AuslJ'alian Mathematical Society Lecture Series. 5
2-Knots and their Groups Jonathan Hillman Macquarie University, Australia
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CAMBRIDGE UNIVERSITY PRESS Cambridge
New York New Rochelle Melbourne Sydney
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Silo Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521371735 © Cambridge University Press 1989
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 Re-issued in this digitally printed version 2008 A catalogue record/or this publication is available/rom the British Library ISBN 978-0-521-37173-5 hardback ISBN 978-0-521-37812-3 paperback
CONTENTS
Preface 1
Chapter 1
Knots and Related Manifolds
Chapter 2
The Knot Group
15
Chapter 3
Localization and Asphericity
36
Chapter 4
The Rank 1 Case
52
Chapter 5
The Rank 2 Case
70
Chapter 6
Ascending Series and the Large Rank Cases
85
Chapter 7
The Homotopy Type of
Chapter 8
Applying Surgery to Determine the Knot
124
Appendix A
Four-Dimensional Geometries and Smooth Knots
142
Appendix B
Reflexive Cappell-Shaneson 2-Knots
145
M(K)
107
Some Open Questions
147
References
150
Index
164
(vii)
Pour Jes noeuds de S2 en S4 on ne sait pas grand chose Preface
Since Gramain
wrote
the above
words
in
a Seminaire
Bourbaki
report on classical knot theory in 1976 there have been major advances in 4-dimensional
topology, by
Casson, Freedman
and
Quinn. Although a com-
plete classification of 2-knots is not yet in sight, it now seems plausible to expect a characteriza tion of knots in some significant classes in terms of invariants related to the knot group. Thus the subsidiary problem of characterizing 2-knot groups is an essential part of any attempt
to classify 2-
knots, and it is the principal topic of this book, which is largely algebraic in
tone. However we also draw upon 3-manifold theory (for the construc-
tion of many examples) and 4-dimensional surgery (to establish uniqueness of knots with given invariants). It is the interplay between algebra and 3and 4-dimensional topology that makes
the study of 2-knots of particular
interest. Kervaire
gave
homological
conditions
which
characterize
high
dimensional knot groups and which 2-knot groups must satisfy, and showed that any high dimensional knot group with a presentation of deficiency 1 is a 2-knot group. Bridging the gap between the homological and combinatorial conditions appears to be a delicate
task. For much of this book we shall
make a further algebraic assumption, namely that the group have an abelian normal subgroup of rank
at
least
1. This is satisfied by
the
groups
of
many fib red 2-knots, including all spun torus knots and cyclic branched covers of twist spun knots. The evidence suggests that if the abelian subgroup has rank at least 2
then
the group is among
these, and
the problem is
then related to that of characterizing 3-manifold groups and their automorphisms. Most known knots
with such groups can be characterized algebrai-
cally, modulo the s -cobordism theorem. However in are examples which are not
the rank 1 case there
the groups of fibred knots, and here less is
known. The other class of groups contains
that
is
of
particular interest
the groups of (spun) classical knots consists of
cohomological dimension
2 and deficiency
1.
(If
some
as
it
those which have
standard conjectures
hold these conditions are equivalent for knot groups). One striking member of this class is the group 4> with presentation , whose
(viii) commutator subgroup is a torsion free rank 1 abelian group. All other knot groups
with
deficiency
1
groups are iterated free
and nontrivial
torsion
free
abelian
normal
sub-
products of torus knot groups, amalgamated over
central copies of Z, and are the groups of fib red 2-knots. We show that any
knot
with such
commutator
a
subgroup)
group (and more generally, whose can
be
cobordism theorem. Together
characterized
algebraically,
these two classes contain
group has
the
free
the
s-
gronps of
the
modulo
most familiar and important examples of 2-knots. However we have by no means
completed
their
classification,
and
the
problem
of
organizing
the
groups outside these classes remains quite open. (The formation of snms and satellites should play a part here). We shall now outline the chapters in somewhat greater detail. In Chapter
1
we
give
the
basic definitions
and
background
results
on
the
geometry of knots and we show how the classification of higher dimensional knots can be reduced (essentially) to the classification of the closed manifolds built from the ambient spheres by surgery on such knots. As far as possible these definitions and results have been formulated so as to apply in
all dimensions.
We have chosen to work in
the TOP category
as our
chief interest is in the 4-dimensional case, where PL or (equivalently) DIFF techniques are not yet adequate. In Chapter 2 we give Kervaire's characterization of high dimensional knot groups, and varia tions on
this
theme: link groups, commuta tor
subgroups of knot groups, centres of knot groups. We also give his partial results
on
2-knot
groups.
Counter
examples
to
show
that
not
all
high
dimensional knot groups can be 2-knot groups were found independently by various people; most of their arguments used duality in the infinite cyclic cover of the exterior of the knot. We review some of these arguments, and we
show
the exterior of a nontrivial n -knot with n > 1 is never
that
aspherical, giving Eckmann's proof via duality in the universal cover. Chapter 3 contains our key result. We show that in contrast to the
theorem
of
Dyer-Vasquez
and
Eckmann
just
quoted
the
closed
4-
manifold obtained by surgery on a 2-knot is often aspherical. If T is the maximal locally-finite normal subgroup of a 2-knot group rr and rr/T has an abelian normal subgroup of rank 1 such that the quotient has finitely many
ends
redundan t)
and holds
if
a
further,
then
ei ther
technical condition rr'
is
fini te
or
(that
may
rr/T = 4>
or
prove
to
be
rr/T
is
an
(ix) orientable Poincare duality group over Q of formal dimension 4. The latter is also true if "IT has an abelian normal subgroup of rank greater than 1.
In the next three chapters we examine these cases separately. In Chapter 4 we determine the 2-knot groups with finite commutator subgroup. All of these can be realized by fibred 2-knots, and many by twist spun classical knots. We show also that if "IT = and T is nontrivial then it must be infinite; in fact we believe that in this case T must be trivial. In Chapters 5 and 6 we consider the Poincare duality cases. further subdivision of cases, according
to
Here there is a
the rank of the abelian normal
subgroup (which must be at most 4). All the known examples with a torsion free abelian normal subgroup of rank 2 derive from twist spun torus knots. The groups of aspherical Seifert fibred 3-manifolds may be characterized as
PD 3 -groups which have subgroups of finite index with nontrivial centre and infinite abelianization. Using
this, we give an algebraic characterization of
the groups of 2-knots which are cyclic branched covers of twist spins of torus knots.
In Chapter 6 we determine the 2-knot groups with abelian normal
subgroups
of
rank
greater
than
2,
and
the
results
of
these
three
chapters are combined to show that if " has an ascending series whose factors are locally-finite or locally-nilpotent, then it is in fact
locally-finite
by solvable. If moreover " has an abelian normal subgroup of positive rank then it is finite by solvable, and we describe all such groups. (We doubt that there are any other 2-knot groups with such ascending series).
In
the last
two chapters
we
attempt
to
recover 2-knots
group theoretic invariants. As we observe in Chapter 1 a knot K
from
is deter-
mined up to changes of orientation and "Gluck reconstruction" by a certain closed 4-manifold "l(M).
We
first
M(K)
try
together with a conjugacy class in the knot group
to determine
the
homotopy
type
of
M
in
terms of
algebraic invariants. The problem of the homeomorphism type may then be reduced to standard questions of surgery. For knots whose group is torsion free and polycyclic we are completely successful, for the surgery techniques are then available to solve the problem. We show also that if the commutator subgroup is an infinite, nonabelian nilpotent group then, excepting for two such groups, the knot is determined up to inversion by its group alone.
Freedman has shown that surgery techniques apply whenever the group is
as in
Chapter 6, but
in general it is difficult
to compute
the
obstructions. On the other hand, for many fibred 2-knots we can determine the (simple) homotopy type and show that the snrgery obstructions are 0, but
it
is
not
yet
known
whether
s -cobordisms with such
5-dimensional
groups are always products. After Chapter 8 the
4-dimensional
there
geometries
are
that
two appendices. The can
be
supported
first
by
considers
some
M(K).
(Among these are some complex surfaces). In the second it is shown that certain CappeU-Shaneson 2-knots are reflexive if and only if every totally positive unit in the cubic number field generated by a root of the Alexander polynomial is a square in tha t field. After there is a list of open questions on 2-knots and related topics. Some of these are well known and very
difficult;
others
are
more
technical
and
algebraic,
but
have
also
resisted solution so far. As the algebra used in this book may perhaps be unfamiliar for many topologists we would like to stress here tha t our principal references have been the Queen Mary College lecture notes of Bieri for homological group theory, and the text of Robinson for other aspects of group theory. I Groves,
Laci
would
like
Kovacs,
to
Peter
thank
William
Kropholler,
Dunbar,
Darryl
Ross
Geoghegan, John
McCullough,
Mike
Mihalik,
Peter M. Neumann, Steve Plotnick, Peter Scott and Shmuel Weinberger for their correspondance and advice on various aspects of this work. I would also like to acknowledge the support of the U.K. Science and Engineering Research Council (as a Visiting Fellow at the University of Durham), which enabled me
to meet
Peter Kropholler and
Peter Scott, and of
the Aus-
tralian Research Grants Scheme, for a grant which brought Steve Plotnick to Macquarie University in mid-1987.
Macquarie University
Chapter 1
KNOTS AND RELATED MANIFOLDS In this chapter are the basic definitions and constructions of the
objects that we shall study. In particular we show how the classification of higher dimensional knots can be reduced (essentially) to the classification of certain closed manifolds. We also give a number of results on the geometry of
these
them in braic).
objects,
for
the
most
part
without
proof, as
we
a crucial way in our arguments later (which are
shall not primarily
use alge-
We shall first state some of our conventions on notation and termi-
nology. Let
Dn = {
<x 1'... x n > in R n: LX
l
~ 1 } be
n -disc and
the
let Sn = aD n +l be the n-sphere. The standard orientation of R n induces an orientation of D n , and hence one of Sn-l by the convention that the boundary of an oriented manifold be oriented compatibly with inward normal last (cf. (RS:
pages 44-45]).
We
shall always
taking assume
the that
these standard discs and spheres have been given such standard orientations. An embedding is a 1-1 map which is a homeomorphism onto its image. All
isotopies of locally
flat
embeddiligs
of manifolds
in
manifolds
shall be ambient isotopies. The inclusion of R n +1 into R n +2 as the hyperplane defined by the equa tion x n +2 = 0 induces the equa toria! embedding
eo:S n -. Sn+l. The interior of a subset N of a topological space shall be denoted by int N. The expression A ~ B means that the objects A morphic
in
some
category
appropriate
to
the
context.
and B are iso-
When
there
is
a
canonical isomorphism, or after a particular isomorphism has been chosen, we shall
write
A
=
B.
(For
instance
the
fundamental
infinite cyclic, and choosing an isomorphism with Z
group
of
a
circle
is
corresponds to choosing
an orientation for the circle). Qualifications shall usually be omitted when there is no risk of ambiguity. In particular, we may abbreviate X(K), M(K) and TTK as X, M and
7T
respectively.
Knots An n -knot is a locally flat embedding K:S n _ Sn+2. also use when
n
the ~
2
terms "classical knot" and
"high dimensional
(We shall
when n = 1, "higher dimensional knot" knot"
when
n
~
3).
Such
a
knot
is
2
Knots and Related Manifolds
determined up
to isotopy
by
its image
K(Sn), considered
as
an oriented
codimension 2 submanifold of Sn+2, and so we may let K also denote this submanifold. Two n -knots KO and K1 are isotopic if (and only if) there is an
orien ta tion
preserving
hKO -= K 1, for such a map h
Thus for instance if rn Sn
homeomorphism
self
b
of
tinct. (Note
that
the
that
such
is isotopic to the identity [KS: page 292].
is an orientation reversing self homeomorphism of
the knots K, rK
unoriented
So+2
rK p may all be dis-
images of K
submanifolds).
The
and K p
knot
K
is
are
the same, considered as
invertible,
+amphicheiraJ
or
-amphicheiral if it is isotopic to K p, rK or -K respectively. An n -knot is trivial if it is isotopic to the unknot e n + 1 en .
By the uniqueness of discs
an n -knot is trivial if and only if it bonnds a locally flat (n+1)-disc in Sn+2. ~
If n
unique j
up
to
4 then each n -knot is isotopic to a PL n -knot which is
PL (ambient) isotopy,
for
then
Hi(Sn +2,K;Z/2Z) = 0
for
= 3 and 4, so the Kirby-Siebenmann obstructions to existence and unique-
ness of PL triangulations vanish [KS: Essay IV.10]. All the examples of 2knots that we shall construct below shall be PL with respect to some triangulation of S4. However as it is not yet known whether all
triangula-
tions of S4 are PL equivalent, and as the 4-dimensional surgery techniques that we wish to use to characterize certain 2-knots have only been established in the TOP category (and then only for a restricted class of fundamental groups), the above definition of knot seems most snitable. The exterior Every (locally flat) n -knot with n = 2 is flat [KS 1975) and so there is an embedding j:Sn XD 2 -
K;' such
tha t
We may assume to isotopy
tha t j
Sn+2 onto a closed neighbourhood N of
and
K
j i Sn X{O}
aN -
j(Sn XS 1) is
bicollared
in
Sn +2.
is orienta tion preserving, and then it is uniqne up
rei SnX{O}. These results (and
their proofs) remain valid when
n = 2, by Quinn's solution of the 4-dimensional annulus conjecture, together
with surgery over Z (cf. [Qu 1982] and [PQ]). The
exterior
of
K
is
the
compact
(n +2)-manifold
X(K) =
Sn+2-int N, which is well defined up to homeomorphism as N is unique up
Knots and Related Manifolds
3
to isotopy rei K. It inherits an orientation from Sn+2, and has boundary homeomorphic to SnXS1.
The interior of X
complement Sn+2-K. The knot group
is homeomorphic to the knot
(X(K». By general posi1 tion, every element of 'irK can be represented by an oriented simple closed
curve in X, and if n class of maps
from
~
S1
is 'irK =
'lr
2 each conjugacy class in 'irK (i.e. free homotopy to
X)
corresponds
to
an unique
isotopy class of
such curves. In particular, any orien ted simple closed curve isotopic to the oriented boundary of a transverse disc (i.e. to {j(s)}XS 1 ) is called a meridian
of K, and we shall also use
this term to denote
the corresponding
elemen ts of 'irK. By Alexander duality H/X;Z) ~ Z erwise. The
meridians are
all
if i = 0 or 1 and is 0 oth-
homologous and
generate
H 1 (X;Z)
(by
the
Mayer-Vietoris sequence for Sn+2 = XuSnXD2) and so determine a canonical isomorphism H 1(X;Z) = Z. The exterior of a
trivial n -knot is homeomorphic to Dn+lxS1.
Conversely if X(K) is homotopy equivalent
to
S1
then K
is
trivial [Pa
1957, St 1963, Le 1965, Sh 1968, Fr 19831. For if 'I is a meridian in the interior of X, it has a product neighbourhood U (as X if X '" S 1 n ~ 3. It is
then
X -int U
is an
s -cobordism and
then easy to see that K
so
is orientable) and a product, provided
bounds a disc in Sn +2. (This argu-
ment works also in the PL and DIFF categories). This criterion for triviality is also correct when n = 1 [Pa 1957) and n = 2 [Fr 19831, although the proofs are different. (Note that there may be PL 2-knots with group Z
which are not PL isotopic to the unknot). The assumption on X
can be
weakened considerably (see below); in particular if n = 1 or 2 an n -knot is trivial if and only if its group is infinite cyclic. Gluck
showed
that
when
n
= 2
the
group
of
pseudoisotopy
classes of self homeomorphisms of SnxS1 is (ZI2Z)3, generated by reflections in either factor and by the map all x
in Sn
and y
in S1, where 8:S 1
t"
given by
t"(x ,y) =
(8(yXx),y)
for
SO(n+l) is an essential map [GI
19621. Browder and Kato extended his result to all higher dimensions [Br 1967,
Ka Dn+1xS 1
19691.
As
(provided
any
self
n ~ 2),
homeomorphism the
closed
of
SnxSl
extends
(n+2)-manifold
across
M(K) =
X(K\.JD n +1 xS 1 obtained from Sn+2 by surgery on K is also well defined,
4
Knots and Related Manifolds
and it inherits an orientation from Sn+2 via X. Since up to homotopy X is the complement of {O}XSI in M, the inclusion of X isomorphism"K
by
"I(M),
characteristic of M is X(M) =
=
general
position
into M
induces an
n ;> 2).
(for
X(Sn +2)_X(Sn XD 2 )+X(D n +1xSl)
The
=
Euler
O. In fact
0, I, n +1 or n +2 and is 0 otherwise, as follows
easily from the Mayer-Vietoris seqnence for M = XuDn+lXSl. (The orientations of Sn +2 and K study
2-knots
determine canonical isomorphisms). We shall in fact
through
K
the
corresponding
closed
oriented
4-manifolds
M(K). There is however an ambiguity when we attempt
M(K). The cocore 'I
from
=
{O}XS 1
C
D n +lxS 1
to recover K
M of the original sur-
C
gery is well defined up to isotopy by the conjugacy class of a meridian in
=
"K
"I(M).
(In fact the orientation of 'I is irrelevant for what follows).
Its normal bundle is trivial, so r has a product neighbourhood, P say, and we may assume that
ax
ways of identifying phisms
SnxSl
of
reversed
the
M -int P = X.
that
jT:
we
obtain
a
across
construction
SnXD2,
of
M
If however we identify new
tw~ essentially distinct
SnxSl = iXS n XD 2 ), modulo
extend
original
(Xu jaSnXD2, Snx{O}). of
with
But there are
self homeomor-
n ;> 2.
provided
we
recover
SnxSl
If
we
(Sn +2, K) =
with X
pair (l:, K") '" (XUjT:SnXD2, Snx{O}).
by means By
van
Kampen's theorem l: is simply connected, and it is easily seen to have the homology of Sn+2. Therefore l: is homeomorphic to Sn+2. We may assume that
the homeomorphism is orientation preserving. Thus we obtain a new
n -knot K", which we shall call the Gluck reconstruction
of K. The knot
K is said to be reflexive if it is determined as an unoriented submanifold by its exterior, i.e. if K" is isotopic to K, rK, Kp or -K. By the work of Browder, Gluck and Kato, if n ;> 2 there are at most two n -knots (up to change of orientations) with a given exterior, i.e. if there is an orientation preserving homeomorphism from X(K 1) to X(K) then
Kl
is
isotopic
to
K, K", Kp
or
K"p.
If
the
homeomorphism
preserves the homology class of the meridians then K 1 is isotopic to K
also or
K". (The long-standing conjecture that each classical knot is determined up
to orientation by
its exterior has finally
been confirmed [GL 19881. The
Knots and Related Manifolds
5
argnment involves some of the deepest results of 3-manifold topology). Thus a knot K
is determined up to an ambiguity of order at most 2 by X(K),
or (if n ;> 2) equivalently by M(K) together with the conjugacy class of a meridian in 'irK. which are provided
not
Cappell and Shaneson gave reflexive
that certain
[CS
(n
the
first
19761. Their method
examples of
works
for
+l)X(n +1) integral matrices exist; at
knots
each n ;> 2 present
such
matrices have been found only for n = 2, 3, 4 and 5. Gordon gave a different
family
of examples when n = 2, all of which are
PL knots
with
respect to the standard triangulation of S4 [Go 19761. Covering spaces and equivariant homology We shall let X(K) and X'(K) denote the universal and maximal
abelian are
covering spaces (respectively) of X(K). Similarly
M (K)
M(K) and
the corresponding covering spaces of M(K). The fundamen tal group of ~
X' (and of M, provided that n
2) is the commutator subgroup
of the
'Ir'
knot group, and by the Hurewicz theorem 'lrl'lr' = H 1(X;Z) = Z. Thus
the
cover X'IX is also known as the infinite cyclic cover of the knot exterior. The
homology
and
cohomology
of
such
covering
spaces
are
modules over the group ring of the covering group, and satisfy a form of equivariant
Poincare duality.
generality.
Let
P
be
a
We
shall
closed
describe
orient able
this
in
somewhat
m -manifold
with
greater
fundamental
group G. Up to homotopy we may approximate P by a finite cell complex [KS: Essay 111.4]. Let H
be a normal subgroup of G
and let PH be the
corresponding covering space. We may then lift the cellular decomposition of
P
to an equivariant cellular decomposition of PH. The cellular chain com-
plex C.. of PH with coefficients in a commutative ring R plex of left
R [GIH l-modules with respect
is then a com-
to the action of the covering
group GIH. Moreover C .. is a complex of free modules, with a finite basis obtained
by
choosing
one
equivariant bomology module
lift of
P
of
each
cell
with coefficients
P.
of
R [GIHl
The is
the
i tb left
module Hi(P;R[GIH]) = Hi (C.. ), which is clearly isomorphic to Hi(PH;R) as an
R -module,
with
the
action
of
the
covering
group
determining
the
R [GIH1-structure. The i tb equivariant cobomology module of P with coefficients
R [GIHl
is
Hi(Hom R [GIH1(C.. ,R [GIH])),
the which
right may
be
module interpreted
Hi(P;R [GIH]) = as
cohomology
of
6
Knots and Related Manifolds
PH with compact supports. If N
module
with
is a
the
-modul~
R
underlying
ng- 1 for g in 0
=
determined by g.a
R [Ol-module we shall let N
right
same
and
the
denote
the left
0 -action,
conjugate
and n in N.
The equivariant homology and cohomology are related by Poincare duality isomorphisms HjCP;R[OIH)) = Hm-j(P;R[OIH» and by a Universal Coefficient spectral sequence with E2 term E~q
= Extj [GIHI (Hp(P;R [GIH»,R [GIH)) => HP+q(P;R [GIHD,
in which the differential d r has bidegree (l-r,r). If J is a normal subgroup of
0
which
contains
H
there
is
also
a
Cartan-Leray
spectral
sequence
relating the homology of PH to that of PJ ' with E2 term
E2 = TorR[GIHI(H (P·R[GIH))R[GIJ)) => H (P·R[OIJ» pq p q" p+q , in
which
the
dr
differential
has
bidegree
(-r,r-l).
There
are
similar
definitions and results for manifold pairs (p,ap) and for (co)homology with more
general
coefficients
[WI.
