(W, W')
(2)
0
...
o
Y,
YbV 0
16(1)
...
0
YJV
319
13
from
is obtained
sign X,
(9
0
...
Xi
X,,
X,,
(9
a
auxiliary are
G(Z, Zv)
X"'-11
0
big enough integer,
coupon closed
on
YVb) )3(
(9
0
product is realized
Proposition
7.3.2.
MTI5, -
TI8,
pends only The
on
proof
yV
(&
IN b)
T
81( Y'180)
...
0
...
X,,-l,,
auxiliary
that Kth
so
H(W, Wv)
0
arc,
0
Ybv
0
0
...
YjV
for each K > N, is
com-
annulus unknotted and unlinked top strand to an untwisted, of T, namely, it as in (7.3. 1). The tangle TN is the truncation a
ribbons
morphism
01Y)
6(1)
ON (9 F (g
by
formed
the Kth
auxiliary
removed for K > N. The first
the tensor
ON
1X01221,
YV
(&
...
coend
under the first
as
W
in T with
plemented with
a
(&
factorizes
morphism
This
G(Z, Zv)
(D
f Here N is
system of maps from the product
dinatural
a
the second coend.
to
arc
morphism. morphism p(&N by the tangle Tk as in (6.3.2). by
followed
the
map V'(M
The obtained
TI9, TDI,
TD2, TD3 * *-TD5
the cobordism is similar
is obtained
I X, Y) *1 TSI
*
1ON
proof
top ribbon
1ON into FON. The second
-+
is invariant
*-TS3
b(a, b).
ME
to the
*
:
and attached via insertion
of
under the
moves
and TS4. Hence, it de-
of Lemma6.3.3.
Proof (Theorem 7.2. 1). Let us first consider the model monoidal) category C from V-Cat-mod (non-strict
functor
(7.2.2)
for
a
modular
-
VC Let cuts.
a
=
(V C, j, #)
a bn
:
graph G E Arcn be decomposed composition Gk 0
Then it is the
implies
into
...
o
_+
QV-Cat-mod.
U Gk by vertical pieces G, U G2 U Arcn in Remark C.2.4 of the sense G, ...
.
that
Vc (G) Thus, the constructed
functor take 6
=
Vc (GI,)
o
o
...
of Ve is strictly compatible with the composition 11 in the 11, # pseudofunctor (VC, 6, #). Obviously, which is well-defined 7.3.2, respects by Proposition
I-morphisms. mapping of 2-morphisms, the vertical composition in the strict sense. of the isomorphism. OG : V(G) Canonicity commutative diagram for composable arc-diagrams So
we
=
the
V'(G)
o
V'(H)
=
-24 V'(G
*0Ht
V(G)
o
-+
V(H)
==
V(G
o
V(G) implies
G and H
H)
tOGoH
OG
o
Vc (Gi).
H)
the
following
320
Generalization
7.
Therefore,
of
the horizontal
the
of
composition (7.2.2). Nowlet
us
a
modular fanctor
composition
arc-diagrams
consider
in
of functors
Arcn. Thus,
V(G) we
bo-andeda-belian
a
is
strictly
constructed
category
compatible
with
the double functor
C from AbCat.
There
VA is the algebra A from V such that C is equivalent to -VA' where C' category of right A-modules. The equivalence induces a ribbon monoidal structure which is modular as well. Applying on C', the model construction to C' we get a exists
an
=
double functor
strict
VC, It
be
can
-
composed
(V C,, [' -1)
the
with
V-Cat-mod
-
=
-+
:
& bn
equivalence
of
Bounded Abelian
_
QV-Cat-mod.
2-categories
VB
k-Categories,
_+
VB'
described
in Remarks 4.3.2 and 4.3.4. It is a strict that is, compatible with 2-functor, and functors, composition of 1-morphisms (V-functors We can respectively). the symmetric monoidal structure from V-Cat-mod to bounded abelian transport via the functor and its quasi-inverse. This is one of the choices for k-categories the monoidal structure. The composition is a strict double functor (7.2. 1)
the
-
VC, This
proves
AbCat,
7.4
=
(Vc,,
[' I[)
Remark 7.2.2.
we
get
a
:
If
&7bn
_4
QBounded Abelian
arbitrary symmetric (7.2. 1) for C' and for C.
we use an
pseudofunctor
k-Categories. monoidal
structure
on
Examples
Weillustrate
our approach in two familiar is examples, where our only contribution the double category picture. Weexhibit the coends F and F, give formuexplicitly las for the integral For such /-t of F. The first example treats semisimple categories. categories we discuss the Verlinde formula and the *-structure of the invariant
part
of F. The second
example
concerns
with
the
example of Departing
an
ordinary
Hopf algebra, H, the categories of modis nothing else but H*
and ribbon structures. from quasi-triangular such algebras, we produce F, which in this case with modified The coend F is also identified with H. The 3-manmultiplication. ifold invariant obtained from this TQFT coincides in this case with the Hennings ules
over
invariant.
7.4.1
Semisimple
Reshetikhin
Modular
and Turaev [RT91] More precisely,
quantum groups.
gories
Abelian
obtained
Categories
of 3-manifolds via proposed to construct invariants ribbon catethey use certain abelian sernisimple from quantum groups at roots of unity as trace quotients. One can
7.4
Examples
321
forget about the origin of these categories and work simply with semisimple modular Weshall describe them as input data for our double functor construction. categories. Assume that C abelian semisimple modular ribbon category. Let C be a k-linear of classes of simple objects is finite. is bounded, that is, the number isomorphism. L the endomorphisms dieach for and Assume also that 1 is simple simple object of of (representatives list the S denote k. We L End vision algebra JLi}i by all simple objects. isomorphism. classes of) The coends F E C and F E Under these assumptions many formulas simplify. =
=
C 0 C take the form
57LOLv,
F=
ELOLv.
F=
LES
LES
Any morphism.
1
-+
F is
of the standard
combination
k-linear
a
morphisms
for
L E S
VL
L
U2 0
OL
:
=
coev
1
louo
L (2) vL
)
2
)
L(&Lv
i ,
F.
iL
F
The
morphisms OL form
a
Any morphism. F
-+
algebra
basis of the commutative
multiplication -+ OL represented
ring of the category C determines the morphism. k (9z Ko (C) -+ Inv F, [L] 1
can
be
Inv F. The Grothendieck
law in Inv F via the
algebra
iso-
of the
mor-
integrals
fF
-
as a
linear
combination
phisms OL where L E S. The and
Ff
.
Therefore,
:
functional,01 01 factors 01
-E4
F :
F
(F
1 satisfies
-+
through
=
L (9 L'
fF
fF )
as
-!LL+ 1, properties
(4.2.3)
of
in
Int F
-1--+
1).
objects, which are simThe morphism. g is a non-zero map between two invertible and 01 can be chosen as a Thus, g is an isomorphism, ple by our assumptions. Similar left integral-functional. reasoning proves that 01 can be chosen as a right Thus, 01 is a two-sided integral. integral-functional. The Verlinde
formula.
The number
322
Generalization
7.
of
a
modular
functor
C)
MV
diMq (M)
is called
that this
Definition
7.4.1.
k
-+
an
object
(& M
1(&UO2)MV (0
MVV
ev
M E Ob C
(Turaev [Tur94]). (The index q q-dimension in the case C Ug-mod.)
with the
=
dimq (M).