For
more
information
on
these
spectral
sequences see [McC). When
K
is
an
P = X(K)
n -knot,
MCK),
Or
0
TrK
=
and
Tr', the group ring Z [TrITr'1 is the ring of integral Laurent polynomials Z[ZI = Z[t,t- 1 1. Since A is noetherian the homology and cohomology
H A of a
finitely
generated free
A-chain
The augmentation module Z free
resolution 0 -
A -
complex are
also
finitely
has projective dimension 1 as it has a short
A -
Z
o.
-
tral sequence for the projection of X'
Therefore onto X
the Cartan-Leray spec-
(or of M
on to
M)
to a long exact sequence (the Wang sequence of the map X' -
Since
X
has
t -l:Hj(X;A) -
the
homology
of
a
circle,
it
follows
tha t
reduces
X):
all
the
maps
H j (X;1\) are surjective for i > O. Therefore they are bijec-
tive (since
the modules are noetherian) and so
all
A-modules.
torsion
generated.
In
particular
the homology
modules are
Hom A(Hp(X;A),A) = 0 for
all p
so
the Universal Coefficient spectral sequence collapses to a collection of short
Knots and Relaled Manifolds
7
exact sequences Ext ~(Hp_2(X;I\),I\) -
o
Ext ~(Hp_l(X;I\),I\) -
HP(X;I\) -
O.
(There are very similar results for H .. (M;I\». The infinite cyclic covering spaces X'
M
and
cally much like (n +I)-manifolds wi th boundary Sn
behave homologi-
and empty respectively,
at least if we use field coefficients [Mi 1968, Ba 1980). If H i (X;I\) = 0 1 E;; i E;;
for
(n
then X'
+1)/2
homotopy equivalent
is acyclic; thus
to SI and so K
if also
is trivial. All
Z
=
Tr
then
is
X
the classifications of
higher dimensional knots to date assume that the knot group is Z
and that
the infinite cyclic cover of the exterior is highly connected. An n -knot K is
r-simple
called
if
X'(K)
is
it is simple.
[en -I)/2)-simple
r-connected;
if
~
n
3
and
the
knot
is
(The word is used in a different sense in
connection with classical knots). Levine classified simple (2q+1)-knots
with
q ;> 2 by means of Seifert matrices [Le 1970); this was reformulated (and reproven) in terms of the duality pairing on the middle dimensional homology by Kearton [Ke 1975). After Kearton and Kojima had each classified some
significant
subclasses of simple
2q -knots
with
q
~
4 by
analogous (but more complicated) pairings, Farber finished application of his classification of stable simple
for
1983).
By
some analogy
r
;> (n +1)/3)
with
guity by
the
classify all
might
with r ;> (n -1)/4 cohomology
I-simple knots
the
hope
of stable rational
that
an
to
homotopy
homotopy
r-simple
finite
pairings
of
in
K
to finite
(Ideally one might
ambiguity
[Fa
type of highly
n -knot
would be determined up
algebra H"(M(K);I\). up
means
task as an
n -knots (i.e. those which are r-
terms
results on
connected manifolds, one "formal range"
in
this
the
ambi-
hope
to
by algebraic invariants,
bu tit is not ye t clear wha t these should be). When n = 1 or 2 it is more profitable to work with the univer-
g (or ii>. When n = 1 the universal cover of
sal cover this
is
a
remarkable
dimensions X
result
of
Papakyriakopoulos
[Pa
X
is contractible;
1957).
In
higher
is aspherical only when the knot is trivial, as we shall see
in Chapter 2. Nevertheless under rather mild assumptions on the group of a 2-knot K
the closed 4-manifold M(K) is
aspherical. (See Chapter 3). This
is the main reason that we choose to work with M(K) rather than X(K).
8
Knots and Related Manifolds
Knot sums, factorization and satellites The sum of two n -knots Kl and K2 may be defined (up to isotopy) as
the
n -knot
obtained
K #K 2 l
as
follows.
D n (±) denote
Let
the
upper and lower hemispheres of Sn. We may isotope Kl and K2 so that each Ki(D n (±» is contained in D n +2(±), K 1(D n (+» is a D n +2 (+),
K (D n (-» 2
K 2 i Sn -1
(as
is
a
trivial
n-disc
the oriented boundaries of
trivial n -disc in
D n +2 (_)
in
and
Kli Sn-l
images of DO (-». Then we
the
let K #K 2 = K l iD n (-)uK 2 iD n (+). By van Kampen's theorem (used several l times)
1I"(K 1 #K 2)
erated
by
a
=
1I"K 1 "Z1l"K 2
meridian
in
where
the
knot
group.
each
amalgamating is
It
not
subgroup hard
to
is
gen-
see
that
X' (K 1 #K 2) is homotopy equivalent to X' (K l)vX' (K 2) and so in particular 1I"'(K 1 #K 2) = 1I"'(K 1)"1I"'(K 2)·
When
n = 1
this
construction
corresponds
to
tying
two
consecutively in the same loop. We say that a knot is irreducible
knots
if it is
not the sum of two nontrivial knots. Schubert showed that each I-knot has an
essentially
unique
factorization
as
a
sum of irreducible
irreducible
I-knots are usually called prime
dimensions
this
is
false
in
general.
shown that every n -knot with irreducible
knots,
length
of
such
needed
for
K
is
(i.e.
and
moreover
factorizations
their
argument
trivial
if
0
is
~
for
X(K)
However
Dunwoody
and
each
knot
19871.
As
criterion for SI),
it
there the
is
a
only
recognizing
applies
with finitely generated commutator subgroup
bound on
with more
terms
than
summands K j must have group Z For each n ;> 3 there factorizations
into
have
also
bound
then K
11"'
on
the
the
result
trivial knot
=
n
when
2,
by
is a 2-knot
has a finite factori-
the Gtushko-Neumann theorem places an upper
the number of nontrivial free
factorization
Fenn
geometric
Freedman's Unknotting Theorem. (It is easy to see that if K zation into irreducibles, for
In higher
3 admits some finite factorization into
[OF a
knots (and so
knots) [Sch 19491.
irreducibles
this
factors bound
of
11"'.
then at
If K least
is a
#Ki
one
of
the
and so be trivial). are n -knots which have several distinct
[BHK
19811.
Essentially
nothing
about uniqueness (or otherwise) of factorization when n = 2.
is
known
Knots and Related Manifolds
9
A more general method of combining two knots is the process of forming satellites. Although this construction arose in the classical case [Sch 19531, where it is intimately connected with the notion of torus decomposition, we shall describe only
the higher dimensional version of [Ka 19831.
Le t K 1 and K 2 be n -knots (with n > 1) and let 1 be a simple closed curve
X(K 1)'
in
with
a
product
homeomorphic to Sn xD 2 and so carries
Sn +2_ int U
onto
a
we
may
product
I:(K 2;K 1')') = bK 1 is called the We also call K 2 a companion
U.
neighbourhood find
a homeomorphism h
neighbourhood
satellite of K 1.
Sn +2-int U
Then
K 2.
of
of K 1 about
is
which
The
knot
K 2 relative
to )'.
If either r = 1 or K 2 is
trivial
then I:(K 2;K 1'") = K 1. The group of a satellite knot may be computed by means of van Kampen's Theorem (cf. Chapter 3). Fibred knots A Seifert bypersuriace for K sion 1 submanifold of Sn+2
is a locally flat, oriented codimen-
with (oriented) boundary
K.
By
a standard
argument these always exist. As we shall make little nse of Seifert hypersurfaces in this book, we shall only ontline the argument briefly. (We shall however use the phrase "Seifert manifold" below in the sense of closed 3manifold foliated that
p:X -
the
Sl
by circles).
Using obstruction
theory, it may
pr2j-l:aX(K) _ Sn xS 1 _ S 1
projection [Ke 1965).
By
topological
extends
transversality, we
be
to
shown a
may assume
map that
the inverse image p -1(1) is a bicollared, proper codimension 1 submanifold of X
[Ou 1982). The union p -1(1)Uj(Sn X[O,IJ) is then a Seifert hypersur-
face for K. If a 2-knot has a Seifert surface which is a once-punctured connected sum of lens spaces and copies of S1xS 2 then it is reflexive [01 19621. In general there is no canonical choice of a Seifert hypersurface. However there is one important special casco An n -knot K is fibred if we may find such a map p every
point of S 1
swept
out
by
is
copies
which is the projection of a fibre bundle (i.e. if a
of
regular the
value
Seifert
of p). The hypersurface
sphere obtained
Sn +2 from
is
then
p -1(1).
which are disjoint except at their common boundary K. The bundle is de te rmined
by
the
isotopy
class
of
the
characteristic
map,
which
is
a
self
10
Knots and Related Manifolds
e
homeomorphism FXeSl =
class of
of the fibre F of p.
Fx[O,II/-,
e
where (f ,0) '" (8({),1) for
is
P
the mapping
(n
f
in F.
The
isotopy
extends to a fibre bundle projection q:M(K) _ SI.
FuDn +1 of q
=
torus of
M(K) fibres over SI
an
all
is called the (geometric) monodromy of the bundle. It is easy to
see that such a map p The fibre
Indeed X(K) is the mapping torns
is called the closed fibre
the closed monodromy. Conversely, if n ;> 2 and
then we may assume that the characteristic map fixes
+2)-disc pointwise, and we see that K
=
is true when n
of K, and M(K)
is fibred. (An analogous result
1 [Ga 1987)). Many of our examples below shall arise as
a result of surgery on a simple closed curve in such a mapping torus. (For instance Cappell and Shaneson construct their pairs of distinct n -knots with homeomorphic
exteriors
by
starting
with
the
mapping
torus
of
a
self
homeomorphism of (SI)n +1). is fibred if and only if "' is free
A I-knot K
[St 19621.
In
high dimensions we may apply Farrell's fibration theorem to obtain a criterion for an n -knot with n ;> 4 to be fibred
[Fa 1970).
This applies
also when n = 3, provided the knot group is in the class of groups for which 4-dimensional surgery
and s -cobordism
theorems are known (cf. [Fr
1983)). Little is known when n = 2. It is conceivable
that
every 2-knot
whose commutator subgroup is finitely generated and torsion free may be fibred.
However
we
shall
show
in
Chapter
8
that
if
the
3-dimensional
Poincare conjecture is true then there are 2-knots whose commutator snbgroup is Z/3Z which are not fibred. If K 1 and K 2 are fibred
then so is their sum, and the closed
fibre of K 1 #K 2 is the connected sum of the closed fibres of K 1 and K 2' However in the absence of an adequate criterion for a 2-knot to fibre, we do not know whether every summand of a fibred knot is fibred. In view of the unique factorization that
there
would be
theorem for oriented 3-manifolds one might hope
a similar
theorem
for
fibred
2-knots. However
the
fibre of an irreducible 2-knot need not be an irreducible 3-manifold. (For instance a spun trefoil knot is an irreducible fibred 2-knot, but its closed fibre is S2xSl#S2XSl). No
nontrivial
2-knot
which
is
fibred
order is reflexive [PI 19861. (See also [HP 1988)).
with
monodromy
of
odd
Knots and Relaled Manifolds
11
Spinning and twist spinning The
first
nontrivial examples of higher dimensional
knots
were
given by Artin [Ar 19251. We may paraphrase his original idea as follows. As
the
half
R3
3-space
about the axis A
=
{
={
<w,x,Y,z> in R4: w = 0, z ;> 0
is
spun
} it sweeps out the whole of R 4, and any
arc in R3 with endpoints On A
sweeps out a 2-sphere. This construction
has been extended in several ways. If K
may choose a small (n+2)-disc Sn+2 n which meets K in an n-disc B such that (B n +2 , Bn) is homeomorphic to the
standard
is
pair.
the (Artin) spin
an n -knot, we
of K
is
=
(Sn+2-int B n + 2 , K-int B n ).
Then
the (n+1)-knot CT 1 K = i(Sn+2, K)oXD 2 ).
The
(n+p)-knot
This
p-superspin
of
makes sense
for any p ;> O. In particular, CTOK = K#-K. If p > 0 then
'lrCT pK
=
'irK.
K
(Sn+2, K)o
Let
is
the
CTpK = i(Sn+2, K)oXDP+1).
In general snperspinning is distinct from iterated spinning (cf.
rCa 1970)). Since the (p+l)-disc evidently has an orientation reversing involution, all p -snperspun knots are -amphicheiral. The p -superspin of a fibred knot is fibred, and the p -superspin of the sum of two knots is the sum o"f their p -supe rspins. For our purposes, another modification devised by 1966]) is of more construction. Let
I
interest. He
incorporated
Fox (cf.
[Fo
a
twist into Artin's original n be an integer and choose B +2 meeting K as above.
Then Sn+2-int Bn+Z = DnXDZ, and we may choose the homeomorphism so that
i(K - int Bn) lies in aDn x{O}.
Let
Pe be
the
self homeomorphism of
D n XD 2
that rotates the D2 factor through e radians. Then n n UPre(K-int B )X{8} is a submanifold of (Sn+2-int B +2)XS 1 homeomorphic to Dn xS 1 and which is standard on the boundary. Therefore
is an (n+1)-knot, called the I-twist spin of K. The O-twist spin is the Artin spin, i.e. r OK = of r I K
is obtained
from
power of (any) meridian
'irK
by
central.
adjoining
the
relation
Zeeman discovered
the
CT
1 K.
The group the
I tb
remarkable
fact
making
12
Knots and Related Manifolds
that
if
r
pOi
then
0
7: r K
is
with
fibred
geometric monodromy
order
of
dividing
r, and the closed fibre is the r-fold branched cyclic cover of Sn+2 , branched over K [Ze 19651. Hence 7:1K is always trivial. The rtwist spin of the sum of two knots is the sum of their r-twist spins. The twist
spin of a
-amphicheiral knot is
-amphicheiral, while
twist
spinning
interchanges invertibility and +amphicheirality [Li 19851. The 2-twist of any knot is reflexive [Mo 1983, PI 1984']. (More precisely, if K then K" trivial
the other hand, if r > 2 then no non-
is isotopic to rK). On
cyclic
branched
cover
of
spin
= 7:2k
an
r-twist
spin
of
a
simple
I-knot
is
reflexive [HP 19881. For other formulations and extensions of twist spinning see [GK 1978, Li 1979, PI 1984', Mo 1983, Mo 19841. In [Mo 19861 it is shown how to represent
twist
spins of classical knots
by means of hyperplane cross
sections (as in [Fo 1962, Lo 1981]). Slice and ribbon knots An n -knot K
is a slice
knot if there is an (n +l)-knot which
meets the equatorial Sn+2 of Sn+3 transversally in K; if the (n+l)-knot can be chosen
to be
trivial
then
K
is
K
is
doubly slice. As
Kervaire
showed that all even-dimensional knots are slice [Ke 19651, this notion is of little interest in connection with 2-knots. However not are
doubly
slice,
and
no
adequate
criterion
is
yet
all slice knots
known.
The
O-spin
uOK = K#-K of any knot K is a slice of the I-twist spin of K and so is doubly slice [Su 19711. An n -knot
K
is
a
ribbon
knot
if it
is
the
boundary of
an
immersed (n +1)-disc t:.. in Sn +2 whose only singularities are transverse double points, the double point set being a disjoint union of discs. All ribbon knots are slice.
It remains an open question as -to whether every slice 1-
knot is ribbon, but in every higher dimension there are slice knots which are not ribbon [Hi 19791. Given such a "ribbon" (n +1)-disc t:.. in Sn +2 the cartesian
product
t:..XDP
(n+l+p)-disc in Sn+2+p.
c Sn+2 XD P C Sn+2+p
determines
a
ribbon
All higher dimensional ribbon knots derive from
ribbon I-knots by this process [Ya 19771.
As the p-disc has an orienta-
tion reversing involution, this easily implies that all ribbon n -knots with
n
~
2 are -amphicheiral. The O-spin of a I-knot is a ribbon 2-knot. Each
Knots and Related Manifolds
13
ribbon 2-knot has a Seifert hypersurface which is a once-punctured connected sum of copies of S1XS2 and therefore is reflexive [Ya 19691. (See [Su 19761 for more on such geometric properties of ribbon 2-knots). An n -knot K is a homotopy ribbon knot if it bounds a properly embedded (n+l)-disc in D n +3 whose exterior W has a handlebody decomposition
consisting
M(K).
The
(n +1)-
dual
and
0,
1
and
decomposition
(n +2)-handles,
connected. (The [GK:
of
definition
Problem 4.221
2-handles. of
and
so
W
The
relative
the that
to
inclusion
of "homotopically
requires only
boundary
this
ribbon"
its of
of
is
W
boundary
W
Minto
for
I-knots
latter condition be
clearly
has
only
is
given
nin
satisfied).
Every ribbon knot is homotopy ribbon [Hi 19791. A nontrivial twist spin of a I-knot is never homotopy ribbon [Co 19831. (See also Chapter 3). Links Knot theory is the paradigm for the general problem of codimension 2 embeddings of connected manifolds in manifolds. Although we do not intend to stray far from our concentration on 2-knots, we shall occassionally point out where the resnlts described below may be extended to more general situations. In particular, similar questions arise in connection with the groups of links and of homology spheres, so we shall describe
these
briefly. A J.I.-component n -link is a locally flat embedding n ... Sn+2. It has exterior X(L) L:J.l.S Sn+2-LXint D2 and its group is
"L = "I(X(L». A link L is trivial if it bounds a collection of J.I. disjoint (n+l)-discs
in
Sn+2.
It
is
split
if each
of
its
components
lies
in
an
(n+2)-disc in Sn+2 which is disjoint from the other components, and it is a boundary link if it bounds a collection of J.I. disjoint orientable hypersurfaces in Sn+2. Clearly a trivial link is split, and a split link is a boundary link; neither implication can be reversed (if J.I. > 1). Each knot is a boundary link, and many arguments with knots that depend upon Seifert hypersurfaces extend readily to boundary links. The notions of slice and ribbon links are natural extensions of the corresponding notions for knots (cf. [H: Chapter II]). A J.I.-component n -link is a boundary link if and only if there is a homomorphism from "L to F(J.I.), the free group of rank J.I., which carries
14
Knots and Related Manifolds
a set of meridians (one for each component) to a free basis for F(Il); such a homomorphism can be realized by a continuous map from X(L) to vllS1, the wedge of Il circles. If n '" 2 the link is trivial if and only if this map is (n+1)/2-connected [Gu 19721. When n
2 the correct criterion for
=0
triviality is unknown: it is plausible that every Il-component 2-link whose group is
freely
generated
by
meridians
is
trivial. (The
condition on
the
meridians is necessary [Po 1974)). Also unknown is a good criterion for a 2-1ink to split, and whether every 2-link is slice. (Even-dimensional boundary n
links
> 1
are
and
always
more
slice
than
1
[Gu
1972]). The
component
exterior of
never
fibres
an
over
n -link
the
with
circle
[H:
Theorem VIlI.4]. Every ribbon n -link with n > 1 is a sublink of a ribbon link whose group is free [H: Theorem 11.1]. An homology m -sphere is a closed m -manifold with the integral homology of Sm. homology tunately (Theorem
More
Il ~ O.
Unforgroups
1 of
dimensional
links of
Kervaire's characterization of high. dimensional knot
m -spheres while
generally, we may consider Il-component
in
an
homology
Chapter 2) extends
link
groups,
a
crucial
(m +2)-sphere
readily
to
theorems in Chapter 3 below is that X(X(L» when Il = I, i.e. when
we
a
assumption
characterization of
in =
with
one
of
our
high
principal
0, which is only possible
are considering knots (in homology 4-spheres).
(We may instead propose the following less standard situations to which our methods
probably
extend
without
difficulty. These are when we consider embeddings of one or several tori SlXS 1 in S3 XS1 or SlXS1XS1XS1, or of pairs of 2-spheres in S2XS 2 ). For recent work on embeddings of other surfaces in S4, see [I:'i 1981, FKV 1988, Li 19881.
15 Chapter 2
THE KNOT GROUP Kervaire characterized
as
the
finitely
which are
presentable
the
groups of n -knots (for 0
groups
each n
~
01G' ;;; Z, H (0;Z) = 0 2
with
3) and
the normal closure of a single element [Ke 19651. These condi-
tions are also necessary when n
1 or 2, but are
then no longer suffi-
cient. The group of a nontrivial I-knot has geometric dimension 2 and has one end IPa 19571. The main concern of this book is with the intermediate case n = 2. In the
this case Kervaire showed also
above conditions has deficiency
1
then it
that if a group satisfying is
the group of a
2-knot.
However not every 2-knot group has deficiency 1. Subsequently several people observed independently and approximately simultaneously that not every high dimensional knot group is a
2-knot
group. Their arguments all
used
duality in the infinite cyclic cover of the exterior of the knot. In
this
chapter
we
shall
review
their extensions and applications. In
these
results
and
various
the next chapter we shall show
of that
by using duality in more general covering spaces we can get much stronger results.
Kervaire's conditions If S is a subset of a group 0
normal closure tains S. The
of S
we shall let «S»O denote the
in 0, the smallest normal subgroup of 0
which con-
weigbt of a group is the minimum number of elements in a
subset whose normal closure is the whole group. If 0 is an element such that «g »0 '"' 0
has weight 1 and g
we call g
then
a
weight element
for 0, and its conjugacy class a weight class for O. As the group " of an n -knot K compact
(n +2)-manifold
it
H 1(X;Z) ;;; Z
and
theorem)
H 2(";Z) = 0
and
is
finitely
o.
H 2(X;Z)
is the fundamental group of a
presentable.
Therefore
(since
it
is
"I"'
the
By
Alexander
;;; Z
cokernel
duality
(by
the
Hurewicz
of
the
Hurewicz
homomorphism from "2(X) to H 2(X;Z), by a theorem of Hopf [Ho 1942]). Moreover " has weight I, for if J.I. is a meridian, represented by a simple closed curve on
ax
then X UJlD 2 is a deformation retract of Sn+2_{o} and
so is simply connected. (Alterna tively X
bounds
a
singular
disc
in
Sn+2
we may observe which
may
be
that
any loop "'I in
assumed
to
meet
K
16
The Knot Group
transversely in finitely many points; "I is then homotopic in X to a product of
conjugates
of
meridians, bounding
Kervaire showed
transverse
discs
near
that, conversely, any group satisfying ~
the group of some n -knot, for each n
3.
these
points).
these conditions is
In fact it is sufficient
"U pK
show that each such group can be realized by a 3-knot, for
to
= "K
for all p > 0, and so we may call such groups 3-knot groups. Theorem 1 [Ke 19651 A group G is a 3-knot group if and only if it
is finitely presentable, G/G' ~ Z, H 2(G;Z) = 0 and G bas weigbt 1. Proof The conditions are necessary, by the above remarks. Let P be a finite presentation for G, with g generators and r relators, and let C(P) be the corresponding 2-dimensional cell complex, with one O-cell, g I-cells and r
2-cells. For each n ~ 2 we may embed C(P) in R n +3. Choose such an
au
embedding and let U be a regular neighbourhood of the image. Then a closed s -parallelizable (n +2)-manifold, and the inclusion of an
n -connected
map,
since
=
"I(C(P» = G
Since
C(P)
a
finite
free
abelian to
"2(aU)'
is of
finite
codimension
2-dimensional complex
onto
n +1
into U is
U.
Therefore
in
and Hk(aU;Z) = Hk(C(P);Z)
rank. Since is
H 2(aU;Z)
has
C(P)
"l(aU) = "I(U)
[Ho
19421.
k C; n.
for
H 2(aU;Z) = H 2 (C(P);Z)
H 2(G;Z) = 0
the
Hurewicz map
Therefore
if
n
is
au
~
3
we
is
from can
represent a basis of H 2(aU;Z) by embedded spheres, which have trivial normal
bundles
as
au
is
s-parallelizable.
spheres we obtain a closed orientable group G
(since
of
Sn+lxS I .
"I(V) ~ G,
sphere, which is
performing
+2)-manifold
V
the surgered spheres have codimension n
has the homology of element
On
(n
we
surgery
on
these
with fundamental ~
3), and which
If we now perform surgery on a weight
obtain
therefore Sn +2 by
a
simply
the
connected
homology
(n+2)-
validity of the high dimensional
Poincare conjecture, and which contains an n -knot (the cocore of the surgery) with group G. 0
The only points at which the high dimensionality was used were where
we
wished
to
use
surgery
to
kill
H 2(aU;Z),
while
retaining
"I(V) ~ G, and when we invoked the high dimensional Poincare conjecture.
17
The Knot Group
The following extension to the case n = 2 is due to Kervaire, apart from the appeal to the subsequent work of Freedman. (Recall that if P is a fingenerators and r
ite presentation of G, with g
relators,
then the defi-
ciency of P is deE P = g - r, and deE G is the maximal deficiency of all finite presentations of G). If G satisEies the hypotheses of
Addendum [Ke 19651
Theorem 1 and
also has a presentation of deficiency 1 then G is a 2-knot group. Proof
/J
Let
be
1-g+r
I-deE P.
H 2 (aU;Z)
=
the
rank
of
H 2 (aU;Z).
/J
Then
1-1+/J = X(C(P»
=
Therefore def P .;;; I, and def P = 1 if and only if
O. In the latter case we do not need to surger any 2-spheres.
Since there is no difficulty in surge ring I-spheres in dimension 4, and since the 4-dimensional (TOP) Poincare conjecture is true [Fr 19821, the above construction gives a knot in S4. 0
It may be shown that any 3-knot group with a presentation of
deficiency 1 is in fact such
a
presentation
D5 u {h />U{h
J}
2-handle" h topy
5-ball
sphere, and
the group of a homotopy ribbon 2-knot, by using to
construct
a
5-dimensional
handlebody
with 11"1 (aW) = "1 (W) ~ G and X( W) = O.
W =
Adjoining another
along a loop representing a weight class for G gives a homoB
with
I-connected
boundary.
Thus
aB
is
a
the boundary of the cocore of the 2-handle h
homotopy
4-
is clearly a
homotopy ribbon 2-knot with group G. It is easy to see that if deE G = 1 then weight G
1 implies
that G/G' ~ Z, and that G/G' ~ Z
implies that H 2(G;Z) O. The con0 and weight G = 1 are otherwise indepen-
ditions G/G' ~ Z, H 2(G;Z) = dent. (For instance, Z-SL(2,5) satisfies
the first
two conditions, but does
not have weight 1 [GR 19621; if G is the (metabelian) group with presentation (i.e. the group of the closed 3manifold
obtained
by
O-framed
surgery
on
the
trefoil
knot)
then
has weight I, but H 2(G;Z) ~ Z, and any finite cyclic group satisfies the last two conditions but not the first). G/G'
Z
and G
If G is a group with G/G' ~ Z
and deficiency 1 then the first
nonzero elementary ideal E 1(G) is principal, and so G' /G"
is torsion free
18
The Knot Group
(cf. Chapter IV of [H», Therefore the group of the 2-twist spin of the trefoil
knot
does not
have deficiency
I, for
it
bas commutator subgroup
cyclic of order 3. Thus the deficiency condition is too stringent in general. Kervaire gave analogous characterizations of the groups of high dimensional links and homology spheres. As the proofs are similar to that of Theorem 1 we shall just state his results. Theorem
A
[Ke
A group G is tbe group of all-component 3-
1965'1
link if and only if it is finitely presentable, G/G' ~ ZIl, H 2(G;Z) = 0 and G bas weigbt Jl, If moreover G bas a presentation of deficiency Il tben it is tbe group of all-component 2-link. 0
The
G
group
of
a
Il-component
I-link
has
Il
weight
and
G/G' ~ ZJ1., but H 2(G;Z) is only 0 (eqnivalently, def G = J1.) if the link is completely splittable [H: Theorem 1.21. If we combine Theorem A with Gutierrez' characterization of high
dimensional boundary links [Gu 19721 we find that the groups of such links are distinguished by
the additional condition that
G maps onto F(Il),
the
free group of rank J1., with a set of weight elements for G mapping to a free
basis
for
F(Il).
If
we drop
the condition on
the
weight
of G
in
Theorem A we obtain a characterization of the groups of links of Il nspheres in homology (n+2)-spheres (for Il
Theorem
B
[Ke
0). In particular we have
A group G is tbe group of an bomology
1969]
m -spbere (for eacb m
~
~
5) if and only if it is
finitely presentable,
G = G' and H 2(G;Z) = o. If moreover G bas a presentation of deficiency 0 tben it is tbe group of an bomology 4-spbere. 0
Theorem B is easier in so far as there is no need to appeal to the b -cobordism theorem (in order to recognize the standard sphere). Little else is known about the gronps of homology 4-spheres. There are many finite perfect groups with presentations of deficiency O. (Note that a perfect
G
group
Hi(G;Z) or q
=
with
a
0 for i
presentation
=
of
deficiency
0
is
superperfect,
1 and 2). For instance the groups SL(2,q) for q
i.e. =
8
a power of an odd prime (q '" 3) are examples of such groups [CR
The Knot Group
19
1980]. It
is not known whether deficiency 0 is a necessary condition. In
particular
can
I-Xlis
1- = SL(2,5)
the
be
the
binary
group
of
an
homology
icosahedral group? (Note
4-sphere,
that
is
j-
where
the
only
finite group that is the group of an homology 3-sphere). The more general manifolds of
V
V
problem of codimension 2 embeddings of n-
in Sn +2 does not lead to new groups unless some component
has nontrivial
first
homology, since Hi(Sn +2_ V;Z)
i = 1 and 2, by
Hi -1 (V;Z) for
is
isomorphic to
Alexander duali ty in Sn +2 and
Poincare
duality in V.
The commutator snbgroup In our later arguments we shall often first identify the commutator subgroup "' of a knot group ", and so we shall reformulate the Kervaire conditions. We say that an automorphism of a group G is meridia-
na} if «g-l(g)lg in G»G = G.
If H
is a characteristic subgroup of
G
then clearly induces a meridianal automorphism of the quotient GIH.
In
particular
the
induced
endomorphism
H 1( O.
Now
H 2(7r';Z)
is
the
cokernel
of
the
Hurewicz map from 7r 2(X') to H 2(X';Z), which commutes with the action of the covering group, and so multiplication by t-l gives an automorphism (namely H 2 (6)-I) of H 2(rr';Z) also. Conversely any such group " has weight I, abelianization Z, and on applying the Wang sequence for the projection of K(rr',I) onto K(7r,l) we find that H 2 (7r;Z) = 0, and so 7r is a 3-knot group. 0
Hausmann and Kervaire have expressed the condition that 7r'X(JZ be
finitely
presentable
in
terms
of
7r'
having
a
"finite
Z -dynamic
The Knot Group
21
presentation" [HK 19781. When "' is an abelian group A
the condition that the automor-
=
phism be meridianal reduces to the action of t -Ion A
HI (A;Z) being
invertible. Moreover in that case the group H 2(A;Z) may be identified with the
AAA
exterior product
[R: page
3341. Levine
and
Weber have
made
explicit the conditions on the pair (A,!) in the following way [LW 1978]. If A
let t.O(A) be
is a finitely generated A-module
the highest common
factor of the ideal EO(A) in A, and for each prime p let t.(A. Then "' is cyclic of order 5, t-l
acts as the identity and H 2("';Z) = 0, so " is a 3-knot group. However it is easy to see that if I( , ) is any nondegenerate Z -bilinear pairing on "'
26
The Knot Group
with values in Q/ Z
then Jet a.,t p) = 4/(a.,p) = -J(a.,P) for all a.,p in rr'. Thus
t cannot be an isometry and so rr is not a 2-knot group.
Centres
Since the commutator subgroup of a classical knot
group rr can
contain no nontrivial abelian normal subgroup [N: Chapters IV,VI, any nontrivial abelian normal subgroup of rr must map injectively to rr/rr' = Z
and
so be central. Burde and Zieschang have shown that in fact the knot must then be a torus knot [BZ 19661. In high dimensions the situation is quite different. Hausmann and Kervaire have shown that every finitely generated abelian group A centre of some n -knot group, for each n that
if P
is a
finitely
presentable
~
3 [HK 1978'1.
superperfect group and
is the
They observe G
is a
knot
group then by the KUnneth theorem Hi(GXP;Z) = HlG;Z) for i " 2, and
If moreover there is an c1ement P in
GxP has centre ((G XP) = CG XCP. P
such that the subgroup [p,PI generated by {(p,xl = pxp- 1 x- 1 :x in P}
is
the
weight
whole of P, and if g element
for
erally if P l' in
Pi
Pi)
CG (i)CP 1(i).
it
shall
then
which is
GXP, Pr
is a weight element
are
r
GXP 1 X
therefore a
for G, then gp
3-knot
group.
is a
More gen-
such superperfect groups (with such elements is
. . xP r
a
3-knot
group
with
centre
. (i)CP r" Thus to obtain the result of Hausmann and Kervaire
suffice
to
take
G
to be
a
knot
group
with
trivial centre (for
instance the group of the figure eight knot) and to show that each cyclic group is the centre of some such group P. For the infinite cyclic group we may take P to be the fundamental group of a suitable Brieskorn homology 3-sphere, for instance the group with presentation , with p may
use
the element represented by a. For the cyclic group of order 2 we I" = SL(2,5) (which
3-sphere), with p
is also
the group of a
Brieskorn homology
any noncentral element. If k :> 3 and F
is any finite
field containing k distinct k tb roots of unity (other than F 4) then SL(k,F) is a finite superperfect group (cf. [M: pages 78,94» with centre cyclic of order k; we may take p in
to be the elementary matrix e £2 (with entry 1
the (1,2) position). Hausmann and Kervaire gave more general construc-
tions and showed that each such A groups.
is the centre of infinitely many 3-knot
The Knot Group They left open the questions as to whether the centre need hI finitely
generated,
Chapters 3 and
and
4
we
what
the
centre
shall show
that
of
a
2-knot
the centre
group
of a
may
2-knot
In
be.
group i.
either torsion free of rank 2 or has rank at most 1. CExamples are known with centre Z2, Z(i)CZ/2Z), Z or Z/2Z). Whether the centre of a 2-knot group must be finitely generated is related to the analogous question for 3-dimensional Poincare duality groups, which seems difficult. ~
If FCp.) is a free group of rank p.
2, and if P l'
finitely presentable superperfect groups with elements Pi as above, then the group FCp.)XP 1 X (for each n
XP r
..•
is the group of a p.-componen t boundary n -link
~
3) with centre (i)CP i . Thus the centre of a p.-component nlink can be any finitely generated abelian group, if p. ~ 1 and n ~ 3. We
shall show p. > 1
in Chapter 3
that
the
centre
of a
p.-component
2-link
with
must be a torsion group.
MiDimizing the Euler characteristic We saw above that a 3-knot group " must satisfy certain homological conditions, notably H 2(";Z)
=
D, and
that if the stronger, combina-
torial condition del" = 1 also holds then " is a 2-knot group. Analogous situations arise in attempting to characterize the groups of 2-links, homology 4-spheres etc. Now every finitely presentable group is the fundamental group of some closed
orient able
4-manifold. Thus
we
may define
a
new
invariant which may help bridge the gap between necessary homological and sufficien t combina torial conditions by qCG)
=
min{XCM)\ M a closed orientabJe 4-maniloJd with "lCM)
=
G}.
Hausmann and Weinberger observed that this invariant is well behaved with respect to subgroups of finite index, since the Euler characteristic is multiplicative in finite coverings, and it is this property that makes qC ) useful. For any space number of fJiCKCG,1);F).
Theorem 4
M
and
M
with coefficients
field F,
and
F
Ie t
for
PiCM;F) be
any
group
G
the
i th
let
PiCG;F) =
Then we have the following estimates.
[HW
1985]
2(1-P1(G;F»+P2(G;F)
~
Let G be a q(G)
~
finitely presentable group.
2C1-del G).
Then
Be tti
28
The Knot Group
Proof Le t
be
M
HI(GjF)
and
manifold
then
any
space maps
H (MjF) 2
by
with
fundamental
onto
If
H (GjF). 2
Poincare duality
group G. Then HI (MjF) =
X(M) =
is
M
an
orientable
4-
These
2(I-PI(M;F»+P2(M;F).
two observations give the first inequality. If P is a finite presentation for G and C(P) is the correspond-
ing 2-complex then we may embed C(P) in R S. regular
neighbourhood
is
=
lTI(V) = G and X(V)
then
a
Proof If M
V
of a
4-manifold
with
2(1-def G), and so we get the second inequality. 0
Coronary Let H be a subgroup which has 2(I-PI(HjF»+P2(HjF) "
The boundary
s -parallelizable
closed
finite index in G. Then
[G:H]q, in
presen ta tion which
each
of
the
relation
form
asserts
the
conjugacy of two of the generators. In this section we shall examine some of the connections be tween these properties. Since the Artin spin of a I-knot is a ribbon 2-knot every 1knot group is the group of some ribbon 2-knot. By an elementary handle sliding argument it may be seen that any ribbon
D
-link (with
D
~
2) is a
sublink of a ribbon link whose group is free [H: Theorem 11.11. It follows that the group of a J.I-component
ribbon
inger) presentation of deficiency J.I,
D
-link (with
knots
[Ya
1969]). Conversely
any
(ribbon)
D
group
(n+3)-manifold
of
the group has
the groups of ribbon 2-
weight
J.I
with
the group of a J.I-component
-link with group free, for each
the group of a homotopy ribbon
:> 2) has a (Wirt-
which is optimal, since
weight J.I. (This was first proven by Yajima for presentation of deficiency J.I is
D
D
D
~
-knot (with
a
(Wirtinger)
sublink of a
2 [H: Theorem 11.31. Since D
~
2) is
the group of a
W which can be built with 0-, 1- and 2-handles only and
which has Euler characteristic 0, such groups also have deficiency l.
30
The Knot Group
Levine
showed
that
if 7r
has
a
presentation
P
such
that
the
presentation of the trivial group obtained by adjoining the relation killing a meridian to P is AC-equivalent
to the empty presentation then 7r is
group of a smooth knot in the standard 4-sphere.
the
According to Yoshikawa,
7r has such a presentation if and only 'if it has a Wirtinger presentation of deficiency 1 [Yo 1982'].
(See also lSi 1980] for connections between Wirt-
inger presentations and the condition that H 2(7r;Z) = 0). A group has (finite) geometric dimension
2 if it is the funda-
mental group of a (finite) aspherical 2-complex, but is not free. Every such group has cohomological dimension 2. It is an open question as to whether every (finitely presentable) group of cohomological dimension 2 has (finite) geometric dimension 2 (cf.
[W': Problem D.4D. The following partial answer
'to this question was first obtained by Beckmann under the further assumption that G has type FL (cf [Dy 1987']).
Theorem
6
Let G
be a
finitely presentable group.
Then
G
has
finite
geometric dimension 2 if and only if it has cohomological dimension 2 and deficiency Pl(G;Q)-P2(O;Q).
Proof Let P and let
be a presentation for G
C(P) be
the
with g
generators and r
relators,
corresponding 2-complex. Then def P = l-X(C(P»
PI (C(P>;Q)- P2 (C(P);Q) "
PI (O;Q)- P2(O;Q),
and
the
=
necessi ty of the condi-
tions is clear. The cellular chain complex of the universal cover C(P) may be
viewed
as a
finite
chain complex of free
left
Z[G)-modules, and so
there is an exact sequence
o -
L
ZIG]g
= 7r 2(C(P»
As c.d.G = 2, the image of Z[G]r lemma. Therefore the inclusion of L Since
c.d.G = 2,
the
in Z[G]g
ZIG] -
Z
is projective, by Schanuel's
into Z[G]r splits, and L
Hattori-Stallings
rank
O.
of L
is projective.
is concentrated on
the
conjugacy class of the identity [Ec 1986], and so the Kaplansky rank of
L
is
Q~Z[G]L
the
dimension
=
0 and so L
of
=
Q~Z[G)L.
If def P = Pl(G;Q)-P2(G;Q)
then
0, by a theorem of Kaplansky. (See Section 2
of [Dy 1987] for more details on the properties and interrelations of the various notions of rank). Hence C(P) is asphericaJ. 0
31
The Knot Group
Suppose now that G has weight p. and a presentation P of deficiency
and
p.,
obtained
by
Ie t
adjoining
element subse t
be
CCP)
G
of
the
corresponding
2-cells
f.J.
whose
to
2-complex.
along
CCP)
normal closure
is
loops
the
The
2-complex
representing
whole
a
p.-
group is simply
connected and has Euler characteristic 1, and so is contractible. Therefore if the Whitehead conjecture is true the subcomplex CCP) must also be aspherical, and so G
is
free or has finite
geome tric dimension 2. (In particular,
this is so if G is a I-relator group Ccf. [Ly 1950, Go 1981)), or is locally indicable [Ho 19821 or if it has no nontrivial superperfect normal subgroup [Oy 1987)).
Thus Theorem 6 and the
Whitehead conjecture together imply
that a 3-knot group has finite geometric dimension 2 if and only if it has deficiency 1, in which case it is a 2-knot group.
If the commutator subgroup of a 2-knot group Tr with deficiency 1 is finitely generated must it be free? This is so if Tr is a classical knot group [N: Theorem 4.5.11 or if c.d.Tr = 2 and Tr' is almost finitely presentable [B: Corollary 8.61.
Sphericity of the eJ:terior The
oustanding property
of
the
exterior of a
classical
knot
is
that it is asphericaJ. In contrast, the exterior of a higher dimensional knot is aspherical only when it has the homotopy type of a circle, in which case the knot must be
trivial [OV 19731. The proof that we shall give is due
to Eckmann, who also showed that the exterior of a higher dimensional link with more than one component is never aspherical [Ec 19761.
(The exterior
of a I-link is aspherical if and only if the link is unsplittable).
7
Theorem
such
1973, Ec
19761
that X(K) is aspberical.
Proof
Let
there
Therefore through
i:aX -
Z
Tr 1CaX)
i.e.
[OV
X
be
and since X
are the
maps map
Hn+l CS 1ih#Z[TrJ)
HD+1CX,aXiZ[TrJ) -
K
is
natural
trivial.
inclusion.
is aspherical i S1
j:ax i#
Then
the
be an n -knot, for some n > 1,
Let K
from and
and
factors
h:S 1 _
HD+1(XiZ[Tr))
so
is
the
H1CXiZ[TrJ) = H1CXiZ)
0
X
to
Since
n
> 1
we
have
through S 1 = KCTr 1CaX),I), with
i
homotopic
H n + 1C8X;i" Z[Tr])
map.
By
Poincare
to
hj.
factors duality,
0, and so Hn+1CXiZ[TrJ) =
o.
32
The Knot Group
By Poincare duality again, H 1(X,oX;Z[IT» - 0 and so HO(oX;i" Z[IT» maps isomorphically to HO(X;Z[IT». But this means that the induced cover of ax is connected and so i.:1T 1(oX)
-+
1T
1(X) is onto. Therefore
IT
and so
Z
-
X '" Sl, since it is aspherical. The theorem now follows from the unknotting criterion. 0
Both Dyer and Vasquez and Eckmann prove somewhat more general results. Eckmann also observes that the full strength of the auumption of asphericity is not needed for
the above theorem. Together [Sw 1976]
and [Du 1985] imply that if in:1Tn (aX)
-+
IT
n (X) is the 0 map then
K
must
be trivial. In
the
M(K) obtained by
such a
knot
next
chapter
we
shall
show
that
the
closed
manifold
surgery on a 2-knot is often aspherical. The group of
has one
end; however Gonzalez-Acuna
and
Montesinos have
given examples of 2-knot groups with infinitely many ends, of which the simplest has presentation [GM 1978]. (This group is evidently an HNN extension of the metacyclic group generated by
{a ,b}
; it may also be viewed as the free product of
an isomorphic metacyclic group with the group of the 2-twist spun trefoil knot, amalgamated over a subgroup of order 3). Weight elements, classes and orbits
Two knots K and K 1 have homeomorphic exteriors if and only if there is a homeomorphism from M(K 1) to M(K) which carries the conjugacy class of a meridian of K 1 to that of K (up to inversion). In fact if M is any closed orientable 4-manifold with
X(M)
'"'
0 and with
IT
-
1T 1 (M) of
weight 1 then surgery on a weight class gives a 2-knot with group Moreover, if t and u are two weight elements and f
IT.
is a self homeomor-
phism of M such that u is conjugate to f .. (t±1) then surgeries on t and u lead to knots whose exteriors are homeomorphic (via the restriction of a self homeomorphism of M isotopic to
f).
Thus the natural invariant to dis-
tinguish between knots with isomorphic groups is not the weight class, but rather the weight orbit: the orbit of a weight element under the automorphisms of the group.
The Knot Group
33
A refinement of this notion is useful in distinguishing between oriented knots. If w is a weight element for
{"(w):,, in Aut(7T), ,,(w) ... w mod
7T'}
then we shall call the set
7T
a strict weight orbit for
A strict
7T.
weight orbit determines a transverse orientation for the corresponding knot (and
its
Gluck
recons truction).