=
Introduce
the set
on
of
number coincides
diMq (MV)
Wehave
coey)MV
0
the dimension
reminds
ObC
U2M
-_
a
biadditive
of isomorphism
function of C:
of
variables
two
s
:
ObC
x
classes
L
V, VM
M
U-2 0
2
UO
SLM
VL
In S
particular,
its
restriction
(SLM)L,MES by
=
Notice
gebra where
that
Inv F n
=
=
I
The matrix
the matrix
so
It has the basis
:
Inv F
x
Inv F
is the matrix
(The Verlinde
:
S
x
S
here L and Mrun
SLM = SML,
w on
SIS
matrix
Homc(1, F).
(SLM)
Lemma7.4.2
a
of notation;
Card S. The form w
S is
to
abuse
s
F induces
is
symmetric. Let OL, L E S, hence, a
bilinear
-L+ Hom(l,
For any
simple objects.
us
Proof
The first
diMq(L),
=
simple
formula
is
VL
SLlSL,MON
straightforward.
R
W)
L E S -and any
SLMSLN-
Since
VM U2 0 E End
vL
the k-al-
Hom(i,w)
F & F)
and N of C we have SLI
consider
it is n-dimensional,
form
of the form w' in the basis
formula).
k, denoted again by over
k
objects
M
7.4
is
a
number,
we can
move
it from the second factor
in the
the first
to
Examples
323
following
computation: SLISL,M(ON M
L
2 U
0
L
M
LV
L
M
L
MV
2 0
2 0
U2 0
MV
MV V LV
N
U20
2
UO
UO
LV
LV
NV
SLMSLN-
This proves
the second formula.
of
The criterion
Proposition equivalent:
modularity In the
7.4.3.
of
Proposition
assumptions
of
5.2.9
Sect.
gives the
Z4.1
in
our
case
following
the
following.
conditions
are
(i) C is modular (w is non-degenerate); (ii) the matrix (SLM)L,MES is non-degenerate; (iii) for all L E S dim, L :A 0 and there exist numbers p' M M E S, such thatfor all L E S we have EMESSLMAIM 6LI,
-=
w' of the non(ii). Semisimplicity implies that the restriction Proof. (i) = F form Inv is also to w non-degenerate. degenerate If the dimension dimq (L) (ii) ==>. (iii). SLI of a simple object L vanishes, to the Verlinde formula implies that S2L M 0 for all M E Ob C. This contradicts =
=
the
assumption (iii) (i).
of
non-degeneracy of (SLM) simple object L E S there
For any
exists
a
number r-L
E k such
that L
L
-2
E AM
UO
MV= r-L
-
MES M
L
L
(7.4.1)
324
Generalization
7.
of
Composing (7.4. 1)(9 ILV
a
modular functor
OL and
with
eVL,
SMLAIM
KL
---:
get
we
-
dinIq
L.
MES
Since SML
SLM,
we
have KL
Composing (7.4. I)o IL
with
v
=
(dinIq L)-'JL,
eVL,
JLI.
=
get the integral
we
F
F
F
F
-OM
def
AM MES
The
modularity
criterion
of
G
Proposition
5.2.9(b)
implies
that
is used
by
side-invertible
w is
and
C is modular. Remark 7.4.4.
Property (ii) Thus, categories. categories.
modular ribbon
Remark 7.4.5.
situation,
his
Specializing
and
our
equation
7.4.3
definitions
(5.2.14)
agree
Turaev for
[Tur94]
sernisimple
and Lemma6.3.3
to the
to define
bounded
sernisimple
that property of the TS 1 (iii) is precisely the algebraic translation Section 2.3.3. Reshetikhin and Turaev find a similar linear equation
we see
Move from
from the Fenn-Rourke numbers VL E Let
Proposition
of
us
move, which
End(L)
determine
=
k
as
the coefficients
tt
involves
defined
besides
in Section
1-im of the
(SLM)
also ribbon
integral-element
E AmOm:I
=
the matrix
4.1.3.
-+
F.
MES
Weuse
equation
(6.3. 1) L
in the form
L'
L
LI
JLM
M The two-sided
M
MV
to 01. Here L and Mvary over S. The right integral A is proportional M, and vanishes otherwise. identity morphism if L Substituting of Om, we rewrite the equation as follows
hand side is the the definition
MV
=
-
L'
L'
M
325
Examples
7.4
M U20
L' For L
1
=
we
that
M9 -, 1
m
-
m
implies
Now return -4
with
(7.4.2)
to
=
M
-
j
-
L
=
11: 1
=
M. If
essentially
tells
(7.4.3)
(7.4.3)
M-+ M.
61M IM:
0. So
=
tt,
1
L'
M
get /-Im
Recall
(7.4.2)
6LM
PM
-+
(7.4.4)
1.
compose that
we
that
equation
with
coev
M' (9 Mwe obtain:
MV
Am
I
1
=
MV =
tim
MV M
with
(7.4.5)
pm = pi is fixed
The normalization
yj -
we
find
which
MV
p,
we can
write
as
M
IL21
2 U 0
1W
-
MV M
diMq(M)-
(7.4.4),
by equation
(7.4.5)
M
MV M
MV M
both sides of
Multiplying
dimq
1: (diMq (M))2. MES
MES
Hence,
(diMq (M))
(/,11)2
2)
(7.4.6)
MES
At least
for
an
Remark Z4.6
Proposition (iv)
for
algebraically Wemay add
7.4.3. each
closed
field
one more
k
we
find
equivalent
pi,
unique
condition
up to
a
sign.
the condition
to
if and
of Sect. 7.4.1 C is modular In assumptions simple L 9 1 we have EMESSLMdiMq M
=
only
0 and
list
of
if
diMq
L
:A
0.
326
Generalization
7.
of
a
modular functor
Remark 7.4. 7 As an application, of a closed 3-manifold, we compute the invariant link is a special A presented by a framed link C with n case JC I components. of a tangle with only closed internal For coloration of the link by a components. objects in C we obtain from (6.2.7) the morphism Tf(,C'P, [Q : [C] -+ 1 as well as the lifted morphism TV* (LIP, [Q : Fon -+ 1 The invariant of the link, defined in is expressed as [RT90] for the given coloration, =
.
7-01
Inserting (2-framed)
cn
...
(L)
expression
the
=
(coevc,
=
(0c,
p
ECESItCOC
=
0...
Ci
expression for
the
from [Tur94]
Complex
coincides
case.
s
different
is invertible
of
k
so
Cx
-+
Positiveness
scalar
=
for
except
reproduces
formula
Inv F possesses
that its columns number is
...
n
be
can not
Card S
=
an
additional
the formulae
(Lemma 7.4.2)
we
homomorphisms
=
Hence, all proportional. dim(C F, hence, there is
XL an
Then Inv F becomes
dim
0-+(XlM'---'XnM)-
XC=C',
to take
:
Inv F
-+
dimq (M)
for
M, the
are
real
positive
numbers
root
of the
so
right
that also A, is hand side of
involution
W)*
Inv F,
=
OLV
product
(OL I OM)
=
[RT91]
calculation
C. From the Verlinde
C-algebra
pi uniquely. in Inv F an antilinear
(Hermitian)
(OL OMI ON)
A
fixes
-*
a
diMq (Cn),rc,...Cn
from
A similar
the dimensions
number. It is natural
One can introduce
and
of the
0M -+(diMq(L))-lSLM=SLM1SLl-
Their
Now we show that
(7.4.6).
get the invariant
C-algebras
X: InvF
real
...
the formula
with
Assume that
characters.
isomorphism
diMq (Cl)
signature correction. for TQFT's.
that the commutative
The matrix
a
we
Cn
...