An
orien ta tion
determined by an orientation for M(K). If K
for
the
ambien t
sphere
is
is invertible or +amphicheiral
then there is an orientation preserving (respectively, orientation reversing) self homeomorphism of M
which reverses the transverse orientation of the
knot, i.e. carries the strict weight orbit -amphicheiral
there
is
an orientation
to its inverse. Similarly, if K
is
reversing self homeomorphism of
M
which preserves the strict weight orbit. Theorem 8
Let G be a group of weight 1 with G/O' ~ Z, and let
be an element of G whose image generates G/O'.
For each g in G let
cg be the automorphism of 0' induced by conjugation by g.
Then
weight element if and only if c t is meridianal; (ii) two weight elements t, u are in the same weight class if and only if there is an element g of G' such that C u - Cg C t Cg- 1; (iii) two weight elements t, u are in the same strict weight orbit if only and if there is an automorphism d of 0' such that C = dc d- 1 u t and d c t d- 1c t-1 is an inner automorphism; (i)
t is a
(iv)
if
t and u are weight elements then u is conjugate to (g"t )±1
for some g" in 0". Proof The verification of (i-iii) is routine. If t and u are weight elements then, up to inversion, u must equal g't
0". Let g"
for some g' in 0'. Since t -1 acts
we have g' = khth -1 t -1 for some h in 0' and k in
invertibly on 0'/0"
- h -1kb . Then u = g't = hg"th -1. 0
An immediate consequence of this theorem is that if t are in the same <Strict weight orbit then c
and u
u have the same order, t and their images in Out(O') are equal. Moreover if C is the centralizer of
ct
in
Aut(O')
then
II(Aut(G' )/Clnn (0'»
~
the
strict
weight
IIOut(O') weight
and
orbi t
classes.
C
of In
contains general
infinitely many weight orbits [PI 1983'). However, if
7T
there
at
most
may
is metabelian
be the
34
The Knot Group
weight class (and hence the weight orbit) is unique up to inversion, by part (iv) of the theorem. Fibred 2-knots
Many of the
most
interesting examples arise
from 4-manifolds
which fibre over the circle; the knots are then fib red knots. The counter example of Farber given above shows that some restriction on torsion is needed for the next result. Theorem
9
Let
71'
be a
torsion
free 3-knot group such that rr' is the
fundamental group of a closed orientable 3-manifold N whose Haken, hyperbolic or Seifert
fibred.
Then
71'
is the group of a
factors are fibred
2-knot with closed fibre N. Proof Let N = P#R where P is a connected sum of copies of SlXS 2 and R
has no such factors. The meridianal automorphism of rr' ~ 71'l(N) may be
realized by a self homotopy equivalence g of N ting Theorem of [La 19741, g
[Sw 19741. By the Split-
is homotopic to a connected sum of homo-
topy equivalences between the irreducible factors of R
with a self homo-
topy equivalence of P. By our assumptions, these maps are homotopic to homeomorphisms.
(Cf. [Wa 1968, Mo 1968, Sc 19831 and [La 19741 respec-
tively). Thus we may assume that g is a self homeomorphism of N. Surgery on a weight class in the mapping torus of g closed fibre N and group
71'.
gives a fibred 2-knot with
0
Since N is determined by rr' and homotopy implies isotopy if N is a connected sum of Haken manifolds and copies of SlXS 2 [La 19741, or is hyperbolic [Mo 19681 or Seifert fibred [Sc 1985, BO 19861, the mapping torus of g
is unique up to homeomorphism among fibred 4-manifolds with
such a fibre. The r-twist spin of a i-knot K
is a fibred 2-knot, and
factors of its closed fibre are Haken, hyperbolic or Seifert fibred.
the With
exceptions for small values of r, the factors are aspherical, and 2 SlxS is never a factor. (See Chapter 5 for a more precise statement).
some
3S
The Knot Group
If
the
group
H 2(rr';Z) == 0
and
so
71'
the
of
a
closed
fib red fibre
S1xS 2 with a homology 3-sphere.
nected if and only if
71"
is
2-knot a
has
deficiency
1
then
connected sum of copies
of
The homology 3-sphere is simply con-
is free. Cochran has shown that if the closed fibre
is #r(S1XS2) the knot is homotopy ribbon, and that conversely the closed fibre of a fib red homotopy ribbon 2-knot is a connected sum of copies of S1XS 2 with a homotopy 3-sphere [Co 1983]. For fibred ribbon knots this follows from an argument of Trace. If V an n -knot K
q of
is any Seifert hypersurface for
then the embedding of V in X extends to an embedding of
== VuDn +1 in M, which lifts to an embedding in M'. Since the image
V
H n +1(M;Z)
in
is
Poincare
dual
to
Hom(7I',Z) == 1M,S1], its image in H n +1(M';Z)
fibred
is homotopy
M'
degree 1 map from
equivalent
to
a
generator
of
H 1(M;Z)
is a generator. If K
Z
the closed
fibre
P,
so
there
is
is a
q to P and hence to any factor of P. If moreover
n == 2 and 71'1(V) is free then each prime factor of
II
must be S1xS 2 or
a homotopy 3-sphere, by Lemma 1 of [Tr 19861. In particular, since ribbon 2-knots have such Seifert hypersurfaces no nontrivial fibred ribbon 2-knot is a twist spin. Ruberman has used a similar idea
together with the fact
that
the Gromov norm of a 3-manifold does not increase under degree 1 maps to show
that no fib red 2-knot with closed fibre
can have a Seifert hypersurface 1987].
(In
particular,
q
V
cannot
such that be
a
q
a hyperbolic 3-manifold is a graph manifold [Ru
Seifert
fibred
3-manifold
or
,r(S1XS 2». He also observes that the "Seifert volume" of Brooks and Gold-
man [BG 1984] may be used in a similar way to show that some fib red 2-knots
whose closed fibre
ribbon knots.
has an SL(2,R )-geometric structure cannot be
36 Chapter 3
LOCALIZATION AND ASPHERICITY Although the exterior X(K) of a nontrivial 2-knot K
is never
aspherical, in many cases the closed 4-manifold M(K) is aspherica!. so for instance if K
This is
is one of the Cappell-Shaneson knots, or (with fin-
itely many exceptions) if it is the q -twist spin of a prime classical knot for some q > 2. In this chapter we shall show that this is true whenever the knot group has a nontrivial torsion free abelian normal subgroup and is cohomologically
i-connected at
latter condition might
fail).
infinity.
The
knot
(We shall group
is
also consider how
then
a
the
Poincare duality
group of formal dimension 4 and orientable type, or PDt-group for short. (We
shall
also
show
that,
conversely,
if
is
TrK
a
PD 4 -group
and
H 1(Tr';Z /2Z) '" 0 then it is orient able and M(K) is asphericaD.
Our argument is based on the idea of embedding the group ring in
Z[Tr]
a
larger ring
R
in which
an
annihilator
for
the
augmentation
module becomes invertible and for which nontrivial stably free modules have well defined strictly positive rank, with duality
and
the
cohomological
having rank
RD
condition
on
Tr
we
D.
then
Using Poincare find
that
the
equivariant homology with coefficients R of a closed orient able 4-manifold wi th group
is concentrated in degree 2 and is stably free
Tr
as an R-
module. Its rank may then be computed by an Euler characteristic counting argument. When
Tr
has a nontrivial torsion free abelian normal subgroup the
existence of such an overring is guaranteed by a remarkable lemma of Rosset [Ro 1984]. This strategy works in considerably greater generality, provided we
forgo
groups
some
have
information
abelian normal
about
torsion.
subgroups,
For instance, although
there
may
be
finitely
solvable
presentable
infinite solvable groups in which no such subgroup is torsion free. In order to get around this problem we may factor out the maximal locally-finite normal subgroup. (This idea is due to Kropholler). The quotient of a 2-knot group by such a subgroup is then usually a PDt-group over Q.
Rosset's Lemma The keystone of the argument of this chapter (and hence of the whole book) is the following lemma of Rosset.
Localinlion and Asphericity Lemma
[Ro
198.. )
I"
Let G be a group witb a
torsion
free abcliu
normal subgroup A, and let S be tbe multiplicative system Z[A]-{O}
In
ZIG]. Tben tbe (noncentraJ/) localization R = S-1Z[G] exists and bas tbe property tbat eacb nontrivial finitely generated stably free R -module bas well defined strictly positive rank, witb R n baving rank n. Moreover R contains ZIG] as a subring, and is nontrivial tben RCj)Z[G]Z
The
prototype
=
flat as a Z[G]-module, and if A is
O. 0
of such
a
result
was
given
by
Kaplansky
who
showed that for any group G the group ring ZIG] has this strong "invariant basis number" property (cf. [K: page 122]). Rosset observes
that
the
mUltiplicative system S satisfies the Ore conditions (cf. [P: page 146]) and so the localization exists and is flat; if a
is a nontrivial element of A
then a -1 is in S and annihilates the augmentation module Z. Beyond this his
argument
follows
that
of
Kaplansky in making use
of properties
of
C· -algebras. (In [Hi 1981] we stated such a lemma for the case when A is central, and tried to derive it algebraically from Kaplansky's Lemma, but there was a gap in our argument). On the evidence of his work on 1-relator groups, Murasugi conjectured that the centre of a finitely presentable group other than Z2 of ~
deficiency
1 is infinite cyclic or trivial, and is trivial if the group has
deficiency > 1, and he showed that this is true for the groups of 1-links [Mu 1965]. (The classical knots and links whose groups have nontrivial centre have been determined by Burde, Zieschang and Murasugi [BZ 1966, BM 1970». As a corollary to our first application of Rosset's Lemma we shall show
that
a
stronger
conjecture
presentable groups, including all
is
true
for
a
large
class
of
finitely
those with a central element of infinite
order. Theorem
1
Let W be a
finite connected 2-dimensional cell complex sucb
tbat G = "l(W) bas a nontrivial torsion Tben X( W) Proof Let ture. Then
~
W
free abelian normal subgroup A.
0, and X( W) = 0 if and only if
W is aspberical.
be the universal cover of W with the equivariant cell struc-
the cellular chain complex of
W
may be
viewed as a
finite
38
LocaIization and Asphericity C2 -
chain complex of free left Z[G]-modules C. = 0 -
where Ci has rank ci' the number of i -cells of W. Since nected, H O(C.) = Z
C1 -
W
Co -
0
is simply con-
and H 1(C.) = 0, while H 2(C.) = H 2( W;Z) = 1I'2( W) is
a submodule of C 2 . Let S be the multiplicative system Z[A]-{O} in Z[G].
Zs
is nontrivial
=
an exact sequence 0 follows
that
Since A
0, and so on localizing the chain complex C. we obtain H 2 (C.)S -
is
H 2(C.)S
a
c O-c 1 +c2 = X(W), which must
C 2S -
stably
C 1S -
free
therefore be
COS -
0 from which it
Z[G]S-module
nonnegative. As
of
rank
H 2(C.) is a
submodule of C 2 , which embeds in C 2S ' it is 0 if and only if its localization is O. Thus W is contractible if and only if X(W) - O. 0
The assumption that to show that
W be 2-dimensional is not needed in order
W aspherical implies that X(W) = 0; this is in fact Rosset's
application of his lemma. Gottlieb obtained the first such result in the case of an aspherical complex whose fundamental group had nontrivial centre [Go 1965]. The
lemmas
of
Kaplansky
and Rosset
have
been
used
in
related
ways in connection with the Whitehead conjecture on the asphericity of subcomplexes of 2-dimensional K(G ,1)-complexes (cf. [Dy 1987] and the references therein). Corollary 1 If a
finitely presentable group G bas a nontrivial torsion
free abelian normal subgroup tben it bas deficiency at most 1. If G bas deficiency 1 and is not Z
tben it bas
finite geometric dimension 2. If
moreover tbe centre CG of G is nontrivial and G is nonabelian tben CG is infinite cyclic and tbe commutator subgroup G' is Proof Let P be a finite presentation for G
free.
and let C(P) be the related
2-complex. Then X(C(P» = 1-del P and so the theorem implies directly all but the last two assertions, which then follow from [B: Theorem 8.8].
As the groups of classical links are all deficiency
)
1,
this
results of Murasugi.
corollary
implies
immediately
0
torsion free and have the
above-mentioned
LocaIization and Asphericily Corollary
2
a
If
39
has deficiency 1 and is nonabelian and O/G'
Z,
then G' is infinite.
= a
Proof Let C be a finite 2-complex with 1T1 (C) were finite then
a
= o.
and X(C)
If A'
would have an infinite cyclic subgroup of finite index.
The corresponding covering space of C would be a finite 2-complex with fundamental group Z
and Euler characteristic
o.
It is easy to see that the
a
universal cover of such a complex must be contractible, and so
must be
torsion free, and therefore infinite cyclic. 0
This corollary does sufficient
to work
with
the
quick proof that if the group
not
really
commutative IT
need
Rosset's
ring I\.
Note
Lemma, that
for
this
it
is
gives a
of a nontrivial classical knot K
has fin-
itely generated commutator subgroup then it has one end. It then follows easily from Poincare duality that
is aspherical.
X(K)
A cyclic branched cover of S3, branched over a knot
connected sum of
the cyclic branched covers of
ITT: rK
prime
K, is the
of K. 2 These are irreducible, and cannot be S1xS [PI 1984]. Therefore the commutator subgroup of
the
factors
is a free product of finite groups and PDt-
groups, and is never a nontrivial free group. (Thus if
ITT: rK
and not Zit has cohomological dimension 4). Since
K
ITT: r
is torsion free
has a cen tral ele-
ment of infinite order, Corollary 1 implies that it cannot have deficiency 1, and so in particular
T: rK
cannot be a
nontrivial homotopy
ribbon 2-knot
(cf. [Co 1983]).
Some of the arguments of the next few chapters may be seen in microcosm
in
next theorem. We shall let ¢> denote the group with a 2>. The centralizer of a normal subgroup A of presentation 1 then TrL does
not have any such pair of subgroups T, group TrL
Proof The
MT
4 and 0 otherwise. Therefore OtT
M. The second assertion then follows from [B: Proposition 4.9].
Corollary
in
is
the
u.
fundamental group of
the closed 4-manifold
obtained by surgery on L, which has Euler characteristic 2-21-'. 0 Corollary
3
If
the
fundamental group of a closed 4-manifold N has an
abelian subgroup A of
finite index then X(N)
;it
0, and if X(N) -
0 then
A has rank I, 2 or 4.
Proof By passing to a subgroup of finite index we may assume that N
is
that Tr 1(N) A and is free abelian, of rank p say. We may clearly assume tha t p > 1. If P > 2 the theorem implies tha t either
orientable, and
X(N) > 0 or N is aspherical, and then p
=
4. If P = 2 then the localized
spectral sequence still collapses, and Poincare duality implies that the only nonzero localized homology module has rank X(N), which therefore must be nonnegative. 0
43
Localization and Asphericity
The manifolds Sl XS 3, SlXS 1 XS 2 and SlXS 1 XS 1 XS 1 have Euler characteristic
° and
fundamental group free
abelian of rank
1, 2
and 4
respectively. (See also Corollary 3 of Theorem 3 of Chapter 7). Corollary
finite (i.e.,
If
.. Tr
Tr
is a virtually abelian 2-knot group tben eitber
is virtually Z) or
is torsion
Tr
is
free and virtually Z4.
Proof It is readily verified tha t a group with abelianiza tion Z Z
Tr'
is virtually
'if and only if its commutator subgroup is finite, while no such group
can be virtually Z2 or Z3. 0
We shall determine completely such 2-knot groups in Chapters 4 and 6. The cohomological conditions In this section we shall show that the cohomological hypotheses of Theorem 3 are automatically satisfied if the group UIT is large enough. If I
is a finitely generated group then e(l)
the number of ends of I. If I
Lemma
1
is infinite H°(J;Z[I])
° also
one end then H1(J;Z [I]) =
0, 1, 2 or
(>=
finitely generated tben Hr(A;F) Proof Let B
=
be a free
>=
° if
i < r. If A is torsion
2.4. 1
a
has
Therefore
>=
°
~
r. If
is finitely generated and
Since B is an FP group and F is free
Hi(B;Z[Bl)(i/)Z[B1F
Hi (B;F) =·If
but A is not
of finite rank s
we may assume B has rank r and if A
B-module, Hi(B;F)
CD
free
0.
abelian subgroup of A
torsion free we may take B = A. as
if I
[Sp 19491.
and finitely generated tben Hr(A;F) ~ Z(1). If r
HP+q(A;F). 0
44
Localization and Asphericity
Theorem
Let I be a finitely generated group with IIJ' ;; Z and
4
which has an abelian normal subgroup A of rank at least 2. Then HS(J;ZII]) =
Proof
By
° for
Lemma
s ~ 2. E~q
the
1
terms
HP(JIA;Hq(A;ZII])) => HP+q(J;ZIJ»
of
vanish
the
LHS
spectral
for
q ~ 2, if A
sequence has
rank
greater than 2, or if it has rank 2 and is not finitely generated. If A
is
finitely generated and of rank 2 then we may assume that it is free, and E~q
=
° for
q ~ 1.
I
Moreover
Theorem 3), so E~2
must
H0(JIA ;ZIJIA])
clude that HS(J;ZII» =
° for
Corollary If a 2-knot group
=
be
infinite
° also.
(cf.
Corollary
4
of
In all cases we may con-
s ~ 2. 0
Tr
has a nontrivial torsion
free abelian
normal subgroup A then the rank of A is at most 4. Proof This is an immediate consequence of Theorems 3 and 4. 0
In Chapter 2 we saw that every finitely generated abelian group is the centre of some 3-knot group. The rank 1 case is somewhat more delicate. In the next theorem we shall let F(a ,tY' denote the set of all words in the second commutator subgroup of the
free
group on
(a
,0. As in Chapter 2, we
shall
Ie t
zA
denote the Z -torsion subgroup of an abelian group A.
Theorem
S
Let I be a finitely generated group with IIJ' ;; Z
and
which has an abelian normal subgroup A of rank 1. If e(JIA) = J' is
= 1 HS(J;ZII]) =
finite. If e(JIA)
entable then
and moreover A ;; Z
° for
or I is
° then
finitely pres-
s ~ 2. If e(JIA) = 2 then IlzA has
a finite normal subgroup N such that the quotient has a presentation . Since
we must have m-II = ±1. Now I
has a subgroup of finite index which maps onto Z
abelian kernel A. Therefore if I
with
is finitely presentable, this subgroup is a
constructible solvable group by (BB 19761 and (BS 19781, and so I ally torsion free. We may then assume
that zA = O. Moreover H
is virtuis then
also finitely presentable, and so it has an equivalent presentation of the form
for some sufficiently large
T.
By a Reidemeister-Schreier rewriting process
(MKS: page 1581 we get a presentation
for H. Clearly H
may be obtained from the group with presenta tion
Localization and Asphericity
46
as a direct limit of a sequence of amalgamated free products, and so contains
this
group
ZlmZ.ZI(m+l)Z
as
a
subgroup.
with
presentation
But
this group maps onto the group via the ·map
sending a 0 to c , a r+l to b and a I' . . . a r
to I, and so can only be
abelian if m or m + 1 is 1. In both cases we then have J I N
When J
=
H ::; 4>. 0
is finitely presentable, the last case of Theorem 5 may
also be extracted from [Tr 19741 or [HK 19781. Is the finite presentability of J needed in the case when e (J I A) = 11 Locally-finite subgronps A
group is said to be locally-finite
if every finitely generated
subgroup is finite. In any group the union of all the locally-finite normal subgroups is the unique maximal locally-finite normal subgroup [R: Chapter 12.11. (We use the hyphen to avoid confusion with the stricter notion of locally-Uinite and normal) subgroup). Clearly
there are no nontrivial maps
from such a group to a torsion free group such as Q. This notion is of particular interest in connection with solvable groups.
Since every finitely genera ted torsion solvable group is finite [R:
5.4.111, if 0
is a finitely genera ted infinite solvable group and T
maximal locally-finite normal subgroup then OIT is nontrivial. it
has
a
nontrivial
abelian
normal
subgroup, which
is
is its
Therefore
necessarily
torsion
free. Thus we may apply the theorems of the preceeding two sections to 4-manifolds with such groups. We shall consider solvable 2-knot groups in Chapter 6. For the present we shall give a more general result. Theorem
6
Let Tr be the group of a 2-knot K and T be its maximal
locally - finite normal subgroup, and suppose Tr has a normal subgroup U such tha t U IT is a nontrivial abelian group. If U IT has rank 1 assume also that e(TrIU)
, so the relations {rl' . .
rq,w} may
be assumed to be words in F(a ,t)". On adjoining finitely many more relations, we find tha t unless
D
1 there is a subquotient of G' (and hence
of Tr) which is free, contrary to Tr being locally-finite by metabelian. 0
We shall see in Chapter 4 that if Tr' is infinite then T is either trivial or infinite. We know of no examples in which T is infinite. PD-groups, asphericity and orientability If M
is an aspherical closed orient able 4-manifold then Tr 1(M) is
a PD: -group. Since we may increase the homology of M without changing the fundamental group by taking the connected sum with a l-connected 4manifold, the converse is in general false. However it is conceivable
that
48
Localization and Asphericity
every PD: -group 0 manifold, N
is the fundamental group of some such aspherical 4-
say. We then have X(N) = q(O). In this section we shall see
that in certain circumstances a 4-manifold whose group is a PD: -group must be aspherical. If f:M -
is an (n-l)-connected degree 1 map between closed
N
orient able 2n-manifolds with fundamental group 0, the only obstruction to
=
its being a homotopy equivalence is Hn (f)
kerU .:7r n(M) -
7r n (N».
Argu-
ing as in Theorem 3 we may show that Hn({) is a stably free module of rank (-l)n(X(M)-X(N», and so f is an integral homology equivalence.
Z[O]-
is a homotopy equivalence if it
We shall next adapt this argument to
a case in which it is not known a priori that the map has degree 1. Theorem 7 Let N be a closed orientable 4-manifold and 0
=
classifying map
and only if
o
f:N -
K(O,l) is a bomotopy equivalence
is a PD: -group and
if
7r
(N). Tbe 1
induces a rational bomology equivalence.
f
Proof As these conditions are clearly necessary, we need only show they are sufficient. Let C., D. complexes of Since
the
and E.
universal covers
HS(O;Z[G)) = 0
for
be
the
equivariant cellular chain
and (K(G,l),ii)
ii, K(O,l)
s < 4,
Poincare
that
duality
respectively.
together
with
the
universal coefficient spectral sequence give an isomorphism of H 2(C.) with
=
HomZ[O](H 2(C.),Z[G)) as in Theorem 3, while Hi(C.)
Since
Q~H2(C.)
trivial
maps
Z[G]
=
As
H 2(ii;Q).
homology
stably free
monomorphically
module
Q[G ]-module
is
K(O,l)
of
Q~E.
to
Q[O],
contractible, is
0 if s
H (C.) 2
the
Q~H3(E.)
by [W: Lemma 2.3]. Since
only
0 or 2.
embeds possible
H 2(N;Q)
f
,.&
which
in nonis
a
induces a rational
homology equivalence the Euler characteristics of Nand K(O,l) are equal. As
these
are
sequence 0 is
also C. -
O. Therefore
the
the
Euler characteristics of C.
D. -
stably
E. -
free
and
D.,
and
as
0 is exact, the Euler characteristic of E. Q[O]-module
H 2(N;Q)
=
Q~H3(E.)
rank 0 and so must in fact be 0, by Kaplansky's Lemma. Thus H 2(C.) and f
is a homotopy equivalence.
the
0
has
=
0
49
Localization and Aspbericity
Suppose
Corollary
tbat G is a PDt-group over Q and tbat tbe cobo-
mology ring H-(N;Q) is generated by Hl(N;Q). Tben N is aspberical. Proof
The
classifying
map
N
from
K(G,1)
to
is
clearly
a
rational
(co)homology equivalence and so the theorem applies. 0
Theorem
Let K be a 2-knot wbose group Tr is a PDt-group over a
8
field F sucb tbat Hl(Tr';F)
O. Tben
,.&
tbe classifying map f
to K(Tr,1) induces a bomology equivalence
from
M(K)
witb coefficients F.
Proof The infinite cyclic covers M
and K(Tr,lY "" K(Tr',l) each satisfy Mil-
nor
3
duality
f':M -
of
K(rr',1)
f'l:Hl(M;F)
formal
dimension
f
of
classifies
Hl(Tr';F)
tr'
with
coefficients
f' l:Hl(Tr';F)
and
F
and
Therefore
the
induced
Hl(M;F)
are
also
Trl(M).
the
lift maps
isomor-
phisms. Since Tr 2 K(IT',l) = 0 the map f' is 2-connected and so Whitehead's theorem ISp: page 3991 implies that f' 2:H2(M;F) - H 2(Tr';F) is an epimorTherefore the map f' 31M 1n:H1(Tr';F) -
phism.
=
since by the projection formula
f' 31M 1nc
H 1(1T';F)
assumption,
H 2(Tr';F)
ISp: 0
,.&
page and
2541. hence
By
f' 31M 1
,.&
O.
On
H 2(Tr';F) is an epimorphism, f' 2(IM 1nf' l(c»
Hl(Tr';F),.& 0, considering
between the Wang sequences of the projections of M onto
K(Tr,l)
we
see
that
f 41M1 '" 0 and so
f
so the
for all c
in
by
duality
map
induced
onto M and K(Tr',1)
induces
isomorphisms
in
(co)homology with coefficients F. 0
Corollary Let K be a 2-knot wbose group Tr is a PDt-group sucb tbat
Tr'
,.&
Proof
Tr".
Tben M(K) is aspberical.
Since
Tr' '" Tr"
there
is
a
field
F
such
that
H 1(Tr';F),.& O.
By
Theorem 8 the classifying map has nonzero degree and therefore induces a rational homology equivalence, so the corollary follows from Theorem 7. 0
As a contrast we have the following theorem.
50
Loca1ization and Asphericity
Theorem suppose
9
Let K
be a 2-knot wbose group Tr bas deficiency I, and
tbat Tr' '" Tr".
Tben Tr 2 (M(K»
'" 0, and Tr is not a PD: -group.
there is a field F such that H 1(M';F) ;:; H 1(Tr';F) is
Proof Since Tr' '" Tr"
nonzero. Milnor duality then implies that H 2 (M' ;F) ,.& O. On the other hand H 2 (Tr';F) = 0 since defTr = 1 (cf. [B: Section 8.51 or [H: page 42]), and so the Hurewicz map from Tr 2 (M) = Tr 2 (M')
to H 2 (M';F) is onto
[Ho 19421.
This proves the first assertion, and the second then follows from the Corollary immediately above. 0
For
2-knot
groups
the
assumption
of
orientability
is
usually
redundant. Theorem tbat
10
Let K
be a 2-knot wbose group Tr is a PD 4 -group sucb
H 1(Tr';Z!2Z) '" O. Tben
Proof
The
classifying
~map
w 1(Tr) = 0, i.e., Tr is of orientabJe f :M(K) -
K(Tr,n
is
a
type.
Z !2Z -cohomology
equivalence by Theorem 8, since every PD 4 -group is orientable over Z!2Z. The orientation character
wI of a 4-dimensional Poincare duality complex
is characterized by the Wu formula w lUx classes
X
=
(J2(x) for all Z!2Z -cohomology
of degree 3 [Sp: page 3501. Therefore wI (M) = f l(w 1 (Tr». Since
M is orientable and f 1 is injective we see that w l(Tr)
=
O. 0
We may use this theorem to give another example of a 3-:knot group which is not a 2-knot group. Let A be a 3X3 integral matrix with detA
-I,
det(A -I) = ±1
and
det(A(2)_1) = ±1.
(Here
A(2)
is
the
induced automorphism of H 2 (Z3;Z) = Z3AZ 3: in classical terms it is the second compound of A. When the first two conditions hold, the third is equivalent to det(A+1) = ±1. It may be shown that there are only 2 such such matrices, up to conjugacy and inversion. Cf. [New: page 52]).
Then A
determines an orientation reversing homeomorphism of SlXS1XS1. The fundamental group of the mapping torus of this homeomorphism is the HNN extension Z3 -A' which is a PD 4 -group of nonorientable type. It is easily
51
Loca1ization and Asphericity
seen to be a 3-knot group, but by Theorem 10 cannot be a 2-knot group. (Cappell and Shane son used such matrices and mapping tori to construct PL 4-manifolds
homotopy
equivalent
but
not
PL
homeomorphic
to
[CS
RP4
1976')). If
every
Tr
(torsion
PD 4 -group?
is
a
free)
PD 4 -group 2-knot
and
group
with
Tr
Is every 3-knot group
is
Tr' = Tr",
still
M
aspherical?
HS(Tr;Z[Tr)) = 0
for
s
~
2
Is
a
which is also a PD 4 -group a 2-knot
group? Finally we shall show that any 2-knot whose group is a PD 4group must be irreducible. Theorem 11 Let G be a PDn -group over 0
witb n > 2. Tben 0
is
not a nontrivial free product witb amalgamation over a cyclic subgroup. Proof If G
is a nontrivial free product 0
infinite index in O. Therefore if 0 A
9.22J.
Moreover
if
A.CB
then A
have
the subgroups
at most n-1, by [B: Proposi-
C is cyclic then b.dOC
~
1.
A
argument (as in [B: Theorem 2.10)) then shows tha t b.d.OO
Mayer-Vietoris ~
max{n -1, 2}.
0
Thus we must have n = 2. Corollary If
and B
is a PDn -group over 0
and B have homological dimension over 0
tion
=
K is a 2-knot sucb tbat M(K) is aspberical tben K is not
a nontrivial satellite knot.
In particular, K is irreducible.
Proof Let K 1 and K 2 be two 2-knots, and let "1 be an element of TrK 1· If "1 has finite order let q be that order; otherwise let q = O. Let
a
me ridian
in
Tr K 2.
Then
(TrK 2/«w q »).CTrK l' where
by
van
Kampen's
Theorem
the amalgamation is over C = Z/qZ
is identified with "1 in TrK 1 [Ka 1983J. 0
w be
TrL.(K 2;K 1,"1) =
and
w
52 Chaptcr 4
THE RANK 1 CASE It is a well known consequence of the asphericity of the comple-
ments of classical knots that classical knot groups are torsion free. This is also true of any 2-knot group which has a nontrivial torsion free abelian normal subgroup of rank at least 2, by Theorems 3 and 4 of Chapter 3. The first examples of higher dimensional knots whose groups have
torsion
were given by Mazur (Ma 1962] and Fox (Fo 1962]. Their examples have finite commutator subgroup, and hence in each of them some meridian is a central element of infinite order. In
power of a
this chapter we
shall
determine all the 2-knot groups with finite commutator subgroup, and we shall also consider the
larger class of groups having abelian normal sub-
groups of rank 1. Most of the groups
7f
with
7f'
finite can be realized by
twist
spinning.
study
Mazur's example. Fox used his method of hyperplane cross sections,
but his knots
This construction was introduced by were later shown to be also
Fox also gave
Zeeman
twist spun knots
in order
to
(Ka 1983'].
another striking example, with group , which is certainly
not even a fib red knot as ' is not finitely generated. If
is a 2-knot group with an abelian normal subgroup A
7f
rank
1
then either
over
a
(e(7fIA
group
is
7f'
= 1) or
(e(7fIA) =
2) or
7f
finite (e (7f1 A) = 0) or
7f/zA
is
a
PDt-group
is an extension of by a finite normal sub-
e(7fIA)
<X>.
After settling the case when
7f'
ite, we shall prove two general theorems. We first show that if A contained
In
7f'
and
if
Next we show that if A
of
moreover
e(7fIA)
1
then
is contained in Tr' and
7f'
Tr'
is
a
is, finis not
PDt-group.
is finitely presentable
then it is a PDt -group with nontrivial centre. Finally we shall show that if e (7f1 A)
2 then
7f
must be .
Cohomological periodicity We saw in Chapter 2 that if the commutator subgroup of a 2knot
group
7f
is
finite,
then all of its abelian subgroups are
cyclic, and
therefore Tr' has periodic cohomology (CE: page 262]. We shall establish this fact directly, in a stronger form, in our next
theorem. (We shall use
full strength of the theorem in Chapter 7). Theorcm 1 Let K be a 2-knot witb group
7f
sucb tbat
7f'
is
finite.
the
The RanIr: I Case
53
Tben M(K) is homotopy equivalent to S3. MC be
Proof Let C be an infinite cyclic central subgroup of 7f, and let the covering space of M with group C. Then 4-manifold
with
homology
fundamental
groups
of
M
r = Z(C] = Z(c,c- 1 ]. onto
MC'
may By
multiplication
group be
the
X(M ) = (7f:C]X(M) = O. The C regarded as modules over the ring Z, and
Wang
c-l
by
MC is a closed orientable
sequence
for
H 2(M;Z)
maps
the onto
projection of
M
itself.
by
But
equivariant Poincare duality and the Universal Coefficient spectral sequence
H 2(M;Z) = Hom r (H 2(M;Z),n. Hence H 2(M;Z) == O. Since 7f 1(M ) C two ends H 3(M;Z);;;; Z and since M is an open 4-manifold
we have has
H 4(M;Z) == O. Therefore the map from S3 to M representing a generator of 7f3(M) is a homotopy equivalence. 0
Tbe commuta tor subgroup 7f' bas cobomological period dividing 4,
Corollary
and tbe meridianal automorpbism induces tbe identity on H 3(7f';Z).
Proof The first assertion follows immediately from the Cartan-Leray spectral
sequence
for
the
projection p
of
M '"
S3
M
onto
357». By the Wang sequence for the projection of M
(CE:
(cf.
page
onto M we see that
the meridianal automorphism induces the identity on H 3(M ;Z). As the spectral
sequence
also
gives
H 3(7f';Z) ~ Coker(H 3(P» ~ Z/ I Tr'1 Z
the
second
assertion is also immediate. 0
We dimensional
may
knot
greater than 3
use
this
groups then
corollary
which
are
not
to
give
further
examples
of
2-knot
groups.
If p
a
is
high prime
ZXSL(2,p) is a 3-knot group with commutator sub-
group the finite superperfect group SL(2,p) (cf. the discussion of centres in Chapter 2). However as
SL(2,p) has cohomological period p-l (if p
iii
1
mod (4» or 2(P -1) (if p ... 3 mod (4» (LM 1978], it can only be the commutator subgroup of a 2-knot group if p
== 5.
The group
group of
knot;
this
the
5-twist
spin of
the
example found by Mazur (Ma 1962].
trefoil
was
zxr*
is
the
essentially
the
The Rank I Case
54
We shall follow· the exposition of Plotnick and Suciu IPS 1987] (rather
than
the
original
one
of
(Hi
1977]) in determining
which
finite
groups with cohomological period 4 have meridianal automorphisms. Let 0(1) be the quaternion group, which has a presentation <x ,y: x 2 = (xy)2 y2 >, be the automorphism which sends x
and y
to y
and xy respec-
and let
(1'
tively.
For each k ;;. 1 let T(k) be the group with presentation
(Thus T(k)' ~ 0(1), and T(k) is a semidirect product O(1)X(1'Zl3 k Z).
The
binary icosahedral group /" has a presentation <x ,y: x 2 = (xy)3 = y5>. Theorem
Let
2
7f
be a 2-knot group with
finite.
7f'
Then
7f'
~ PXZlnZ
where P ~ 1, 0(1), T(k) or /" and (n ,2: P:) = 1. Proof All the finite groups with the property that every abelian subgroup is cyclic are listed, in 6 classes, on pages 179 and 195 of (Wo]. As meridianal
automorphism
induces
meridianal
a
of
the
commutator
automorphism on
the
subgroup
quotient
of
by
a
knot
any
the
group
characteristic
subgroup, we may dismiss from consideration those which have abelianization cyclic of even order. There remain types 1,11,111 and V. Each
,
phism
=
on
the
a jb k
abelianization.
(1'(a r) then implies
(1'
Therefore
that irk _ ir
= 1
-
presenta tion
with
and
rn
1
must induce a meridianal automor-
(k,n) = (k-1,n) = 1.
where
(m,n(r-1)
where
mod (m). A meridional automorphism
(1'(b)
metacyclic,
(1'(a) = a i The
where
condition
mod (m), and so r
=
(i ,m) = 1 that
and
(1'(bab- 1 ) =
1. Therefore
the
group is cyclic of odd order. Each group of type II has a subgroup of type I of index 2, and has a presentation
where
p2
iii
rq - 1
!IS
1 mod (m),
q+1 ... 0 mod 2 u and q2 _ 1 mod
n = 2u v (n).
for
some
u ;;. 2
and
odd v,
55
The Rank I Case
The condi tions (m ,n (r -1» = 1 and r n _ m
1 mod (m) imply
is odd and (m ,n) = 1. The subgroup generated by {a ,b
2u
tha t
} is the unique
maximal subgroup of odd order, and so is characteristic. Therefore a meridianal automorphism on such a group induces a
meridianal
automorphism on
the quotient, which has a presentation
2
only
u
=
I, c
elements
2
of
=
u b 2 -1 ,cbc- 1
order
2u
in
=
this
bq>.
quotient
are
the
odd
powers of b, and so no automorphism can be meridiana!' Therefore we may assume that u = 2. The subgroup generated by a
is also characteristic. Any meridia-
nal automorphism of the quotient by this subgroup must map b to bee for and so must map b 4 to b 2e (q+1). Therefore (q+l,v) = 1 and the
SOme e
then imply that q
congruences above
!!!!
1 mod
Hence the quotient is
(v).
isomorphic to O(1)XZ/vZ. Thus a meridianal automorphism of a group of type II with such a presentation must map a, b 4 , b V and c to a f , a8b4e, ahbvc and aib v respectively.
b 4 ab -4
=
{prY ..
be
Since
ar
frY,
and 4e fr ..
it
must preserve the 1 cac- = a P , we obtain
fr 4
and
frY
!!!!fp
the
group is
the
further
congruences
mod (m). As moreover
1 and p2 .. 1 mod (m) and (4(e -l),v)
mod (m). Therefore
relations
I, we
=
isomorphic
to
find
that
(f
r
,m) must
... p
iii
1
O(1)XZ/m vZ, and mv is
odd. Each group of type III is an extension of a group of type I by
0(1), and has a presentation
where
the
subgroup
order
n
of
b
is
an odd
0(1) is characteristic, the
multiple
quotient
of
admits
3. Since a
the
Sylow
2-
meridianal automor-
phism; since it is of type I it must be cyclic, and so r = 1. Let n
=
3k s
where s is not divisible by 3. Then the subgroup generated by ab 3 is cenIral and of order MS. The subgroup generated by {x,y,b s } is isomorphic to
56
The Rank 1 Case
T(k),
and
(ms,6)
so
the
whole
group
is
isomorphic
to
T(k)XZ/msZ,
where
1.
The
only groups of
type
V
that
we
need consider are
direct
products IXSL(2,p) where I is of type I, p is a prime greater than 3 and
(I I 1,1 SL(2,p) I) = 1.
If
such
a
direct
product
(with
factors
of
coprime
order) admits a meridianal automorphism, then so do its factors. Therefore I
is cyclic of odd order.
As remarked above, the cohomological dimension
of SL(2,p) is greater than 4 if p > 5. This completes the theorem. D
It is well known that each such group is the fundamental group of a 3-dimensional spherical space form IWo]. In particular it has
trivial
second homology. Therefore each meridianal automorphism of such a group can be
realized by some
3-knot group. Plotnick and Suciu study n -knots
which are fibred with fibre a punctured spherical space form, and show that if n > 2 then "' must be cyclic IPS 1987]. Meridianal automorphisms Having found the groups with cohomological period 4 which admit
some
meridianal
phisms (up
to
We shall show
automorphism, we must inversion
next determine
and conjugacy in
that in fact
the
outer
all such automor-
automorphism group).
there is in each case just one 2-knot group
with a given finite commutator subgroup. Throughout this section " shall be a 2-knot group with finite commutator subgroup. Our results shall be developed in a number of lemmas and then summarized in a theorem. The first lemma is self-evident. Lemma 1 If a group G
HXI witb (IHI,III) = 1 tben an automorph-
ism '" of G corresponds to a pair of automorpbisms "'H and "'I of H and I respectively, and'" is meridianal if and only if "'H and "'I are.
D Lemma 2
Tbe meridianal automorpbism of a cyclic direct
factor of rr' is
tbe involution.
Proof The
endomorphism
Is ]:x_x s
of
the
cyclic group of order m
is
a
The Rank I Case
S7
meridianal automorphism if and only if (s -I,m) is
a
direct
factor
of
",
then
it
is
a
=
=
(s,m)
direct
1. If the
summand
of
group
"'I""
=
HI (M(K);A) and so Theorem 3 of Chapter 2 implies that s 2 "" 1 mod (m). Hence we must have s
Lemma
3
iii
-1 mod (m). 0
An automorphism of 0(1) is meridianal if and only if its
image in Out(O(1»
equals that of
17
or
17-
1.
Proof It is easy to see that an automorphism of 0(1) induces the identity on 0(1)10 (1)' if and only if it is inner, and that every automorphism of
0(1)10(1)'
lifts
to
one
0(1).
of
Therefore
Out(O(1»
=
Aut(O(1)IO(1)').
Moreover as 0(1) is solvable an automorphism is meridianal if and only if the
induced
automorphism
0(1)10 (1)'
of
represented by the images of
17
is
meridianal.
The
latter
are
and 17- 1 in Out(O(1». 0
A more detailed calculation shows that every meridianal automorphism of 0(1) is conjugate to
Lemma
4
17
or
17-
1 by an inner automorphism.
All nontrivial automorphisms of I" are meridianal. Moreover
each automorphism is conjugate
to its inverse, and Out(J") = Z/2Z.
Proof An elementary calculation shows that if an automorphism of a group
o
induces the identity on OleO, then it is the identity on all commuta-
tors. Therefore if 0
is a perfect group the natural map from Aut(O) to
Aut. Then every subgroup of finite index in (1) = is
Cm)
to
some
m
each
f
from G = "ICY) to (m) for some
finite
kernel. Then the integral homology groups of the infinite cyclic covering space Y determined by G = f- 1 C<SCm» are finitely generated A-torsion modules, and H 2 Ca;Z) is Proof
Since
Y
is
compact
and
finite cyclic of odd order.
A
is
noe therian,
the
groups
HiC}>;Z) = HiCY;A) are finitely generated as A-modules. Since Y is orien table, XCY) = 0 and H 1CY;Z) has rank I, H CY;Z) is finite and the rest of 2 the first assertion follows from the Wang sequence for the projection of }> onto Y. By Poincare duality and Hopf's theorem H 2 Ca;Z) is a quotient of Ext ~ Ca/(:}',A).
CCf. Theorem 3 of Chapter 2). By assumption there is an
exact sequence 0 -
T -
ala' -<Scm) ~ N(t _2m) -
0 where T is a fin-
Therefore Ext ~ CGIG',A) ~ N(t -2m) and so H 2 CG;Z) is a quotient of A/C2 m t-l), which is isomorphic to Z[K) as an abelian group. ite A-module.
Now
GlkerCi) ~ Z[K)
also, and
H 2(Z[K);Z) = Z[K]AZ[K]
334)) = 0, and so by the LHS spectral sequence for
a as
Cby
[R: page
an extension of
Z[K] by kera) we see that H 2 Ca;Z) is finite. But a finite quotient of Z [K) is cyclic of odd order. 0 Lemma pi.
11
Let A
be an abelian group and F a
2, considered as a
product
field of characteristic
trivial A-module. Then the map induced by cup from H 1CA;F)AH 1CA;F) to H 2CA;F) is injective.
65
The Rank 1 Case
Proof We may identify H1 CA ;F) with HomCA,F) and H 2 CA;F) with {w:A 2 _FlaZw = 0}/{a1 [1[:A_F} where a1 [Ca ,b) = [Ca)+ [Cb)- [Cab) and aZwCa,b,c) = wCb,c)-wCab,c)+wCa,bc)-wCa,b) for all a, b, c
in A
maps [
in H 1CA;F) is
and w. The cup product of two elements [
and g
and set
represented by the function sending Ca,b) in A2 to [Ca)gCb) in F. There
is
a
natural monomorphism J.I
from
H 1CA;F)AH 1CA;F)
to
,,2CA ,F) Cthe space of skew symmetric bilinear maps from A2 to F ) given
J.lq:. [iAgiXa,b) = 'L({iCa)giCb)-[iCb)giCa».
by
H 1CA;F) are such that 'L[iUgi
from A
=
to F such that 'L[iCa)giCb)
that
=
[i' gi
in
hCa)+hCb)-hCab) for all a, b in A.