XL:InvF--+C,
are
(6.7.5),
C.
P)T,* ('CSP
0
E
n
Yl
conclude
into
=
P(,Csp) (1-t
This
...
coevcn)T/(L'P, [C]) E End(l) Ocn)Tf*(,C"P)
...
3-manifold:
-rwc)
factor
0
0
a
finite
=
4M)
dimensional
Hom(L (9 M, N)
L,M
commutative =
dim
E
S.
Hilbert
Hom(L, M'
0
algebra.
N)
=
Indeed,
(OL I OMON)
7.4
Examples
327
theory of finite dimensional commutative Hilbert algebras we know that idempotents in the algebra Inv F are self-adjoint (only in that case the scalar product is be that is, XL(0*) a *-morphism, can Hence, X XL(0)positive definite). In L 1 and for Therefore, SLMV/SL1 1, we have s1l particular, L-mlL-1.
From the
=
dimq (M) This proves the
Proposition Corollary
=
dirnq (Mv)
s1mv
For any ME C its dimension Wecan choose yj
Examples of Semisimple
7.4.2
=
=
Ti-m
=
diMq (M)
-
following
7.4.8. 7.4.9.
=
dinIq (M)
is
a
real
number
positive.
Categories
Modular
In their original and Turaev use as algebraic input data the paper [RT9 I] Reshetikhin theory of the quantum deformation U representation Uq (S12) of the Lie algebra sl(2, C), where q is a root of unity. They construct the invariant as a trace over Uequivariant morphisms, and prove the necessary modularity condition concerning of the braided pairing. the non-degeneracy is drawn by Turaev in his book [Tur94], The general picture where 3-maniand TQFT's in the sense of Atiyah are constructed fold invariants from sernisimple modular categories. of certain subcatHe shows how to obtain the latter as quotients of a modular Hopf algebra by the ideal of trace-negligible egories of representations morphisms. defined partition function The heuristically of the Witten-Chern-Simons theory k and with a compact, connected and simply connected gauge group G is at level S-matrix. constructed identified, data, with the rigorously e.g., via the corresponding 2" invariant,ru (M) for U Uq (9). Here q exp( ), 9 is the Lie algebra of G, =
=
and
=
number of g. h. of modular To make the verification
k+hg
is the dual Coxeter
troduce
in
[TW93] the notion
of
a
easier, Turaev and Wenzl properties quasimodular Hopf algebra, show that such
inan
algebra produces invariants of 3-manifolds similar to a modular one, and prove that Uqg at a root of unity is quasimodular for a Lie algebra g of the series A, B, C, and D. The proof uses the structure of the algebra, generated by the braiding automorphisms of tensor powers of some modules. For the A series this is the Hecke algebra; for the B, C, D series the generalized Hecke algebras appear, which are now called Birman-Wenzl-Murakami algebras, see also [BW89] and [Mur87]. The results of Andersen [And92], combined with the results of Turaev and WenzI [TW93], prove modular for 9 of series A, B, C, and D in the sense of [Tur94]. setting C does not have to be related to any Hopf algebra at all so that, and TQFT's. given such examples, our theory does in fact imply new invariants Constructions that do not use Hopf algebras can be found in the field of operator algebras, such as the invariants by Ocneanu, Evans and Kawahigashi [EK95] or by Xu [Xu], who use the theory of subfactors and algebraic quantum field theory. that
Uqg is
In
our
328
In all
lar
Generalization
7.
of
however,
cases,
there
groupoids,
whose
put for
construction.
modular
functor
the invariants
Finally,
category.
a
are
are
identified
generalized
with
via
ours
Hopf algebras,
for
a
unitary
instance,
moqUquantum
categories of modules are modular and might be used as an inof subfactors in They naturally appear in the classification some von Neumann algebras, and Vainerman [NVOO]. Further examsee Nikshych ples of sernisimple modular categories can be obtained by applying various quotient and orbit constructions to known categories, see [FK93]. 7.4.3
our
Further
A natural
Related
refinement
Constructions
of the extension
(0.3.1)
framby considering This 2-framings implies via [M, SO(3)) -+ H'(M, Z/2) a spin structure on M. Hence, Mis naturally presented by a bounding 4-manifold, with a compatible spin structure and, therefore, number. Thus, the extension even intersection is restricted to a subgroup of index in the following between short 16, resulting exact map sequences:
ings rather
than
in
is obtained
3-manifolds
of the
f2spin 4
with
n
spinGbb
comers.
)
Gobn
)
Gobn
16t n
S 24
Here,
S?
spin
denotes the
3-cobordisms. from [KM9 and
1]
generalize
The inclusion
us
interesting
also mention
the WRT-invariants
[Saw99].
in
much
structures on
the
spin
Our constructions
complication
allows
us
to
can
of framed be drawn
of
presentations cobordisms with spin
work with
to
balancing property. of Spin TQFT's
number of
the category
structures
modular
cate-
theory developed given in [Mas97] and
The as
combinatorial
approaches that give insights into the Witteninvariants. Blanchet, Habegger, Masbaurn and Vogel construct from the Kauffman bracket entirely in [BHMV95]. In their arti-
computational,
Reshetikhin-Turaev
spin
to constructions
here
CD,,b n
group and for manifolds with
without
of
spin assumptions
gories with relaxed in [BHMV95] also leads [Be198].
cle
spin cobordism
A surgery calculus and was made explicit
TQFT functors
structures.
Let
Gob
a
technical
as
well
more as
geometrical
[KM9 1] Kirby and Melvin manage to show that for U Uq (S12) and small k the of the 3-manifold and -ru can be given as summations over spin structures =
invariant
is congruent
to classical
invariants
logical quantum field theory, based by Frohman and Kania-Bartoszyfiska
such
as
the Rohlin
on a concrete
or
Casson invariants.
group G = SO(3), The generalization
A topo-
is constructed
of the identi[FKB96]. presented by Murakami in [Mur94] provides the first link of the Witten-Reshetikhin-Turaev invariants finite to general type invariants as defined Ohtsuki in and H. J. Murakami and [Oht96] Murakami, by by Le, Ohtsuki in [LMM095]. This is presently under investigation by many people, see for example Garoufalidis and Levine [GL98] and references therein. fications
with
the Casson invariant
in
Examples
7.4
Hopf algebras
Quasi-Triangular
7.4.4
Let H be
which
finite
a
is
Hopf k-algebra.
dimensional
invertible
an
satisfying
the
an
R-matrix, of Drin-
relations
R13R23
(A (& I)R (1 oA)R
R13R 12
RA(a)R-1
A'Pa for any a E H, so (H, into a braided category. For
a
finite
map v 0
V
:
is
The
It makes C
quasi-triangular. braiding is c(x
(9
y)
-+
VO
(V
=
__!L+ V(4V)).
decomposable
V
:
into
a
4
U-2 -1,
2
U1
Theorem 7.4.10
(Drinfeld
of thefollowing
The element
g is
The notion
of
[RT90].
us
-+
1,
V(-4V)).
bijections
U4
:
V
The maps ul
7
U4
(e(q)
grouplike
--!L+
VV
V
and U4
are
UT2
)
V.
given by the
action
2(R')R"
1, Ag
=
=
-y(ul)-',
.9
g), andfor
g (9
any
=
a
UlU4E H we have
Let a
Hopf algebras a
Hopf algebra
ribbon
recall
ribbon
was
proposed by
structure.