= 0 for all a,
b
in
abelian. Since J.I is a monomorphism, 'L[il\gi must be
o.
0
Then J.lq:.[iAgiXa,b)
=
Suppose
0 in H 2 CA;F). Then there is a function h
hCba)-hCab)
For more general groups G of characteristic 2 or the ring Z
A
since
A
is
and for coefficients including fields
the kernel of cup product may be related
to the subquotients of descending central series for G [Hi 19871. Lemma finite
12
Let rr be a 2-knot group which is an extension o[ by a
normal subgroup N.
Proof Let K Le t A
be
since N
Then N -;;; O(1) or is trivial.
be a 2-knot with such a group rr, and fix a meridian t in rr.
a maximal abelian
subgroup of N
and Ie t G 1 = C rrCA). Then
is finite and normal in rr the index m = [rr:G 11 is fini teo In par-
ticular t m is in G
and G 1 maps onto Z, with kernel 8 1 say. Let G be Then G has fini te index in the subgroup of rr generated by COl and t m 1
rr and so there is an epimorphism Moreover by
the
page
8
1061.
from G
onto Cm), with kernel A.
[ -lC$cm» is abelian, and is an extension of SCm) -;;; Z[*1
=
finite
[
abelian group A, and so is isomorphic to
Now
H 2 C8;z) is cyclic of
odd order, by
other hand, by [R: page 3341 it is isomorphic to fore A
A~Z[*1,
Lemma
by
[R:
10. On
the
AM~CA and let H the
right E-module
given
above
we
with
may
be
the conjugate action. From the three resolutions that e 1 F2 = e1He = e 1 Hf = O. (In fact
compute
eO F 2 = e 1 F 2 = 0, which is equivalent to the fact that the group has one
end).
Then
Poincare
duality
and
the
Universal
Coefficient
spectral
sequence give H 3 = 0 in either case, and an exact sequence
Now
since
the
skew
field
of
fractions
L
is
flat
H p CL(i5J=.C.) = L(i5J=.Hp' and so is nonzero only if p
as
a
right
module,
= 2. But since M has
Euler characteris tic 0, which is also the Euler characteristic of L(i5J=.C,. and therefore
of L(i5J=.H., we
e OH 2 = 0 Csince
e OH 2
may conclude
the e 2H f note
0 also. Therefore
contained in HomCL(i5J=.H 2 ,L» and e 1 H2 - 0 in which e 2 H 1 - HI
term has order 4 Cas an abelian group). But this is absurd as
right
=.-modules
= Ell f
that
L(i5J=.H 2 -
HomCH 2 ;E.) is
we have a short exact sequence 0 the middle
that
for
e2He
=.lIe = =./Ct 2 +tCa-1)+(a-1)3,a 4 -1)=.
= =.ICa 4+a 2+ 1,t +a 3+a 2+a )=.
instance
e 2Hf
contains
are
E/EnI f
each =
infinite. CTo E/Ca 4+a 2+1)
and
see as
a
this sub
The Rank 1 Case
69
E.-module). Thus there can be no such 2-knot. Therefore by Lemma 12 we must have rr = . 0
Corollary If the quotient of a 2-knot group rr by its maximal locally-finite
0
normal subgroup T is then T is either trivial or infinite.
In fact we believe that in this case T must be trivial, but have not ye t been able to prove this. The modules He and H f 0(0 presented by
1
group [Mi
of
19751.
cyclic fundamental groups.
The
M(p,q ,r)
The
is
triples
finite (2,q,2)
remaining 6 such
lead to 3 distinct manifolds, with groups 0(1), T easily
from
these
observations
and
Dunbar's
give
if
and lens
only
triples (with (p,q) = =
work
T(1) or 1-. It
that
no
if
spaces, with
2-knot
1)
follows whose
IV
The Rank 2 Case
group has commutator subgroup O(1)XZ/IlZ for some
11
> 1 cln hI
Iwhl
I
spin. (Note that the meridional automorphism of such a commutllnr IUItIIlIllI' has order 6). The fundamental group of M(2,3,6) is nilpotent; III Ih. ,lIh,'I aspherical Brieskorn 3-manifolds are finitely covered by circle bundl ..
\lVI'r
surfaces of hyperbolic type, and so their fundamental groups do nlll
l'dvlI
abelian normal subgroups of rank greater than 1. If p-1+ q -1+ r -1 ~ 1 then
the Seifert
fibration
r
and
= cp'q'
where (a,cq')
= (b,cP') = (p,q) =
exceptional fibres of mUltiplicity p', a and
1 exceptional
fibre
of
M(p,q,r) i~
Let p = ap', q -
essentially unique. (Cf. Theorem 3.8 of [Sc 1983']).
M(p,q,r)
hoa
b
exceptional fibres of mUltiplicity
(i'
of mUltiplicity
c, and
1.
so
Then
II.,'
the
triple {p, q, r}
is
determined by the Seifert structure of M(p,q ,r).
Theorem 7
Let M be
tbe r-fold cyclic brallcbed cover of S3, brallcbed
over a 1-knot K, and suppose tbat r > 2 alld tbat "1(M) bas a subgroup of but is Ilot
fillite illdex witb Ilolltrivial celltre alld illfillite abeliallizatioll virtually abeliall. Tbell K is a
torus kllOt, alld is determilled
ulliquely by M alld r. Proof The assumptions imply that "1(M) has one end, and so M is aspherical. If it were a connected sum then one of the summands would have to be
a
homotopy
branched over a
sphere, knot
and
would
be
summand of K.
a But
cyclic any
branched
cover
such homotopy
S3,
of
3-sphere
must be standard, by the proof of the Smith conjecture. Therefore
M
is
irreducible, and almost sufficiently large, and so must be Seifert fibred [Sc 19831. Now since M is not a flat 3-manifold we may assume that there is a Seifert fibration which is preserved by the automorphisms of the branched cover [MS 19861. Since r > 2 the fixed circle (the branch set in
M)
must
be a fibre of the fibration, which therefore passes to a Seifert fibration of X(K).
Thus K must be a torus knot [BZ 19671, and so M is a Brieskorn
manifold. The uniqueness follows as in the above paragraph. 0
If "I(M) is virtually abelian and infinite then a similar argument shows that M is irreducible, hence flat, and that only r
3 or 4 is possi-
ble. It is not hard to show that only one of the six closed orientable flat
80
The Rank 2 Case
3-manifolds is a cyclic branched cover of S3, branched over a knot, and the order of the cover must be 3. (See Theorems 6 and 8 of Chapter 6). Theorems 2 and 7 together with this observation imply that a classical knot k such that
7ft: rk
has centre of rank 2 and whose commutator subgroup has
a subgroup of finite index which maps onto Z
(for some r ;;. 3) must be a
torus knot. It is in fact clear from the tables of [Du 19831 that if the r-
fold cyclic branched cover of a classical r
;;. 3 then either the
and r
=
knot
is Seifert
fibred
for some
knot is a torus knot or it is the figure eight knot
3. All the knots whose 2-fold branched covers are Seifert fibred
are torus knots or Montesinos knots. (This class was fi,rst described in [Mo 19731, and includes the 2-bridge knots and pretzel knots). The number of distinct manifold there
knots can
whose be
2-fold branched cover is
arbitrarily
large
[Be
19841.
are distinct simple 1-knots whose
a
given Seifert
Moreover
for
fibred
each
r
3;;. 2
r-fold cyclic branched covers are
homeomorphic [Sa 1981, Ko 19861. Pao has observed that if K of finite order r
is a fibred 2-knot with monodromy
and if (r ,s) = 1 then the s -fold cyclic branched cover of
S4, branched over K, is again a 4-sphere and so the branch set gives
new 2-knot, which we shall call
the
a
s -fold cyclic branched cover of K.
This new knot is again fibred, with the same fibre and monodromy the s tb power of that of K
[Pa 1978, PI 19831. Using properties of S1-actions on
smooth homotopy 4-spheres, Plotnick obtains the following result. neorem
[PI
19861
Modulo
tbe 3-dimellsiollal Poillcare cOlljecture, tbe
class of all fibred 2-kllots witb periodic mOllodromy is precisely tbe class of all s - fold cyclic brallcbed covers of r-twist SpillS, wbere 0 < s < r alld (r,s) = 1. 0
Here ·periodic monodromy· means that the fibration of the exterior of the knot has a characteristic map which is of finite order. It is not in general sufficient that
the closed monodromy be represented by a map
of finite order. We shall call such knots brallcbed twist SpillS, for short. If we homotopy
restate
3-spheres
we
this
result
in
may
avoid
explicit
terms
of
twist
dependence
spinning knots on
the
in
Poincare
The Rank 2 Case
conjecture. directly
81
In our application in the next
that
theorem we are able
the homotopy 3-sphere arising
to show
there may be assumed to be
standard.
Theorem 8 A group
a
wbicb is Ilot
virtually solvable is tbe group of a
2-kllot wbicb is a cyclic brallcbed cover of a
twist SpUIl torus kllot if
alld ollly if it is a 3-kllot group, a PDt-group witb celltre of rallk 2, some Ilollzero power of a
subgroup of Proof If K
a'
weigbt elemellt beillg celltral, alld
bas a
fillite illdex witb illfillite abeliallizatioll. is a cyclic branched cover of the r-twist spin of the (p,q)-
torus knot then M(K) fibres over the circle with fibre M(p,q ,r) and monodromy of order r, and so the r tb over the
power of a meridian is central. More-
monodromy commutes with the
natural circle
action on
(cf. Lemma 1.1 of [Mi 1975]) and hence preserves a Seifert follows is
that
therefore
rr 1(M(p,q ,r»
M(p,q ,r)
fibration. It
the meridian commutes with the centre of rr 1(M(p,q ,r», which also is a PD
central
in
t -group
a.
Since
(with
the
above
exceptions)
with infinite cyclic cent re and which is virtu-
ally representable onto Z, the necessity of the conditions is evident.
a
Conversely, if
is
such a
group
group of a Seifert fibred 3-manifold, N over N
a'
is
the
fundamental
say, by Theorems 2 and 6. More-
at
is "sufficiently complicated" in the sense of [Zi 19791, since
not virtually solvable. Let t
be an element of
the whole group, and such tha t til we may
then
assume minimal).
Then
a
whose normal closure is
is central for some positive
G
is
11
(which
a/ is a semidirect product of
=
Z/IlZ with the normal subgroup G'. By Corollary 3.3 of [Zi 19791 there is a fibre preserving selfhomeomorphism phism of lifts
a'
determined by
1"
of N
inducing the outer automor-
t, which moreover can be so chosen that its
to the universal covering space
N
together with
generate a group of homeomorphisms isomorphic to
N
corresponding to
the
image of
t
in
a
fixed point set by Smith theory. (Note that point set of the map
1"
O.
the covering group The automorphism of
has a connected
N
1-dimensional
= R 3). Therefore
the fixed
in N is nonempty. Let P be a fixed point. Then P
determines a crosssection "'I = pxS 1 of the mapping perform surgery on "'I to obtain
a 2-knot
torus of
with group
a
t".
which
We is
may
fibred
82
The Rank 2 Case
with monodromy (of the fibration for the exterior
X)
of finite order.
We
may then apply the above theorem of Plotnick to conclude that the 2-knot is a branched twist spin of a knot in a homotopy 3-sphere. monodromy
respects
the
Seifert
fibration
and
leaves
the
Since the
centre
of
G'
invariant, the branch set must be a fibre, and the orbit manifold a Seifert fibred homotopy 3-sphere. Therefore the orbit knot is a fibre of a Seifert fibration of S3 and so is a torus knot. Thus
the 2-knot is a branched
twist spin of a torus knot. 0
It shall follow from the work of Chapter 6 that if rr is a virtu-
ally solvable 2-knot group in which some power of a weight element is central then either rr' is finite or rr is the group of the 2-twist spin of a Montesinos knot K(O I b;(3,l),(3,l),(3,±1»
(with b even) or of the 3-twist spin
of the figure eight knot or of the 6-twist spin of the trefoil knot. We might hope to avoid the appeal to Sl-actions implicit in our use of Plotnick's theorem by a homological argument to show that
t'
has
connected fixed point set. The projection onto the orbit space would then be a cyclic branched covering, branched over a knot, and we might then be able to show that the orbit space is simply connected, by using the fact that the normal closure of t fibred and
t'
in G is the whole group. Since N
is Seifert
is fibre preserving, the branch locus would then be a torus
knot in the standard 3-sphere.
However we have been unable thus far to
establish this connectedness directly. If p, q and r are relatively prime then M(p,q ,r) is an homology
sphere and the group rr of the r-twist spin of the (p,q)-torus knot has a central
element
rr -;:;; rr'XZ, and rr'
which
maps
to
a
generator
has weight 1. Moreover if t
of
rr/rr'.
It
follows
that
is a generator for the Z
summand, then an element b of rr' is a weight element for rr' if and only if bt is a weight element for rr.
This correspondance also gives a bijec-
tion between conjugacy classes of such weight elements. If we exclude the case (2,3,5) then rr' has infinitely many distinct weight orbits, and moreover there are weight elements such that no power is central [PI 1983]. Therefore we may obtain many 2-knots whose groups are as in Theorem 8 but which are not themselves branched twist spins, by surgery on such weight classes in M(p,q,r)XS l . Is there a 2-knot group which contains a rank 2 free
abelian
normal
subgroup
and
for
which
no
nonzero
power
of
any
The Rank 2 Case
H,'
weight element is central? If K
is a 2-knot with group as
aspherical, and so the
theorem implies
in Theorem 8
that it
then M(K) IA
is homotopy equivalenl
In
M(K 1) for some K 1 which is a branched twist spin of a torus knol. II h a well known open question as to whether homotopy equivalent aapherlcill closed manifolds
must be
homeomorphic. In
M(K) and M(K 1) must be that
if
K
is
homeomorphic.
fib red,
Chapter
8 we
topologically s -cobordant. Here
with
irreducible
fibre,
then
M(K)
shall lee
Ihal
we shall Ihnw and
M(K 1) are
This is a version of Proposition 6.1 of [PI 19861, startinK
from more algebraic hypotheses. Theorem
Let K be a
9
fibred 2-knot wbose group rr bas centre of
rank 2, some power of a weigbt element being central, and sucb tbat rr' bas a subgroup of tbe
finite index witb infinite a beJianization. Suppose tbat
fibre is irreducible. Tben M(K) is bomeomorpbic to M(K 1) wbere K 1
is some brancbed twist spin of a torus knot. Proof Let F F
be the closed fibre and :F-F the characteristic map.
Then
is a Seifert fibred manifold, as above. Now the automorphism of F con-
structed as in Theorem 8 induces the same outer automorphism of rr l(F) as , and so
these maps must be homotopic. Therefore they are in fact isoto-
pic, by [Sc 1985, BO 19861. The theorem now follows.
The
closed
fibre
of
any
fib red
2-knot
0
with
such
a
group
is
aspherical, and so is a connected sum F#P where F is irreducible and P is a homotopy 3-sphere. If we could show that the characteristic map may be isotoped
so
that
the
fake
3-cell (if
there
is one) is carried onto itself
then we would have K = K 1 #K 2 where K 1 is fibred with irreducible fibre and K 2 has group Z. We could then use Freedman's Unknotting Theorem to sidestep the assumption that the fibre be irreducible. Other twist spins We
may
also apply
Plotnick's
theorem in
attempting
to
under-
stand twist spins of other knots. As the arguments are similar to those of Theorems
8
nnd
9,
except
in
that
the
existence
of
homeomorphisms
of
84 finite
The Rank 2 Case order
and "homotopy
implies
isotopy"
require
different
justification,
while the conclusions are less satifactory, we shall not give proofs for the following assertions. (Cf. also Theorem 9 of Chapter 2). Let G be a 3-knot group which is a PDt-group in which some nonzero power of a weight element is central. If G'
is
the
fundamental
group of a hyperbolic 3-manifold and the 3-dimensional Poincare conjectnre is true then we may use the Rigidity Theorem of Mostow [Mo 1968] to show that G is the group of some branched twist spin K
of a simple non-
torus knot. Moreover if K 1 is any other fibred 2-knot with group G and hyperbolic fibre
then
M(K 1)
is homeomorphic
to
M(K).
In particular
the
simple knot and the order of the twist are uniquely determined. Similarly if G' is the fundamental group of a Haken, non-Seifert fibred
3-manifold and
we may use
the
[Zi 1982]
3-dimensional
to show
Poincare conjecture
that G
is
the
is
true
then
group of some branched
twist spin of a prime non torus knot. If moreover all finite group actions on the fibre are geometric then by [Zi 1986] the prime knot and the order of the twist are unique. However we do not yet have algebraic characterizations
of
the
groups
of
hyperbolic
or
Haken
3-manifolds
comparable
to
Theorem 6. (Question 18 of [Th 1982] asks whether every closed hyperbolic 3-manifold is finitely covered by a surface bundle over the circle). We raise the following questions. Is a 3-knot group which is a PD: -group in which some nonzero power of a weight element is central
the group of a branched twist spin of a prime knot? If K
is a 2-knot
such that rr' has one end and some power of a weight element is central in rr then is M(K) homeomorphic to a cyclic branched cover of M(T: rk) for some prime classical knot k and r ;;. 2, and if so are k and r unique?
85 Chapter 6
ASCENDING SERIES AND THE LARGE RANK CASES All but two of the 2-knot groups with abelian normal subgroups
of rank greater than 2 have commutator subgroup Z3 (as in the examples of Cappell and Shaneson)j the exceptions are virtually abelian of rank 4. We shall prove this after considering the more general class of groups with ascending series whose factors are either locally-finite or locally-nilpotent. If rr
is
such a 2-knot group and
T
subgroup then rrlT is in fact either Z length 4.
In the
latter case
is its maximal locally-finite normal or
. Then rq is a torsion free 2-stage
q = Z
(generated by
Z2 (generated by the images of x
the
and y).
image of z) and
Every finitely gen-
erated torsion free nilpotent group of Hirsch length 3 is isomorphic to Z3 or to r q for some q.
Lemma
1
Let D be a
torsion subgroup of GL(2,Q). Then D is
finite,
of order a t most 12.
Proof Let E
be a
finite
subgroup of D. If L
ue(L) is a lattice invariant under E, so E
is
a lattice
in
Q2
then
is conjugate to a (finite) sub-
group of GL(2,Z) and hence has order at most 12 (cf. [Z: page 85)). D
is locally finite
[K: page
subgroups and so also has order at most 12. 0
Theorem 2
As
105], it is an increasing union of its finite
Let J be a PDt-group over Q with J I/,
Z
which has
Ascending Series and the Large Rank Case
88
an ascending series witb factors eitber locally - finite or locally -nilpotent, and suppose tbat J bas no nontrivial locally - finite normal subgroup. Tben I' is a
finite extension of Z3 or of r q (for some natural number q).
Proof Let B be the Hirsch-Plotkin radical of 1', and let b be the Hirsch length of B. As in Theorem 1, B
~
b.d'OB
hence
~
~
b.d.OI'
abelian, of
torsion free
and b
is at least 2.
< b.d'OJ = 4 by [B: Theorem 9.22], and so
Since [J:I'] is infinite, b.d.OI'
b
is
3. Suppose that b = 2. Then B is locally-abelian,
rank 2. Let E
be
the
preimage in J
of the
maximal
locally-finite normal subgroup of JIB and let a:E-Aut(B) be the homomorphism
induced
by
conjugation.
Then
B
is
central
in
ker(er),
and
Schur's
Theorem then implies that B = ker(er) (as in Theorem 1). Moreover im(a) is a locally finite subgroup of Aut(B), which is in turn isomorphic to a subgroup of GL(2,O).
Therefore by Lemma 1 im(er) is finite and so EIB is a ~
b.d'OB + 1
(assuming I'IB is locally-finite) ~ 3 which is absurd, as J is a
PDt -group
finite
solvable
group.
I'
Now
~
E, for
over O. Let F be the preimage in I' group of I'IE. Then solvable and b.d'OF
FIE ~
is nontrivial
b.d.OI'
~
otherwise
b.d'OJ
of the maximal abelian normal subMoreover
F
is
the Hirsch length of F
is
at
and
3 and so
torsion
free.
most 3. Since it is at least b+l, it must equal 3 and so e-qual b.d'OF. Moreover I'IF must be a torsion group. This has two contradictory consequences. On the one hand FIE has rank 1 and so is central in I'IE. Using Schur's Theorem and the maximality of EIB we find that F
=
1'. On the
other hand [J:F] = co so c.d'OF ~ 3 by [St 1977]. Since F is solvable of Hirsch length 3 and virtually torsion free, this implies that F must be finitely generated [OS 1981]. But then I'II"E impossible
for
a
=
FIP ~ Z(J)z(FIP), which is
group (JIE) with infinite cyclic abelianization. Theretore
b = 3.
Since c.d'OB
0;;;
3 by [St 1977], b = c.d'OB and so B is finitely
generated [B: Theorem 7.14], and therefore must be isomorphic to Z3 or to
r q for some O. Finally let j be an element of J which is not in 1'. Then the subgroup generated by Bu{j} is a poly-Z group of Hirsch length 4 and so has finite
index in J
by [B: Theorem 9.221. Therefore J
group and B has finite index in 1'. 0
is a poly-Z
89
Ascending Series and the Large Rank Case
If we knew a priori that J was virtually poly-Z, we could sim-
plify the proof of this theorem by observing that every sub- and quotient group is then also virtually poly-Z. Abelian HP radical
In this section we shall show that if the Hirsch-Plotkin radical
H
of
the
commutator
subgroup J' = rrlT
of
the
quotient
of
a
2-knot
group rr by its maximal locally-finite normal subgroup T is Z3 then J'IH is
isomorphic
to
J'IH must be
a
finite
subgroup of SL(2,Z).
trivial, the four-group
V
We
shall
then show
that
or the tetrahedral group A 4 . In
particular rrlT is solvable (of derived length at most 4). If moreover rrlT is torsion free then rr' ;;;; Z3, with just two exceptions. As we have found no convenient reference listing the finite subgroups of SL(3,Z) we shall derive what we need in our next lemma. Since any
integral
matrix
of
finite
order
is
diagonalizable
over
the
complex
numbers the corresponding cyclotomic polynomial must divide the characteristic polynomial of the matrix. Thus in particular the finite cyclic subgroups of SL(2,Z) or SL(3,Z) have order I, 2, 3, 4 or 6. In fact the only other finite subgroups of GL(2,Z) are the dihedral groups of order 2, 4, 6, 8 or 12 [Z: page 851. (Note that -I is the only element of order 2 and determinant I, but that there are two conjugacy classes of elements of order 2 and determinant -1).
Let E be a
Lemma 2
finite subgroup of SL(3,Z). Then the order of E
divides 24, and E is either cyclic, dihedral, a semidirect product of ZI8Z or D8 with normal subgroup Z/3Z, or is A4 or S 4.
Proof If p
is an odd prime, A
with (p,q) = 1 then (/+pr A l
eo
is a 3X3 integral matrix and k = P v q modulo .
PEj, jE
If p is a meridianal automorphism of
°6 ,
then it must induce a
meridianal automorphism of 061A6' and so we must have
p
_
modulo E. Conversely any such automorphism is meridianal, for that 06 modulo «g-Ip(g)1 g ill 06>1>0
j
or j-l
it implies
is a perfect group, and therefore 6
99
Ascending Series and the Large Rank Case
is trivial as 0 6 is solvable. There are 32 elements in the cosets jEuj-1ii of Out(06). The centralizer of j in Out(06) is generated by ap and j, and has order 12. The distinct cose ts of this centralizer in Out(O 6) are represented by
n,
a, "I, a "I,
i, ia, i"l, ia"l}. Conjugating j
and j-1 by these
elements we get 16 distinct elements of jEuj-1ii, which all give rise to the group with presentation 1, txt- 1 = xy, tyt- 1 = x>.
However this group cannot be a PDt-group, as already the subgroup generated by {x 2 , y2, z2,
t}
is nonorientable. The
elements ja and
jp
also
have centralizers of order 12 and their conjugates exhaust the remaining 16 elements of jEuj-1ii. Each of ja and jp is conjugate to its inverse (via i), and so
the groups 0(+) and 0(-) that they give rise
to are distinct.
Moreover these antomorphisms are orientation preserving on A6 and hence on 0 6 (in fact
ja =
(ja)4)
and so 0(+) and 0(-) are PDt-groups.
In each case A6 is an abelian normal subgroup of rank 3, while the subgroup generated by A6U{t6} is an abelian normal subgroup of rank 4. Since the characteristic polynomial of t
acting on A6 is X 3 -1, the only
candidates for normal subgroups of rank less than 3 contained in A6 are (essentially) (t -1)A , generated 6
erated by {x 2y-2 z 2}.
by
{x 2y2, x 2 z- 2 }
and
«2+t +1)A 6 ,
gen-
It is easy to see that neither of these groups is
even normal in O(e )'. Therefore any abelian normal subgroup B
of O(e)
such that A6nB has rank less than 3 must map injectively to the abelianization, and so have rank 1. Such a group must be central. and (0(-) are
generated by
t3
and
t 6 x- 2 y 2z -2
In fact (0(+)
respectively. This com-
pie tes the proof of the theorem. 0
The group 0(+) is the group of the 3-twist spin of the figure eight knot. On the other hand, it can be shown that no power of a weight element is central in 0(-) and so it is not the ·group of any twist spin, although it is the group of a fibred 2-knot, by Theorem 9 of Chapter 2.
100
Ascending Series and the Large Rank Case
Nilpotent HP radical try to analyse rr'IT
We may
in a similar fashion when it has
r q for some q.
Hirsch - Plotkin radical isomorphic to
Let I be a PDt -group over Q
Theorem 9
with IIJ' ;:;' Z and which
has no nontrivial torsion normal subgroup. If the Hirsch -Plotkin radical H
of J' is isomorphic to r q
for some q
~
1 then
either J'
=
H
or the
quotient J'IH is isomorphic to Z/3Z or V. In particular, I is solvable, of derived length at most 4. Proof Let j
be an element of I
which is not in J'. Since H is normal in
I, the subgroup generated by Hand j is a poly-Z group of Hirsch length 4, and so has homological dimension 4. Therefore it has finite index in I (by [B: Theorem 9.22)) and so J'IH is finite. Let a:I-Aut(HICH) be the homomorphism determined by the conjugation action of I on H. Then H is contained in K = J' nker(a) and has finite index there. Since CH ;:;' Z, it is central in K and as HICH is central in KICH we may apply Baer's extension of Schur's theorem [R: 14.5.11 to
conclude
trivial. (by
that
[K,K'1
finite.
Since
it
is normal
in
is a nilpotent normal subgroup of J'
Therefore K
maximality
is
of
H).
a(J')
Since
is
a
finite
I, it must be and so K subgroup
H of
Aut(HICH) ;:;' GL(2,Z), it is cyclic or dihedral, of order dividing 24. Since it is the commutator subgroup of 11K it must admit a meridianal automorphism, and so must be trivial, Z/3Z or V. 0
We
shall assume henceforth that rr is
torsion free. Then rr' is
the fundamental group of a closed orientable Seifert fib red 3-manifold with a Nil-structure, by [Sc 19831 and [Z: Section 631. (Recall that Nil is the nilpotent Lie group of 3x3 upper triangular real matrices). These are either circle
bundles
over
the
torus
or have
fibres, of type (ai,Pi)' with !:.ai- 1
=
base
S2
and
3 or
4
exceptional
1. Each circle bundle over the torus is
the coset space of Nil by a discrete uniform subgroup isomorphic to for some q
~
r q'
1. We shall treat these cases first.
If is an automorphism of
r q , sending x to xaybzm and y to
101
Ascending Series and the Large Rank Case xC yd zn
for some a, . . . n
integral
A
matrix
in Z
~)
= (:
represents
r qlCr q = Z2 , and every pair Z2
an
determines
ordered
pairs
is
(A,~)
in GL(2,Z) and
to
GL(2,Z)
ker(p) ~ Z2.
On
which
(A,~XB,v)
messy:
(A,~)
sends
factoring
out
~
of
= (m,n) in
r q . The mul tiplica tion rule for such = (AB,~B+<detA)v+q
where w(A,B) is biquadratic in the entries of A
Aut(rq)
automorphism
induced
the
with A
automorphism of somewhat
to zad -bc. The
then it must send z
the
A
to
and B. is
w(A,B»
The map p from
a
homomorphism,
inner automorphisms,
and
represented by
q.ker(p), we find that Outer q) is a semidirect product of GL(2,Z) with the normal
(ZlqZ)2,
subgroup
[AB,~B+(detA)v),
where
[A,~)
with
multiplication
is the image of
given
(A,~)
by
[A,~)[B,v)
in Outer q).
=
In partic-
ular, Outer 1) = GL(2,Z).
Theorem
10
Let
7r
be a 2-knot group with
7r'
~
r q. Then q is odd
and the image of the meridianal automorphism in Outer q) is conjugate to [A,O), where A is one of
1) (11 -1)0' (11 01) or (01 -1). 1\ (1 2'
If
q > 1 only the latter two matrices determine meridianal automorphisms. (Moreover they represent mutually inverse automorphisms). Proof If
rq1crq
(A,~)
is
meridianal
then
so
are
the
induced
automorphisms
of
= Z2 and of z(rq/r q ,> = ZlqZ. Therefore Idet(A-I)1 = 1 and
detA-l is a unit modulo (q), so q must be odd, and detA = -1 if q > 1. The
characteristic
polynomial
of
such
a
2X2
matrix
must
be
one
X2_ 3X + 1 , X2_X+l, X2_X-l or X2+X_1. The corresponding fields
of
Q(.,fj)
and Q(v='J) each have class number 1 and so the conjugacy class of such a matrix is determined by its characteristic polynomial [New: page 52). Thus A
is
must be conjugate to one of the above matrices. The inverse of
[A -l,-(detA)~A -I),
so
[A,~)
[A,~)[A,v)[A,~)-l = [A,~+(detAXv-~)A -1)
[A,~(A -(detA)I)A -l+(detA)vA -lJ.
Since
A -(detA)I is
invertible,
[A,v)
is
conjugate to [A,O). (The remark in parentheses now follows as the latter two of the given matrices are mutually inverse). 0
102
Ascending Series and the Large Rank Case Every automorphism (A,J.I) of r q
is orientation preserving. This
follows, with some effort, from the criterion of [B: page 177]. Alternatively, any self homotopy equivalence of a nontrivial S1-bundle over an orientable surface of positive genus is orientation preserving [Fr 1975]. Each of the groups allowed by Theorem
10 is
the group of some
Theorem 9 of Chapter 2. We shall see q = 1 and detA
fibred 2-knot, by
in Chapter 8 that, except when
is positive, the knot realising such a group is unique up
to changes of orientation.
The group of the 6-twist spin of the trefoil
has commutator subgroup r l' In all the other cases the meridianal automorphism has infinite order and the group is not the group of any twist spin. Among the Nil -manifolds which are Seifert fibred with base S2 only
those
with 3 exceptional fibres of
type (3,Pi)' with Pi=±1, are of
interest to us, as in all other cases the quotient of the fundamental group by its Hirsch-Plotkin radical is cyclic of even order, and so the fundamental group does not admit a meridianal automorphism. (In particular, there is no 2-knot
group
7T
with
7T'
an
extension of
V
by
some r q)' Such 3-
manifolds are 2-fold cyclic branched covers of S3, branched over a Montesinos link K(O: b;(3,P1),(3,P2),(3,P3» [Mo 1973]. Up to change of orientation, that P1 = P2 = 1. If b
we may assume
does not admit a meridianal automorphism.
is odd then
the first homology
Therefore b must be even, and
so the link is a knot. These knots are all invertible, but none are amphicheiral [BZ: Chapter 12E]. (Among them are the knots 935 , 9 37 , 946 , 948 , 10 74 and 10 75 ), Let
7T(b ,£)
be
the
group
of
the
2-twist
of
spin
K(O: b;(3,l),(3,l),(3,£».
be a 2-knot group such that 1T' is a (torsion extension of Z!3Z by r q , for SOme q ~ 1. Then q is odd, and
Theorem
11
Let
isomorphic to 7T(q +2+£,£), for Some
Proof By the
free)
7T
£
7T
is
= ±1.
remarks above, we may assume
that 1T' is the fundamental
group Of the 2-fold cyclic branched cover of K(O: b;(3,1),(3,1),(3,£» for some even band
£
= ±1.
Therefore it has a presentation of the form
Ascending Series and the Large Rank Case
A Reidemeister-Schreier calculation normally generated
by
r b-2-E' Hence b
images
the
q+2+E, and q
shows
that
the
of rs- 1, rt -1
G
let
subgroup
of
index
3
is isomorphic to
and h
The centre of rr' is generated
is odd.
by
the image of h, and 3 . If we
103
1T'I u = r- 1s
has and
a
presentation v = sr- 1 we see
that G also has a presentation . Proof Clearly A has rank 1, and An T = zA. If TlzA ITIA
the
were infinite then
would have one end, and so H 2 (IT;O[ITJ) '" 0 [Mi 1986], contradicting theorem. Therefore ITlzA
is a finite
extension of 4>, and so zA
has
finite exponent, e say, by Lemma 3. Since eA is then a torsion free rank 1 abelian normal subgroup of IT the same argument shows tha t T must be finite, and hence trivial by Theorem 6 of Chapter 4. 0
106
Ascending Series and the Large Rank Case
The argument
for our next
theorem depends on an unpublished
result of Kropholler on solvable PD-groups over Q. Theorem
13
Let
be a
7T
virtually solvable 2-knot group witb maximal
locally - finite normal subgroup T. Suppose tbat 7TIT is a virtually poly-Z group of Hirscb lengtb 4. If
7T
also bas an abelian Dormal subgroup A
of rank> 0, tben it is a solvable PDt -group, and rr' is virtually
nil-
potent. Proof Clearly AnT - zA, so AlzA
is isomorphic to a
torsion free
sub-
group of 7TIT, and is thus free abelian. Theorems 3, 4 and 5 of Chapter 3 the'n imply that 7TlzA
is a PDt-group over Q. Since it is virtually solv-
able it must be virtually poly-Z [Kr 1987], and so zA has finite exponent, e
say, by Lemma 3. Since eA is a free abelian normal subgroup of
7T
of
rank > 0, the theorem follows from our earlier work. 0
When
is virtually nilpotent no assumption on abelian normal
7T'
subgroups is needed, and the classification can be made explicit. Theorem
14
Let
7T
be a 2-knot group witb virtually nilpotent commuta-
tor subgroup. Tben eitber and
£
=
±1, or
7T'
:::
7T
:::
from
to
Ext~+l(M.r> which are functorial in M. Similarly if A,. is the chain COmpie x 0
-
-
1\
1\
0 (concen t ra t ed in degrees 0 and
then there is a
1)
chain homotopy equivalence of A,. Qi) B,. with B,., and we obtain maps from Hq(B,.;f> to Hq+l(B,.;r> which are functorial in B,.. Suppose B j = 0 for j
now
the
above
H 1(B,.;I\)
is
B,.
-
complex
H 1(Hom r (B,.,I\»
as
7/
r-chain
free
the
{X in Home(Bo'Z) I
xa1
Ext ~ (Z ,I\)~HO(jj,.;Z)
Let represented
Z
= O}
by
such
that
a
H 1(B,.;I\)
of
the
[Ba
our
Hq +l(B,.;n
Hl(B,.;I\~Hq (B,.;f)
factors
through
pairing
Then
7/
is
the
of
analogous
1980'1.
from
in
class
~ Z. Now we may also
generator
under
from B,.
£,.
the
£l:Bl .... I\.
Ext ~ (Z,I\»
of
image
to
(=
be
7/
Ext~(Z,I\~HO(jj,.;Z) ;:; Z,
degree
complex
and there is a chain homomorphism
1\,
image of a generator of H 1(A,.;I\) obtain
a
< 0 and HO(B,.) = Z. Then the quotient map from BO onto
HO(B,.) factors through to
that
Since
pairing
from
Ext ~ (Z,I\) =
Ext~(Z,I\~Hq(B,.;f) and
we
may
to
identify
our
raising maps from Hq (B,.f) to Hq +l(B,.:n as those given by cup
product with
7/.
PD-fibratioDI If
the
homotopy
fibre
of
a
map
from
a
PDm -complex
to
a
PDn -complex is finitely dominated, then it is itself a PDm - n -complex. (For a nice proof in the case when all the spaces are homotopy equivalent to finite
complexes
see
[Go
1979]).
We
are
interested
in
obtaining
such
a
result for maps from PDt-complexes to S I, under weaker, purely algebraic hypotheses.
112
The Homotopy Type of M (K) Let E be a connected PDt-complex and let 0 S1 induces an epimorphism { .. :o
that the map {:E -
Then we may identify the homotopy fibre of {
"1(E). Suppose
=
-
with kernel H.
Z
with I!, the covering space
of E associated to the subgroup H, and we may recover E up to homotopy type as the mapping torus of a generator r.I! -
I!
of the covering group.
is a PDt -complex then it is finitely dominated [Br 1972] and so H
If I!
finitely presentable [Br 1975]. Moreover H .. (I!;Z) is then
must be almost
finitely generated and so X(E) "" O. We shall assume henceforth that these two conditions hold. Let C.. be the equivariant cellular chain complex of the universal covering
E.
space
HP(C.. ;r>
Then
H 4 _p
Since H 1(C.. ®rA) = H 1(I!;Z) "" HIH group,
E
of
0, and since
only nonzero homology modnles are HO(C.. ) = Z
abelian
[E]
=
complex by [Wa 1967]. So henceforth we shall assume and
from
is infinite HO(C,,:r>
0 also.
S3 (as in Theorem
are
Homr(H 1(C.®rA),A)
=
O.
An
and
-
is
a
is 1-
PDt-
is infinite,
0 also, so
the
n "" H 2(C.. ) = "2(E).
is fini tely generated as an
elementary
computation
then
shows that H1(C.:A) is infinite cyclic, and generated by the class 1/ introduced in the previous section. By Poincare duality for E with coefficients A the
H 3(I!;Z) "" H 3(C.. ®rA) is infinite cyclic, generated by Thus Hj(I!;Z) = Hj(C"®e Z ) 1/n[E]. Hj(C.. ®r A ) is fin-
abelian group
the class [I! ]
=
itely generated over Z
i
=
2.
-
and so is a torsion A-module, except perhaps when
On extending coefficients to Q(t), the field of fractions of A, we
may conclude that H2(C.. ®rA) has rank X(E) = 0, and so is also a torsion A-module. Poincare duality and the Universal Coefficient spectral sequence then imply that H2(C.. ®rA) ~ Ext ~ (H 1(C" ®rA),I\), and so is finitely generated
and
torsion
free
over
Z. Therefore I!
satisfies
Poincare duality
with coefficients Z, i.e. cap product with [I!] maps HP(C.;Z) isomorphically to H3_p(C.®eZ)
=
H3_p(C.. ®rA) [Ba 1980'].
The Homotopy Type of M (K)
113
By standard properties of cap and cup products, to show that I! satisfies Poincare duality
[I! I gives isomorphisms show that the map Tip cup product with
TI
with coefficients S, i.e.
HP(C.. ;S)
from
that cap product
H 3 _p 5 then the commutator sub-
is a classical knot
is as in the corollary, and M(T rK) fibres over S1. next
result
improves
upon
Corollary
3
of
Theorem
3
of
Chapter 3.
Corollary
3
Let M be a closed orientable 4-manifold with X(M)
and free abelian
=
°
fundamental group. Then M is homotopy equivalent to
S3 XS 1, S2XS 1 XS 1 , or S1XS 1 XS 1 XS 1. 0
It can be shown that any such manifold M
is homeomorphic to
one of these standard examples [Kw 19861.
Groups of finite geometric dimension 2 In dimension
this
section
2, deficiency
we
shall
1 and one
assume end.
that
has
G
finite
geometric
For any (left) r-module
N
let
e q N be the left r-module which has the same underlying abelian group as the right r-module Ext~(N,r) with the conjugate left r-module structure. Since c.d.G = 2, the ring r
q > 3. Moreover if rg -
has global dimension 3 and so e q N =
r -
the augmentation r-module
Z
-
° is
° for
a partial projective resolution of
then the kernel at
the left, L
say, is projec-
tive. Since G has a finite 2-complex as an Eilenberg-Mac Lane space and deficiency 1, we may assume that L is free of rank g -1. Since G has one end,
eO Z
resolution
e 1Z
°- r
=
e 3Z
-
e Oe 2 Z = e 1 e 2 Z
e 3e 2 Z
nonzero,
the
and
Aut(Z) = ±1.
by
° and so transposing this e2 ° for e 2 2 2 '" ° and e e Z In
rg _ rg -1
functoriality
Z
_
Z.
=
of
Z.
e 2 (_)
we
resolution gives We
e 2 Z is Aut(e 2 Z)
particular, have
a
then see that
The HomolOpy Type of M (K) Theorem 4
117
Let M be a closed orientable 4-mani[0Id witb X(M) = 0 and
sucb tbat G = "l(M) bas
finite geometric dimension 2, deficiency 1 and
one end. Tben "2(M) ~ e 2 Z, and tbe isomorpbism is unique up to sign.
Co -
0
be
the
cellular
chain complex of the universal covering space M. Then C. is a complex of finitely
generated
free
r-modules
left
whose
homology
H•
is
=
H.(M;r) = H.(}ii;Z). By Poincare duality and the Universal Coefficient spectral
e 2 H 2 == Z 0
Hj
sequence
and
=
j
unless
0
is
there
or
== 0
an
2,
sequence
exact
- - - - - - - - e 2Z
eOH
H2
e1H of
e 3 H 2 == 0,
2
left
r-modules
functor e .(-) to this exact
Applying the
O.
2
while
sequence, and writing P for eO H 2 , we obtain a twelve term exact sequence 0
e1p
that 0
e 3 e 3Z
eOp
e 2p
0,
e 2H
2
Using the above obse rva t ions, we find
O.
there and that 3 2 2 e p O. e e Z
submodule of a e 2H 2 = e 2 e 2 Z
free
tive, and so H2
P~e2Z.
module
and
r
an
is But has
exact
e 3p
global
Z, it follows that e 2 p == 0 also.
0
since
dimension
sequence
P
is
3.
Since
a
Hence P is projec-
Therefore we have exact sequences (1)
and (2)
where Z2 is the module of 2-cycles, which is projective, since (from (5» is a third syzygy. Now P~e 2 Z
also has a resolution of the form (3)
O.
Applying Schanuel's lemma to Z
given
above,
we
find
Z2~rg-l~cl~r = c2~rg~cO
it
(1)
and (3) and to (2) and the resolution of
c4~rg~z2 == r~c3@p~rg-l
that and
and
therefore
r~c3~rg -l~p~rg -l~cl~r. But X(M) == 0, so C4@C2~CO == C3~Cl. Thus
there
is
a
finitely
generated
free
r-module
F = c3~cl@r2g
such
that
118
The HomOlOpy Type of M (K)
= F~P. H2 = e 2 Z
This
F
implies
that
P = 0,
by
Kaplansky's
Lemma,
so
and the first assertion follows from the Hurewicz Theorem. The
isomorphism is unique up to sign since Aut(e 2 Z) - (±1). 0
Is this theorem still true under the formally weaker assumption that G has cohomological dimension 21 It is not yet known whether each such
group
has
geometric
dimension
(Cf.
2.
Theorem
of
6
Chapter
2).
Whether such a knot group has one end is related to the question of Kerva ire as
to whether a
nontrivial free
product with abelianization Z
can
have weight 1 [Ke 19651. Corollary 1 The cellular chaill complex of M(K) is determilled up to
chaill homotopy equivalellce over r by G.
Proof
There
are
exact
sequences
0 -
B1 -
C1 -
Co -
Z
0
-
and
n - O. Schanuel's lemma implies tha t B 1 is projective, since c.d.G - 2. Therefore C 2 ~ Z 2~B 1 and so Z 2 is also projective. Thus C. is the direct sum of a projective resolution of Z jective resolution of n ;;; e 2 Z
In
general
there
and a pro-
with degree shifted by 2. 0
may
be
an
obstruction
to
realizing
a
chain
homotopy equivalence between two such chain complexes by a map of spaces (cf. [Ba 1986».
Corollary
2
Let K alld K 1 be 2-kllots with group G.
3-collllected map f o:M(K l)-illt D4 f :M(K 1)
-
M(K), thell
f
M(K). If
f0
Theil there is a
extellds to a map
is a homotopy equivalellce.