IV for any finite
dimensional
all
morphisms
dimensional,
following
definition
and Turaev [RT90].
and Turaev
H-module V. One can prove that the map -2
commutes with
Reshetikhin
from the category of modules. Assume that C = HU-2 V-1 = = U-2V Then there is a morphism. U-2 1 0 1
it, starting
xv
is finite
in any
V.
2
[Dri90]).
-y(R")R,
Ribbon
mod has
WV
as
2
4
__!L4
of the two
V,
U
7 4(a).
=
7.4.5
2
elements:
ul'=
gag-'
_2 4 VVV
V
:
UT
(V
=
_!0.4 V(-4V)
product
hand, in C
On the other
-4
'qV are
comod- H*
=
X.
2
VVV
)
rigid braided category there are morphisms uo Composing them, we get linear bijections
They
H-mod
=
R21.y 0
=
H-module V as for any vector space there is a canonical V11 such that (v, y) = (y, V20 (V)) for v E V, y E Vv- Its 2
V04
gives
square
R)
dimensional 2
linear
The
H has
Assume that
R E H 0 H,
element
[Dri87]
feld
V
329
:
a
U0
2
)
W
V
Vo
V
xx (9 xy and xX2 = gx. If H xx(gy that xv is the action of a grouplike element x of H. version of the original definition given by Reshetikhin
and satisfies
we deduce
is
V
330
Generalization
7.
Definition
7.4.11.
of
modular
a
grouplike
a
element
X2 for
any
In the
xa>c-1
9,
dimensional
of finite
category
have canonical
U2 0
isomorphisms
identify following
quasi-triangular
a
Hopf al-
^/2 (a)
=
V
:
modules
over
VVV' U2(V) 0
_+
a
=
ribbon
V2(XV) 0
Hopf algebra =
XV2(V), 0
we
which
these modules.
to
The
=
is
E H such that
;,c
E H.
a
we use
(H, R, ;-C)
Hopf algebra
A ribbon
gebra (H, R) and
ftmctor
shows that,
indeed,
ribbon
Hopf algebras
produce ribbon
cate-
gories. Theorem 7.4.12
(cf.
egory H-mod is
by
cation
the central V
If (H, R, x)
[KR931).
ribbon
a
braided
=
-(V)
Here
=
The braided
C
=
H* be its dual
R'-y(R")x
f,
H*,
g E
Consider
element
-/(V)
1,
explicitly
algebra =
x, y -E
linear
the braided a
[Swe69a].
H, where Af iL : vL 0 L
if H* is
(,)
pairing
a u
equipped with U
O'f
V
(g
f(j)
0
f(2)
Hopf algebra
ribbon
=
is the
U(1) X) U(2), Y) 7
coproduct
in H*.
H*, la 0 lb -+ taLbI where taLb is the matrix basis (lb), that is, taL is a linear function on H b E H. The maps
coadjoint
('T(f(1))f(3),
h become homomorphisms of
H-module structure
UV(2)
u E H, f E H*. The vector space H* with this H-module structure denoted F. The maps h are homomorphisms of H 0 H-modules if H* is with the following H 0 H-module structure
for u,
v
E
H, f
be denoted F.
E H*.
0
V) J
The vector
=
our
and let
H* 0 H -+ k satisfies
for
(U
(7.4.7)
V.
-+
the
=
:
U, XY)
=
for
.
F and the coend F for
dimensional
U) X(1)) (9) X(2)),
(1 a, U.1b)
(R 21R12)
=
Hopf algebra
finite
The
maps
=
Av
F and the coend F
of the H-module L with
given by (U, taLb) H-modules
R'; 12 P, (aM), these varieties the associated morphism is zero as well. Consequently the Frohman-Nicas TQFT is an example of a non-semisimple TQFT as defined in [Ker98b] using a modified composition law expressing funcin the case that connected cobordisms are glued over unconnected surfaces. toriality modular category associated to the TQFT in [FN91] is given The non-semisimple
homology
of the
,
=
=
in
[KerOO].
phenomena occur for the TQFT's that Frohman and homology in order to obtain SU(n) using intersection from the traditional the deviation for knots. They circumvent Casson type invariants Inof cobordisms. to "monotonous" subcategories composition law by restricting it turns out that the higher order knot invariants can be computed from terestingly obtained from the U(l)-TQFT, the Alexander polynomial suggesting that a similar holds for the TQFT's themselves. assertion of In [Don99] Donaldson uses similar TQFr methods to give an interpretation Alexanrelated of homology circles, which is closely to their a Casson invariant Similar
Nicas
non-semisimple
construct
in
[FN94]
342
A.
der
polynomial.
connections
From
Quantum Field Theory
a
non-trivial
groups. Also here non-trivial interior
into
we
R(Z)
he considers
have the
homology
of
U(1)-theories. to Milnor's
moduli
space
the vector
spaces
a
bundle and defines
invariant
a sum
Witten
of
Instead
in
to Axiomatics
non-semisimple
characteristic
vanish.
Moreover
Similar
TQFT's
this
are
that
M(Z) as
their
SO(3)-
homology
cobordisms
TQFT essentially
considered
of
to relate
with
decomposes the Seiberg-
torsion.
Inspired by ideas of Donaldson and Segal Fukaya [Fuk99] defines the relative homology groups of a 3-manifold with boundary using the an analogous double category picture as the one we use here, only in one dimension higher. To a surface Z he associates a category Co(Z) Cag(R(Z)) of Lagrangian submanifolds with G with G SU(2) (or,(hg(M(Z)) SO(3)). The morphisms associated to the 3-cobordisms between two Lagrangian manifolds are given by the Floer The required cathomology groups HF(Lj, L2) of the corresponding 3-manifold. Floer
=
=
=
C (Z) = egory is then the A' -category of chain complexes. The relative Floer the solutions corners.
of certain
anti-selfdual
Func(Co (Z), Ch),
homology equations on
where Ch is the category constructed via
subsequently the bounding
is
4-manifolds
with
Categories
B.1 Double Double
are
defined
required
of
is
as
with respect
to
with respect
functors
are
A double
B.M.
operations, composition 2.
a
to
each other,
such that all
Specifically,
another.
maps of definition
structure
the formal
follows:
Definition
1.
as
a
to be distributive
category
one
by Ehresmann [Ehr63a] as a generalization They 2-category and, hence, also of strict monoidal categories. which are class of morphisms equipped with two compositions, have been introduced
categories
of the notion
and Double Functors
Categories
B. Double
(a, b)
-+
a
respectively,
is
category
b)
q, b and (a, such that
a
class
-+
o,
a
Z
equipped
with
two
multiplication and vertical
the horizontal
b, called
(Z, o,). ofmorphisms ofa category, denoted Z h- Similarlyfor 1 -morphisms defined as Uh Ae subclass of junits OfZh} is stable Z the subclass of horizontal under vertical 1-morphisms Similarly, composition. under horizontal compositions. I units of 0, 1 is stable U; law) If all products b q, a, d c , c, c o, a, and d o,, b are defined, (Interchange then thefollowing expressions are defined and equal (Z, o,)
is
a
class
vertical
=
=
3.
(d 4.
Denote
by
This
definition
which is called
c)
o,
(b
c,
a)
=
(d
o,
b)
(c
q,
o,,
a).
the targeth (resp. source, and targetj with each morphism its source or target in Oh (resp.
sourceh
associating
o,,
and
sourceh
o
source,
=
source,
o
sourceh
targeth
o
source,
=
source,
o
targeth;
sourceh
o
target,,
=target,
o
sourceh
targeth
o
target,
=
o
targeth
suggests a
the
2-morphism,
sourceh sourceh
sourceh
or
(source,
2-arrow,
or a
f)
ft (target,,
diagrammatic
following a
target,
targeth
target,
(source,
t argeth
f f)
presentation
2-cell.
?
targeth
(target,,
f) f
f)
maps 1) -+ in
for
Z,
Then
0j.
f
E
0,
344
Double
Categories
interchange
law 3)
B.
The
that
asserts
we obtain
multiplication
assumes a
same
one or
remaining
and then the
t
a
t opposite notations
For
a
source,,
g
g
For
are
t
b
t
t
d
t
is easier
o, b and b e,
the
interchange
d)
e,
to translate
a
=
start
=
(a
o,
c) a
source,
a
=
tg
op,
(b
o,
horizontal
sourceh
gtarget,
into a
using the use
these
d). 1-morphism,
1
i
i
j
E
U;)
- 4 targeth
+
sourceh
g
formulae
o, b. Weshall
law reads
sourceh
g
It
one.
i
targeth
targeth
arrow. (resp. j) collapses to a vertical (resp. horizontal) uh n q which are both horizontal and vertical units objects the 2-cell collapses to a vertex.
a
A
with
units
Let %be gory of
(c
we
horizontal
E
x
B.1.2.
Definition
phisms
*,,
whether
diagram below. perform all horizontal with the compositions
in the
first
for g
Elements
objects.
b)
g target,,
target,
a
we
g -
gt The 2-cell
9,
indicated
as
1-morphism, g E Uh, (resp. reduces to presentation
vertical
diagrammatic source,
notation
b o,
For instance,
synonyms.
as
(a the
C
The diagrammatic laws multiplication
Notation.
situation
element in 0, whether
and then the vertical
direction
in vertical
the
and Double Functors
2-category respect
is to
a
double
vertical
category
composition
for which all vertical (ie. objects).
Ehresmann [Ehr63b] associates Q9A. It has the class of morphisms
2-category.
quintets
F
A G
C
'k
F,
with
called
are
1-mor-
it the double
cate-
B
JG' D
from 9A. There are two composi2-morphism (natural transformation) of morphisms, namely, horizontal and vertical pasting. In particular, of quintets QV-Cat-mod in the 2-catwe use the double category in G We are interested the R, and G' egory V-Cat-mod'. R,8 representing action of the symmetric group defined in Section 4.3.3. The double subcategory with V-Cat-mod! itself 1 and G' G 1 is the 2-category where
tions
a
on
is
this
a
class
=
=
=
=
double
(strict)
There is
sitions.
categories is a map compatible with and vertical and with horizontal and target, compoweak version of a double category. However, double functors between two double
functor
and vertical
horizontal
no
admit
weak versions.
tions:
horizontal
source
(4i, (1)
a,
Z is
it
:
them here, etc.
to
natural are
a
a
triple
consisting
A
categories. of
as
well
Our main
as
related
objects
in the weak
double functors
priori
Let (t and Z be double
B.2.1.
0)
describe
transformations
going
We are
and vertical
ies, namely, extended TQFT's, Definition
345
pseudofunctors
B.2 Double A
pseudofunctors
Double
B.2
sense.
pseudofunctor
(double)
no-
of stud-
4i
and (t -+ Z commuting with the maps sourceh, source, targeth, I-morvertical to P horizontal or I-morphisms (in particular, maps targetv phisms of the same kind and objects to objects); a
(2) for
map !P
pair of
any
o,
-composable horizontal
,P(C),
sourcev
F -G -morphisms A hor+ B hor+ C a ver-
sourceh(aFG) OFe ,!PG, targetv (aFG)
aF,G with
2-cell
invertible
tically
I
(aFG)
=
=
=
P(A), targeth(aFG) (P(Foh G). It is visualized
as
PA
PA
!PC
4iF
-PG
013
)
)
aFG
aF,G
;P(F*hG)
(3) for
any
pair
!PC
vertical
of o,-composable
!PC F
A
I-morphisms
t3
G
C
a
ver
ver
OF 0, OG, invertible 2-morphism flF,G with sourceh(PF,G) horizontally 4i(A), and targetv (OFG) O(F sh G), source, (OFG) targeth PF,G) =
=
=
0 (C).
It is visualized
as
!PA 0A
OA
PF,G
OFe,,OG 11
tP(F%,G) OC
(PC
-PA
=
PFt M
OGt
#F,G
O(F*,,G)
OC
0C such that
(i)
the 2-cell
OF) (ii)
aFG collapses
whenever F
the 2-cell
(resp.
PF,G collapses
OF) whenever F (resp.
to a
G)
is
to
G)
is
horizontal an
a
aFG
=
OG(resp.
aF,G
0G (resp.
#FG
=
object;
vertical
an
morphism
object;
morphism PFG
=
=
346
(iii)
Double
B.
any three
for we
Categories
-composable horizontal
c
1 -morphisms
(aF,G
4W)
oh
which is visualized
aF*hG,H
*,,
( PF
`
-PF
)
4iG
M
OC
4
11
-PF
)
I
OB
PF
-PH
aFGehH)
9,
OD
IIIH
any three
have the
o,
PG
)
(PH
OC
OD
1
&5(G*hH)
IPB
Ov
11
aFG*hH
P(F*hGehH)
OA
-composable
vertical
1 -morphisms
OD F
A
)
G)
B
ver
H
C
)
ver
4W)
0,
which is visualized
Oh
F*,,G,H
the mirror
as
=
reflection
(!PF *,,8G,H)
Oh
of the previous
flFG*,,H diagram with respect
diagonal;
any
pair (a, b) of
oh
-composable
F
t
t
a
F*hG
b
t
44
t
K
naturality
as
sh
fib)
9.
a
shb
HehK
propeny
(!Pa which is visualized
2-morphisms
G
H
have the
D
ver
cocycle property
(,8F,G
we
aG,H)
aGH
11
(v) for
oh
1
OF
OA
the
'D
aF*hG,H
OA
to
H
-hor+ -
-P(FohG*hH)
,PA
we
C
-Zo-r+
OH
-P(FohG)
0 4
for
G
B
-h -or+
as
aFG
(iv)
F
A
cocycle propeny
have the
4A
and Double Functors
aH,K
=
aFG
*.!P(a
ehb),
t
B.2 ,PF
PG
Pa
Pb
-PH
45K
t
aFG
P(FehG)
t
,P(H*hK)
for
pair (a, b) of
any
Ft have the
naturality
which is visualized
iPb
Remark B.2.2.
Pb)
e,
ao,
b
He,,K
40h
PF,G
PH,K
!P(a
sh
9,
b),
as
45 F
PH,K -PG
Fo,,,G
property
tPH
!Pa
.[,(H*hK)
tK
b
( Pa
(PFt
b)
sh
tH ev
we
P(a
-composable 2-morphisms
o,
a
Gt
347
(PG
PF
all,K
(vi)
pseudofunctors
Double
I
OK
When e is
OF,G
'PH*vK
a
I
4iG
2-category, all
and
( P,
vertical
a,
0)
P(a
P(F*vG)
:
e
morphisms
-+
0 is
e,
a
b)
O(H*vK)
double
objects.
I
pseud-
Hence, the
by (ii) 0 with the notion of a 2-pseudP : e -+ 0 coincides pseudofunctor % C 0 consists of 2-morphisms where the 2-category ofunctor!P : (t -+ %[B6n67], are objects. whose horizontal source and target then
ofunctor
=
of
B.2.1
Compositions
a pair composition
Given
are
double
notion
a
11, since
of double
of double
(-P,
pseudofunctors
pseudofunctors
a,
P)
=
(9A
93 and V'
V
93
e)
:
93
-+
(t,
their
348
Double
B.
is defined
and Double Functors
Categories
follows.
as
The map !P is the
composition
of maps 9A
93
C Other components
are
V'VA
P"VF*hP"0'G
a0,F,,PIG a F, G
a
=
lpl IF,-PIG
V1 (a/FG)
e,,
(P"VA
,P"(VF6hP'G)
1
F,
4IFVG I
V'VF*lv
VIVC
V14VA
VWG 0
'PIF,PIG
P
4WVA
I
5PII(4,G)
(P' 'F*,,VG)
"
45114PIC
proof
4
I
(VI( VA
The
1
G)
VIVA
OF,G
4,11!DIC
PII'PI
VIVC
satisfy
that these 2-cells
to
cocycle
and
I
(F*hG)
VIVC
naturality
conditions
is left
to the
reader.
Functors
B.2.2
categories
from double
case, when the target particular for us. Let especially interesting
A
Let 0 be
-P
(!P, a
double
a
#)
a,
map 0
:
:
0
0
category, Q9A is
-+
-+
2-categories
to
double
and let
9A be
triple
consisting
a
is the category of quintets, is to this case. general definition A (double) 2-category. pseudofunctor
category
specialize
us
a
the
of
Q9A commuting with the maps
sourceh,
targeth,
and
source,
targetvq
2-isomorphism -composable a 2-isomorphism o, -composable a
o,
-2+ 0(F o, G) is given for F and G, -2 o : P(G) + !P(F o, G) is given for #F,G O(F) 1 -morphisms F and G, vertical aFG
:
4i(F)
o!P(G)
any
pair
any
pair of
of
1 -morphisms
horizontal
such that
W aFG (ii) #FG
(iii)
for
=
=
1 whenever F
or
G is
an
1 whenever F
or
G is
an
object; object;
horizontal any three o,-composable there is a commutative tetrahedron:
1-morphisms
A
F -
+
B
C
H
D
Double
B.2 45 G
M
!PC
pseudofunctors
349
(PC
!PB
0110, 4W
PF
OA
(iv) (v)
horizontal,
similar
axiom with
for any
pair (a, b) of oh-composable
t
a
b
a
a
commutative
t
t 0
a
0,
0
we
b
have
a
10
iP(KqhH)
o,-composable
2-morphisms
tH GqF
Gt
tK
commutative
t
CiG,P P(GqhF) (GOhF)
of
a
prism:
H
pair (a, b)
Ft
b oh
KqhH
P(K%H) for any
o,,
Goh F
/;a OiKH (vi)
OD
(P(HohGohF)
replaced with vertical, 2-morphisms
IV
aK, K,
PH
0
K
H
have
1$.
G
F
we
q,,
PF
OA
OD
P(HohGohF)
t
=
prism:
b o,
a
Ko,,H
r
350
Double
B.
\,, 1P (Gc ,
13
Categories
and Double Functors;
I'K
F
F)
to
F)
Gc ,
A
0.0
KoH o, H
.tb +
+
Proposition
(-P,
a,
fl)
fiinctor following
B.2.3.
Z
:
-+
Suppose that 0 is a double category, Q9A is a double pseudo/unctor, and (TI,
(=, a', 0')
Then the composite
Z
:
Q
-+
is
a
9A and 93
J)
9A
:
-+
are
93 is
doublefunctor
2-categories, a 2-pseudogiven by the
pastings:
T/,P F
TIOA
F
A
Gt
B
C
TV (,P EoP F)
tE
a
F-+
TI-PG
TliPE
D
H
TIOB
TV (-P HoP G)
TI(fic
2.
pair of o,-composable horizontal 1-morphisms 2-isomorphism aIFG is given by the pasting For any
A
G
F
B
C the
TIM
\41"
45 F, 51PF,
TI!PA
Proof.
#FIG is similar Ow T'#F,GThe
Corollary
proof
B.2.4.
to
is
the above with
analogous
to
the
Let 9A and 93 be
functor, 6FG : TI(F) o TI(G) ( P, 6, J) : Q9A -+ Q93
TI!PC
4TOf,g TnP (F oh
3.
G
horizontal,
case
of
2-categories.
TI(F
o
G).
G) oh,
TIaFG replaced
2-pseudofunctors Let
(TI, 5)
Then there
:
is
with
vertical,
[B6n67]. %-+ 93 be a
double
a
2-pseudo-
pseudofiinctor
TI F
TIA
t,
G
C
F-+
E
4
T, G
E)
H
IFE
TI(HoG)
TID
WH
Apply Proposition
Proof.
(Id, 1, 1)
Double transformations
There
are
kinds
two
horizontal
Definition
B.2.5.
horizontal
transformation
(ii)
identity
double
a
horizontal
a
2-cell
Let
transformations
of natural
and vertical
The vertical
formations.
(i)
to the
B.2.3
(pseudo)functor
(!P, a,#)
Q%-+ Q%.
:
B.2.3
namely,
TVB
TI(EoF)
t
!P:
__4
F
A
351
pseudofunctors
Dadile
Let
ones.
us
only
here
pseudofunctors,
the horizontal
(4i, a"5, #'5) and (T-1, aT, gq') be double functors A : 0 -h-,-+ TI : (t -+ 0 consists of A
I-morphism
PA
:
---+ hor
TlAfor
1 -morphism
AF for each vertical
AA
!PA
-PFt
F
object A of
every
A
:
Q
)
B
ver
TIA
tTIF
AF
M such that
B is an object, A (a) if F (b) for every 2-cell a of (t =
then
A
Ft B
naturality
property
holds
trans-
similarly.
defined
ones are
double
between
consider
AF
=
H
a
K
AA is
a
C
tG D
horizontal
1-morphism;
0. A
352
Double
B.
PH
PA
tP
!Pa
!PB F
(c) for A
)
G
PD
)
qfK
G
B
)
ver
AV
tqlG
AG
!PC
V)
:
ft
tTfF
AF M
TD
TA
=--
tTIG
T/a
TIB
A
TV
Tv D
K)
B.2.6.
#;,G!P(F*,G)
(5Pt3
I
PGt
TIC
A modification a
T1 A
PFt
PC
A -+ B is such
PA
OA
tTIF
AF
TIB
Definition
)
=
TIC
)
C we have
TIA
,PGt
tVIG
AG
TIH
Tf A
ver
OA
,PFt
!PA
Tf C
!PC
)
OFt
and Double Functors
Categories
collection
#;gr,,GP(F*,,G)AF*,,GIP
I
transformations of horizontal of 2-morphisms
TIC
I
I
m:
A
-*
-4
p
T1 A
4A
1
Ac
(PC
-PC
(F*,,G)
MA
PA
Tf A ILA
thatfor or,
in
every
vertical
I-morphism
F
A ver
B we have mA9, AF
detail, TI A
!PA
iPFt
MA
P 4
OFt
AA
AF
(PB
TA
tTIF
TI A
-PA
TI B
4iB M13
...........
AB
tqlF
AF
AB
-+
TIB
AF
0..
MB,
tangles
C. Thick
bicategory
CA Monoidal It
is
rather
involved
task
various
functors
which
a
tween
tangles
of thick describe
to
use
the
relations
between
isomorphisms
To make them accessible
coends.
we
be-
introduce
bicategory of thick tangles as a kind of a free monoidal bicategory generated by a self-dual object. An analogy would be to introduce the free braided category, of unoriented the as generated by a self-dual tangles category thick the much is of simpler than its braided object. Naturally, bicategory tangles and Langford [BL98a]. the of Baez 2-tangles proposed by counterpart category braided the free semistrict in of is that a [BL98b] They prove 2-tangles category similar monoidal 2-category with duals on one unframed self-dual object. Very geometric 2-categories appear in the theory of knotted surfaces in 4-dim space as developed by Carter, Saito, Fischer, and others, see for example [CRS97], [CS98], and the monoidal
with
duals
-
[Fi94]. Simipresented by generators and relations. which tangles is presented by generators, include the generator-object 1, 1-morphism generators, 2-morphism generators bethere is a list of of a few 1-morphism generators, tween compositions and, finally, relations This list of relations between 2-morphism generators. enables us to present this monoidal category combinatorially. At the next step we modify the set of I -genThat allows us to reduce the list of 2-generators and the list of relations. erators. The latter form of the category (equivalent to the previous ones) is especially easy in the 2-category To give such a representation of abelian categories. to represent monoidal category. it suffices To achieve this result to give a bounded abelian we
larly,
of tangles The category the monoidal bicategory
prove
a
be
can
of thick
few lemmas about coends.
bicategory ofPlanar Thick Tangles PTT has nonobjects. The 1 -morphisms from k to .1 are smooth oriented comintervals distinguished pact surfaces X with boundary OX equipped with disjoint j < k, i', , : 1 -4 OX, 1 ,o. The functors !P : PT9 (k, 1) -+ UPT9(k, 1) and T1 : UPT9(k, 1) -+ PT9 (k, 1) are and quasi-inverse to each other equivalences of categories,
C.2
Representation
Nowwe associate a
bifunctor
with
from thick
Theorem C.2.1.
tangles
of thick
by
categories
abelian
a rigid monoidal, bounded abelian category C graphs to bounded abelian categories.
There is
a
(C,
0, a, 1', r,
trifiinctor
UPT9
--4
'M
T
=
V-Cat
CO-
F-+
Id
:
C
-
Unit
:
f--+
Inv:
C
F--+
6)
F--+
Coend
C
-+
k-vect -+
-+
C,
kI
XHom(l,
k-vect,
X)
:CNC-*C xCC
a,
,YF-4
:
CzCzC
C,
x1zxV
k
COC a
10(9 11
0
COC k-vect
1+--+
C0
k-vect
UnitZ1
4C
NC
C
t1d
t
COC
4C C
C 0 k-vect r
119Unit"
S-
COC
tId C
YEC F--+
(1
F--+
(Inv
YZ(YVOX)
N Inv)
YEC
Z
1)f(XOY)Zyv
.)x
n)X
1)
366
C.
Mck
proof
The
is based
LemmaC.2.2.
F'(X) is
The
f
=
isomorphism
_
detail,
0
Y)
Y',
0
10(lyv
YV
z
the upper
the lower
C 0 C,
-+
(X)
=
Yz
(YV
(YV
0
0
X)
:a
fYEC (X
0
Y)
)
f
z
XV)
0 X
ZEC
(ZV
z z
0
X).
in
row
OCO!14) (X
10(lyv
0
Y)
z
(X
0
Y)v
0 X
tixoyox
0
W
F11'2
F121
(9
X211
row.
Let Y E C. The considered
fZEC
C
:
fYEC (XOY)Z(Xoy)v(&x
X11
W:
F"
F"
Ocoev
10iyt Proof.
-+
morphism
xoyzyv
induces
F'
w :
YEC
(X
Y)
0
lemmas.
YEC
YEC
(X
following
the
on
given by thefollowing
f In
tangles
Y(DZMZv
isomorphism
is
(11ZHOM2,3 014)
-+
XEC
(
ZEC
xzx)zf
(Y(2)z)zzv
XEC
XzXVOY. WhenY is
phism W:
Hom(X,
injective,
coincides
fZEC
Z)
is exact in X and Z, and the above isomor-
XIZEC
y 0 z z zV
f f The above
Y0
with
XZHom(X,YOZ)OZ'
-+
X,ZEC
X0
Hom(Zv,
Xv
(D
X,WEC
morphism
XZ can
be
Hom(W, Xv
presented
o
Y)
&
in another
Y)
(&
Zv
W_fXEC
XzXVOY.
-
form via the exterior
of the dia-
gram
f
ZEC
10coev,
f
.
y(&ZgZV
ZEC
fX,ZEC XoHom(X,Y(&Z)ZZV
-2 4
t
yoZZZVq)yV(&y
f
Z,XEC
XZHom(Zv,XvOY)OZv
(C.2.1)
-2:+ f ZEC(y(gZ)M(y(g)Z)V(g)y
fXEC XZXV(g)y
Here
and
a
in such
P are defined
a
way that ev
X0Hom(X,Y0Z)NZV01
tangles by
of diick-
Represcritation
C.2
the
diagrams
ZZv woevy
4
il
0
Z)
z
zV
it z
0
f
Z)
0
I
iYOZOY
(Y
z
Z)v
0
The left
mutativity
zV
XzXVOY
valid
for
XEC
(X
any
f
:
X
X 0 Xv E C 0 C
M (Y &
of
Commutativity
=
Proof.
deal with the
I ED n
The
left
z
:
Inv
isomorphisms exact
following
0
-!: 4
Z)v
f
functors,
diagram
of the coend F
the definition
F)
M)
f
and
Y0 Z Z
(Y
0
Z)v !L24 F).
the lemma.
(INInv
two arrows
z
by
N(YOZ)V
Diagram (C.2. 1) implies
(10
Y
have
(Y
Thefollowing
LemmaC.2.3.
0 1
we
X 19 Xv
(X
Com-
F
Z)v =
a.
zV
X
Y 0 Z E C. Indeed,
-+
of
by the definition equation
Y
F
that
XEC
square of diagram (C.2. 1) is commutative of the right square of (C.2. 1) follows by the
X
z
f
(& Y
commutative.
are
f
J(yOZ)VOy
(YOZ)Z(YOZ)VOY
-,
YV 0 Y
ZEC
(Y
zt
fZEC(y(DZ)
fX,ZEC XoHom(X,Y(&Z)NZv
(Y
(Y(&z)zzV(DYVOY
t
ix'z
367
categories
abelian
011) (IF12'OF2113)
X YEC
(X n are
z
made
it suffices is commutative:
XV 0 Y z
explicit
to prove
for
YV)
-+
-+
f
ZEC
in Lemma5. 1.11.
projective
IF coincide:
X and
z z
Since
injective
zV.
we
Y
tangles
Thick
C.
368
It
XOev
X0Hom(Yv,XV)OYV The
X Z-Inv(Xv
object
0
XNXV
---+
Y)
Yv is
0
direct
a
sum
it suffices To prove the above diagram, Xv 0 Y and represent it in the form f : X
--+
Diagram (C.2.2),
Then X0
Inv(Xv
0
Y)
restricted
0
Yv,
to
nothing
is
the
subobject
YZYV
)
1iY
ix
)
of diMk
of X 0 Yv. 1
evSYV
X0Hom(XY)ZYV
X[9Inv(XVOY)(9YV
up
Y and
ft
f
X Z
ZZZV
InV(XV
pick
to -+
fZEC
(C.2.2)
0
an
Y) copies
Yv
Xv. X Z Yv -*
0 Yv
-+
of the coend in
else but the definition
f
element
(5.2. 1).
functor is fixed on 1-morphisms by the Proof (Theorem C2.1). The constructed it strictly commutes with the tensor product and the compofollowing requirement: Wehave to prove diagram (C. 1.5) for w using the calculation sition of 1 -morphisms. made in LemmaC.2.2.
(X
(2) Y 0
fZ
That is,
Z Z ZV
101_,v
Ocoey4
X0
fZ
Y (2) z cg
zV
(2)
YV (2) Y
--+
f UX(DUOU a4 f WzWV(OX(&Y) ((XOY)O f zzMzV f Z(XOYOz)NzVOYVOXVOXOY f WzWVOXOY). X(g
U
f
101uv
uzuvoy
Ocoe
V
W
0
XV oxoy
191ZV
-4
--+
Ocoe
=
W
-
-+
This is obvious
coevx,&!:
(,
(1
-4
wevy
since
(X(OY)VO(X(OY)_YVOXVOX0Y) 0COeVX ly '4 YV (& YV (g Y _ YV (0 1 (2) Y v
_
Lemma C.2.3
(C. 1.3)
are
obviously
RemarkC.2.4. is also the
contains
If
a
thick
composition
proof
the
satisfied.
graph
of Relation
(C. 1.4).
XV (2) X (&
Relations
Y).
(C. 1.2)
and
This proves Theorem C.2. 1. is
of relevant
composed of subgraphs, functors.
the
corresponding
functor
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Index
IR-b
r g lplan R2
40
6bb
-
68, 73,76 18,19 24,26
C,ob2 Gob
&7b
7gl
b)
Gob cn 2
24
144
IFR 152 r gS2 138 7©9 1 S 2© 139 rogl S;* S2 150 -rg lKi S2 116 Tgl ,
18
A -
119
137
2
J 1L 7©91S2
68, 75
Cob(a,
125
r g 1well-pos R2
18,20
COB2
226, 272
comultiplication
map COB -+ Cob 296 T
-
74
7©gltgl"
168
22,157 UPT9 362 Y
To
189
X",
V-Cat
189
V-Cat-mod
189
V-Cat©
.T,31 .F,l
191
V
188
VA
YC F
[1,2]-7:0 [1,2]yl
W+ 9
70
li- 9
70
Int
a
191
Oh
H
Diff e
229
71 (M,,) M(g, a/b)
40
219 226
40
XEC
262 f H H f fHl Hfl f
357
k
353
1
2,8,268
218
279,331
20
6u q 45ur02
36 51
Emhd+
40
,Fsq
+
"+© j
ev
70
FTT
222
c
MPTT 357
X
4,243
219
coev
167
Pb
248
5, 343 5,343
o,,
166
X P7-9
246
21
191
71
Isl+
249 248
AbCat
270
gg ts 9c Inc+
247
23
23,33 11, 173 199,204,206,207,211
116 7©gl 118 ©rgIR2 7- g Idec-proj R2
v
223
r
218
188
-r
220 (Dop u20 223
123
2
Ul
223
229
Index
378
U40
braid
framed
223
1-2-Slide
40
group
algebra
Frobenius
197
19
114
1-2-cancellation
Slide
197
2-Handle
Slide
1550 handle
344
2-category 2-framing
69
admissible
tangle
Cancellation
General
I-Handle
149
Move
51
53 decomposition 53 generic handle trading 113, 177 38, 70 handlebody 332 Hennings invariant 226 Hopf algebra 227 Hopf pairing Hopf-Link Move 150 horizontal 1-morphism (1-arrow) horizontal 5, 343 composition 156 in Tg1
handle -
-
-
embedded
S©
in
108 117
1]
[-1,
x
antipode arc-diagram associativity
136
226, 274 314 218
constraint
balanced
223
category 190
Beak Lemma
P-Move birth point bounded
-
148
-
189
8,218,243
-
-
-
336
15 2
cobordism
270
F
270
76
Annulus
death
Move
149
190
101© 152
enveloping
category
double
functor
evaluation
small
257
sernistrict
245, 249 257,259
symmetric
category
262
monoidal
functor
218 219
of monoidal
functor
51
function
190
Lemma
227
category
219
Fenn-Rourke
Move
fill
30, 74
152
4,270
357
8, 2 76
2-category
semistrict
morphism
Tangles
Thick
category
braided
Morse
229
36-38
group
monoidal
4,343
category
38
243, 259
345
Hopf algebra
essentially
class
stable
monoidal -
-
product
transformation
functor
internal
Marked Planar
189 tensor
double
dual
190
101 101
n-Move
-
point Deligne©s
auxiliary external
modular
286
manifold
Dovetail
352
Karoubi
-
294
surface
coloring Connecting
dinatural
351
transformation
modification
mapping 219
coevaluation
critical
-
-
group 262
F
colored
89
Principle Trajectories Independent 229, 279 integral-element 229,280 integral-functional law 5,343 interchange -
3,335
theory
cobordism
coend
78
Obb
interval
Segal©saxioms
cobordant
-
229
3
Chem-Simons
-
idempotents
split
with
CFT
e6b
horizontal
222
222 braiding 57 bridged link 202 diagram
category
in
-in
category category
braided
4, 343
01-Move 02-Move 03-Move
115 150 148
integrals
object
of
Planar
Thick
Planar
Thick
Graphs Tangles
230 357 353
219
Index
pseudofunctor pure braid
345
-
-
5, 344
quintet
recombination relative
invariant
Reshetikhin-Turaev
10, 320
-
-
-
-
auxiliary
product
tensor
-
105
-
110-112
102
315
extended
7
-
Triangle
through
106
TS I
105
top
ribbon
224 rigid Hopf algebra
ribbon
twist©
signature Signature
-TS3**
145 140-141
Moves
318
218
object
unit
Useful
Planar
Graphs
Thick
362
246
V-category V-natural Verlinde
232 151
vertical
69
Cancellation
Move
split idempotent functor splitting standard standard standard
handlebody
standard
surface
closure connected
152
54
Smale cancellation
vertical -
in
-in
32
-in
70, 84 6,20
247
transformation
formula
322
1-morphism composition 155 7 gl
(1-arrow) 5, 343
©
6DB &7b -
229
71 cobordism
247
V-functor
228
Hopf algebra
side-invertible o--Move
113-115
Moves
219
category
self-dual
TS3*
-
TSl**
151
Move
190
TS5 Moves
330
223
Ribbon-TS3
Lemma
I
TS1*-TS3*Moves
224
category
strictly
-
TS1©6
.
axioms
enhanced
102
318
218
283
Atiyah©s
internal
rigid
140
TD5** Moves
-
TQFT
106
closed external
ribbon -
112-113
TD5* Moves
TIl-TIllMoves 102
-bottom -
144
TD5 Moves -
TD3**
ribbon -
balls
-
TD3*
4
cobordism
94, 200
ribbons
TDI
32
110,129
Moves
56
Delm
surgery
functor
Reidemeister
52
surgery
40
group
71
cocycle well-positioned
Wall
Wilson
lines
76 77 77
tangle 6,336
119
4,343
379