Proof In each case H 3(G;"2(M» is trivial, since c.d.G "" 2, and so the first k -invariant is O. The existence of a map f
0
which induces isomorphisms on
"1 and "2 now follows as in Theorem 1. Since H 3UJ;Z) "" H 3(M;r) = 0,
"3(M)
= r W("2(M»
(where r W(-) is here the quadratic functor of White-
head [Wh 1950)), and therefore f
0
induces an epimorphism on "3' and so is
The HomolOpy Type of M (K) If
3-connected. homology
I
extends
of
the
=
0 for j
H/M(K);r»
10
then it must
universal ~
119
covering
induce isomorphisms on
spaces,
the
since
is a homotopy equivalence. 0
3. Therefore I
Thus the major task in determining the homotopy type of M(K) is to decide when 10 finitely
generated
extends. When 7fK is such a knot group and
then
it
must
be
free
[B:
Corollary
8.6J,
and
7f'
is
then
7f
determines the homotopy type of M, as we shall show next. Free kernel We shall now assume that G is an extension of Z generated
free
normal
subgroup,
and
shall
adapt
our
by a finitely
earlier
notation
without further comment. Theorem
5
Let M be a closed orientable 4-manilold with X(M)
and suppose that the map I:M G
=
7f 1 (M)
to Z
S 1 induces an epimorphism
with kernel H a
Iree group 01 rank r.
simple homotopy equivalent to a PL 4-manilold N with
fibre
which
I..
Then
=
0
I rom M is
fibres over Sl
#r(SlXS2) and which is determined up to PL homeomorphism
by its lundamental group G. Proof If r = 0 then the argument of Theorem 1 shows that M
is homo-
topy equivalent to S3 XS 1. Therefore we shall assume that r > O. Since G then has
a nontrivial finitely generated free
normal subgroup of infinite
index, it has finite geometric dimension 2, deficiency 1 and one end. Therefore n = e 2 Z, by Theorem 4, and so eqn = Z if q = 2 and is 0 other-
=
wise. In particular, Homr(n,r) e 2Z
=
Ext
f (Z,r)
is isomorphic to Ext
also have Ext~(ii,a) = Z sequence
for
0 and so 1/1 is an isomorphism. Moreover,
H"(C.. ;a)
if q
we
H3(c.. ;a) -;;; Z. Therefore
C..
then
J (Z ,a)
as a (left) a-module, so we
1 and is 0 otherwise. From the spectral find
that
H 2 (C.. ;a) = H 4(C.. ;a) = 0
and
is chain homotopy equivalent (over a) to a
complex which is concentrated in degrees 0, 1, 2 and 3. It then follows that the map from H 3(C.;€I) to H3(C.;Z) induced by the augmentation of €I
The Homotopy Type of M (K)
120 onto Z Z.
is onto, and so is an isomorphism, as each module is isomorphic to Similarly the map from H 4 (C.. ;r> to H 4(C.. ;/\) induced by the projection
of r onto 1\ is also an isomorphism. There is a commutative square
in which the vertical maps are the isomorphisms just described and the horizontal
maps
are
given
by
cup
product
H .. (C.. ;I\) is finitely generated over Z
Since H finite
is free, K O(H) =
PDt -complex.
simple
As
1/.
remarked
earlier,
and so the lower map is an isomor-
Therefore the upper map is also an isomorphism.
phism [Ba 1980'1.
a
with
Now
Wb (H) = 0 [Wa 19781 and so M
is
such a complex is determined up
to
homotopy type by its fundamental group and a class 1977). Therefore
is homotopy equivalent
M
t"
in H 3(ff 1(M );Z) [He
to ,T(S1XS 2 ). As every self
homotopy equivalence of ,reS 1XS2) is homotopic
to a PL homeomorphism
(which is unique up to isotopy [La 1974», M is homotopy equivalent to the mapping
torus
of
N
such
a
homeomorphism.
Finally
any
such
equivalence is simple, as G is in Waldhausen's class Cl and so
homotopy Wb (G) "" 0
by Theorem 19.4 of [Wa 19781. 0
If K
is
the
Artin spin of a classical fibred
knot
then
M(K)
fibres over S1 with such a fibre. However not all such fibred 2-knots arc: obtained in this way. (For instance, the Alexander polynomial need not be symmetric [AY
1981». A problem of interest
here
which
was
raised by
Neuwirth [N: Problem PI and is still open is: which groups are the groups of classical fibred knots? Apart from the trefoil knot group and the group of the figure
eight knot
rank 2 and GIG'
=
Z
there are just
two groups G
with G
ciency 1 they are 2-knot groups. So also is
the group with presentation
<x,y I x 2 y2x2 = y> which is the group of a fibred knot in
homology 3-sphere M(2,3,1l), but which is not knot [Ra 19831.
free of
[Ra 19601; as they each have weight 1 and defi-
the Brieskorn
the group of any classical
The HomolOpy Type of M (K)
121
Quasifibres and minimal Seifert hypcrsurfaces Let M be a closed orientable 4-manifold and
f:M -
S 1 a map
an
epimorphism
f.
transverse over p
in Sl. Then
? =
f -l(p) is a codimension 1 submanifold
N
?X[ -1,11.
which
with
induces
a
product
o± W
neighbourhood
from
!:>
maps Z
4>
may
be
viewed
as
an HNN
its subgroup 2Z. It is
onto
extension
therefore in
4> =
Waldhausen's
class CI, and so Wh(4)) = O. This is also true of knot groups with and of all classical knot groups example of a finitely
presentable
tWa
19781.
torsion free
(In fact
z .. 4>'
TT'
free,
there is no known
group which has
nontrivial
Whitehead group). Theorem 6
Let M be a closed orientable 4-manifold with
group 4>. Then any homotopy equivalence
fundamental
f:N ... M is homotopic to a
homeomorphism. proof Although
4>
is
not
square
root
closed
accessible, Cappell's
splitting
theorem holds for it, by the remark on page 167 of [Ca 19761, so there is still an exact sequence
L (Z) ... L (Z) ... L 5(4)) ... L 4 (Z) - L 4(Z) where 5 5 the extreme maps are essentially l-L .. (4)). (Cf. pages 498 et seq. of [Ca
140
Applying Surgery 10 Determine the Knot
19731, or [St 1987]). Since L 5( to Z induces a map to the corresponding exact sequence lor Z, considered as an HNN extension of the trivial group. A diagram chase now shows that
L 5(4))
L 5(Z) = L 4(1) = Z
to
is
isomorphism.
Similarly
L 4(4)) =
is s -cobordant to id M' and
2 the map f
L 4(Z) = L 4 (1). Now by Lemma
so f
an
the induced map from
is homotopic to a homeomorphism. 0
Corollary A
ribbon knot K with group 4> is determined up to orientation
by the homotopy type of M(K). Proof Since 4> is met abelian, there is an unique weight class up to inversion, so the knot exterior is determined by M(K), and since K is a ribbon knot it is determined by its exterior. 0
This
corollary
applies
in particular
to Examples
10
and 11
of
Fox [Fo 19621. Ribbon 2-knots are -amphicheiral, but no 2-knot with an asymmetric Alexander polynomial can be invertible. Thus as oriented knots these examples are distinct. Is there a 2-knot with group 4> which is not a ribbon knot? Theorem 7 Let K be a 2-knot such that
Then K is s-concordant to a
TT'
is a
free group of rank r.
fibred knot with closed fibre #r(SlXS2),
which is determined (among such fibred knots) up to changes of orienta lions by
such
TT
together with the weight orbit of a meridian. Moreover any
fibred knot is reflexive and homotopy ribbon.
Proof By Theorem 5 of Chapter 7 M(K) is simple homotopy equivalent to a PL 4-manifold N which fibres over Sl with fibre #r(SlXS2), and which is determined among such manifolds by its group. As the group
TT
is square
root closed accessible, abelianization induces an isomorphism of L .. (TT) with L .. (Z).
Therefore by Lemma 2 there is an s -cobordism
We may embed an annulus A a meridian for K
= Sl X[O,11
in Z
to such a fibred knot
from M to N.
so that MnA .. Slx{O} is
and NnA = Sl X{l}. Surgery on A
s -concordance from K
Z
K l'
in Z
which is
then gives an reflexive [Gl
Applying Surgery to Detennine the Knot
141
1962] and homotopy ribbon [Co 1983]. 0
Corollary The O-spin of an invertible
fibred knot is determined up to
s -concordance by its group together with the weight orbit of a meridian. Proof The O-spin of a classical knot is -amphicheiral and reflexive. 0
If the 3-dimensional Poincare conjecture holds then every fibred
2-knot with
TT'
free is homotopy ribbon [Co 1983]. Is every such group the
group of a ribbon knot?
142 Appendix A
Ponr-Dimensional Geometries and Smooth 2-Knots
A smooth 2-knot is a smooth embedding K of S2 into a smooth homotopy 4-sphere Eo
The manifold
M(K) obtained
by surgery on K
is
then a smooth orient able 4-manifold, and if it fibres smoothly over Sl we shall say
that K
supports
a
is a smoothly fibred 2-knot. In a number of cases
4-dimensional geometry.
There
are
in
fact
19 classes
M
of 4-
dimensional geometries. In the light of the role that algebraic surfaces are currently that
playing
some of
in 4-dimensional differential
these
manifolds in fact
admit
of
interest
nonsingular complex
topology
it
is
analytic
structures. (These cannot be algebraic or even Kihler, as P1 (M) is odd. See [Wa 1986] and the kiewicz,
Inoue,
references
Kato
and
there, in particular to
Kodaira,
for
more
the work of Filip-
details
on
4-dimensional
geometries and complex surfaces). If K
then M
is a branched twist spin of a torus knot or a simple knot
fibres over Sl with fibre a closed irreducible 3-manifold with a
geometric structure (of type S3, SL, Nil 3, H3 or E3) and monodromy having finite order and nonempty fixed point set. The manifold M finite
cover
which
has
the
corresponding
4-dimensional
then has a
product
geometry
(S3 XE 1, etc.), and so M admits this geometry alone, if any at all. We shall show
that
the
only
other geometries
manifolds M(K) are Sol ~, Sol
t
that
may
be
realized
by
such
4-
and Sol,!,m±l for 2m±1 ~ 11, and possibly
H4 or H 2(C) (although we know of no examples of the latter types). A closed 4-manifold admits at most one geometry, and homotopy equivalent
geometrizable
manifolds
must
have
the
same
geometry,
by
Theorem 10.1 of [Wa 19861. The 4-dimensional geometries not already mentioned are
S2XS2, S2XE2, S2XH2, E2XH2, H2XH2, S4, p2(C), Nil4. and
F4. The last of these cannot be realized by any closed 4-manifold, and so
cannot occur here. Since the universal cover geometries S2XS2, S4
and
M
is an open 4-manifold the
p2(C) cannot occur. If
M i~
not contractible
then the geometry must be a product geometry, S2XE2, S2XH2 or S3 XE1. Of these, only S3 XE 1 admits discrete uniform actions of groups with infinite cyclic abelianization. All of the remaining geometries have contractible models. Most of the geometries of solvable type can be realized by smooth 2-knots with
Four-Dimensional Geometries and Smooth Knots M
aspherical and "
solvable. The results below in such cases follow
our determination of conjunction with
143
the
virtually
torsion free
from
solvable 2-knot groups in
the description of the discrete
uniform subgroups of
the
isometry groups of the solvable 4-dimensional geometries given in Section 2 of tWa 1986). If M has a geometry of type S3 XEl then "' is finite and
M
is
R 4 -{a}. If M is a complex surface (compact com-
diffeomorphic to S3 XR
plex analytic manifold of complex dimension 2) and "' is finite then M is a Hopf surface, i.e. over M cyclic,
is
T(1)
is analytically isomorphic to C 2 -{O} (Kodaira). More-
M
then determined up
to diffeomorphism by", and "'
must be
or /- (Kato). If M has a geometry of type E4 (i.e. is flat) then it is deter-
mined up to diffeomorphism by", which is virtually Z4. Since " isomorphic to
G(+)
or
G(-),
must be
there are just two such manifolds. Neither sup-
ports a complex structure. If M
has a geometry of type
Nil 3 XEl
then it
is fibred with
fibre of Nil 3 type and monodromy of finite order, and so is diffeomorphic to
M(K 1)
where
K 1
is
either
the
2-twist
spin
of
a
Montesinos
knot
KCOI b;(3,l),(3,l),(3,±I» for some odd b or the 6-twist spin of the trefoil knot. These manifolds admit complex surface structures as secondary Kodaira surfaces. The
discrete
uniform
subgroups
abelianization if and only if m -D
of
Sol,! ,D have
infinite
cyclic
±1. The manifold M(K) has a geometry
of type Sol,!,m±1 if and only if K is a Cappell-Shaneson knot and all the roots of its Alexander polynomial are positive, in which case 2m±1 ;;. 11. If K
is a Cappell-Shaneson 2-knot whose Alexander polynomial has one posi-
tive root and two negative roots then M(K) admits no geometric structure, but its 2-fold cover has a geometry of type Sol,!,D for some m, D. These manifolds do not support complex structures. However Shaneson knots
the
geometry of type Sol of
type
SM'
six
distinct
manifolds
arising
from
whose Alexander polynomial has only one
(Cf.
J,
the real
Cappellroot
have
and all admit complex structures as Inoue surfaces
Section
9
of
[Wa
1986».
These
manifolds
may
be
Four-Dimensional Geometries and Smooth Knots
144
distinguished by the Alexander polynomials of the corresponding 2-knots (cf. Table 1 of [AR 1984]). If M has a geometry of type Sol
t
then it is fibred, with fibre
a coset space of Nil 3 and monodromy of infinite order. One such manifold has
a
family
of complex
structures
as
an
Inoue
surface
of
SJ,t'
type
depending on a complex parameter t; the others form an infinite family, all having complex structures as Inoue surfaces of type S
N'
This family has a
"universal" member: each surface is a quotient of the universal surface by a free action of a finite cyclic group of odd order. Since any discrete uniform subgroup of Nil4 has abelianization of rank at least 2, this geometry cannot occur.
M
If
is contractible but
IT
is not
solvable
then the geometry
must be one of H 4 , H 2(C), H3 XEl, SLXE 1, H2XH2 or E2XH2. The last three of these do not infinite
cyclic
admit discrete uniform actions of any group with
abelianization,
and
geometry of type H4 or H3 xE l
so
cannot
occur.
No
manifold
with
can be a complex surface. (According to
Bogomolov, the Hopf and Inoue surfaces are the only nonelliptic complex
PI = 1 and P2
surfaces with Any
M(K)
= 0).
admitting one of the above geometries other than H4
or H 2 (C) is fibred, with
fibre a geometric 3-manifold. The TOP classification (up to Gluck reconstruction) of the knots of type Nil 3 XEl, E 4 , Sol04,
Sol
t
and So/,!,m±1 given in Chapter 8 then gives also the smooth classifi-
cation. In the present context however it is natural
to require that
the
weight class be represented by a cross-section of the (geometric) fibration. (Recall
also
that
monodromy has
by
finite
Plotnick's
theorem
any
order and nonempty
fibred
2-knot
fixed point
set
twist spin, if the 3-dimensional Poincare conjecture is true).
whose is
closed
a branched
145 AppcndiI B
Reflexive Cappell-Shaneson 2-Knots
Let replacing A The
be
A
characteristic
where a
a
matrix
in SL(3,Z) such
that
polynomial
of
is
On
det(A -1) = ±1.
by its inverse if necessary, we may assume that det(A -1)
..
1.
[ (X) .. X3- aX2+(a -OX-1 a ' is the trace of A. It is easy to see that fa is irreducible and A
then
has either 0 or 2 (distinct) negative roots. We shall assume
that
fa
has
one positive root Al and two negative roots A2 and A3. (This is so if and only if a is negative). Since the eigenvalues of A are distinct and real, there is
a matrix
P
in GL(3,R) such
A
that
= PAP- 1
commutes with
A
is
the diagonal
then jj .. PBP- 1
matrix diag[Al'A 2 ,A 3 ]. If B in GL(3,Z) commutes with A
and so must also be diagonal (as the A;'S are distinct).
Suppose that jj = diag[P1,P2,P3]. Then the criterion of the footnote in [CS 1976] for a 2-knot with rr' = Z3 and meridional automorphism A
to be
determined by its exterior is that P2P3 should be negative. On replacing B by -B if necessary, we may assume that detB = 1 and the criterion then becomes
PI < O. Let
F
be
the
field
integers in F. We may view
O[X]/(f a)
0 3 as
and
let
be
OF
the
ring
of
a O[X]-module, and hence as a 1-
dimensional F-vector space via the action of A. If B in GL(3,Z) commutes with
A
preserves
then a
it
induces
lattice,
detB = NF10(u(B».
itself
arises
in
(Note
however
this that,
and
an so
automorphism determines
Every unit way. for
a
of unit
which maps
In particular, instance,
this
this
the is
vector
u(B)
of
space
OF.
which
Moreover,
image
of
so
OF .. Z[X]/(f a).
if
Z[X]/(f a)
[ -22 = (X-2)3+7(X-2X4X+3>+7 2
is
to
in
the square of a maximal ideal of Z[X] and so the ring Z[X]/(f -22) is not integrally closed [Hi 1984 D. Let u be the embedding of F in R in F
which sends the image of X
to AI. Then if P and B are as above, we must have u(u(B»
Thus if the criterion of [CS 1976] holds, there is a unit
u
=
Pl.
in OF such
that NF10(u) = 1 and u(u) < O. Conversely, if there is such a unit, and if OF = Z[X]/(fa) then there is such a matrix B.
146
Relexive CappeU·Shaneson 2·Knots We may reduce the question of the existence of such a unit to
a standard problem of number theory as follows. Let U be the group of all units of OF' let UU be the subgroup of units whose image under U is positive, let U+ be the subgroup of totally positive units, and let U 2 be the subgroup of squares of units. Then U ~ (±l)XZ 2, since F is a totally real cubic number field, and so UIU 2 has order 8. The unit -1 has norm -1 and the element Al is a unit of norm +1 in UU which is not totally
positive (since its conjugates are A2 and A3). It is now easy to check that there is a unit of norm +1 which is not in UU if and only if U+ = U 2 , i.e. if and only if every totally positive unit is a perfect square in U. When the discriminant of fa
is a perfect square in Z, the field
F is Galois over Q, with group z/3Z. If moreover the class number of F
is odd, then U+ = U 2 , by [AP 1967]. In particular, if A
has
trace
-1
then the characteristic polynomial has discriminant 49, the ring Z [X1/( [ -1) is the full ring of integers, and the class number is 1 (cf. [AR 1984 D, so the corresponding 2-knot
is determined by its group, up to a
reflection.
(In this case the polynomial [ -1 determines the conjugacy class of A, by the theorem of Latimer and MacDuffee [New: page 521, and so determines the group among metabelian 2-knot groups).
147 Some Open Questions In
the
tradition of
[N]
and
[GK], we list here a
questions which we have not been able to settle.
number of
Of course some of these
questions are well known to be very difficult. 1. Is the Disk Embedding Theorem valid over arbitrary fundamental groups?
In particular, are s -concordant 2-knots isotopic? 2. What can be said about smooth
2-knots? In particular, is every fibred
2-knot isotopic to one which is smooth in the standard smoothing of S4? 3. When is a 2-knot fibred? Is this so if it has a minimal Seifert surface and the knot group has finitely generated commutator subgroup? 4. Is every PD 3 -group with nontrivial centre the fundamental group of a Seifert fib red 3-manifold? 5. Is every PD 3 -complex X such that e("l(X» 6. Le t N be a 3-manifold, and
~
CD
a connected sum?
a homotopy 3-cell in N such tha t
contains no fake 3-cells. Can every self homeomorphism of N so as to leave
~
N-~
be isotoped
invariant? In particular, is this so if N is either aspheri-
calor has free fundamental group? 7.
Let
M
be
a closed 4-manifold with
"3(M) be finitely
generated?
(If
X(M) =
0 and "2(M) = O.
so then e("l(M»
Must
1 or 2, and so either
M is aspherical or it is finitely covered by S3XS l).
8. Let M a
be a closed 4-manifold with X(M) = 0 and such that "l(M) has
finitely
presentable
corresponding
covering
normal space
a
subgroup
H
PD 3 -complex?
wi th In
quotient
Z.
particular, if
Is
the
e(H) =
1
must M be aspherical?
9. Le t M
be a closed orient able 4-manifold and f:M ... S 1 a map which
induces an epimorphism on fundamental groups. When can
[
be homotoped
148
Some Open Questions
to a map
transverse over 1 in SI
and such
that one (or both) of the
pushoff maps from F = [ -1(1) into M -FX(-E,E) induces a monomorphism on fundamental groups? 10. Is the homotopy type of M(K) uniquely determined by "K if c.d." = 2? In particular, is this so when " ~ ? 11.
Is
there
a
2-knot
K
such
that
M(K)
is
contractible
but
Dot
homeomorphic to R 4? 12. Is every 2-knot uniquely factorizable as a sum of irreducible knots? 13. Is every homotopy ribbon 2-knot a ribbon knot? In particular, is every 2-knot group " with "' free the group of a (fibred) ribbon knot? 14. If the group " of a fib red 2-knot has deficiency 1, must "' be free? 15. Show that if r > 2 the r-twist spin of a nontrivial I-knot is never reflexive. 16. If the centre of a 2-knot group is nontrivial, must it be Z/2Z, Z, Z(i}(Z/2Z) or Z2? Is the centre of the group of a 2-link with more than
one component always trivial? 17. Must a virtually locally-finite by solvable 2-knot group with one end be torsion free? 18. Is there a 2-knot group which has a rank 2 abelian normal subgroup contained in its commutator subgroup? 19. If a 2-knot group " has a rank 1 abelian normal subgroup A
such that
e("IA) = 1 must it be a PDt-group?
20. Is every 3-knot group which is a PDt-group in which some nonzero power of a weight element is central the group of a branched twist spin of a prime classical knot?
Some Open Questions
149
21. Does either of the conditions "c.d.G other,
G
when
is
a
finitely
=
presentable
=
2" and "del G group
with
imply
J.l"
~ ZJ.l
GIG'
the and
H 2(G;Z) = O?
22. If the commutator subgroup of a 2-knot group is finitely
generated,
must it be finitely presentable? coherent? 23. Which A-modules and torsion pairings are realized by HI (X(K);A)
for
some 2-knot K? 24. Find criteria for a 2-knot to be doubly slice. In particular, must a 2knot with rr'hr"
finite and hyperbolic torsion pairing be TOP doubly slice?
A related question is: does every rational homology 3-sphere with hyperbolic linking pairing embed (TOP locally flat) in S4? 25. What can be said about the fibred 2-knots which are the links of isolated singularities of polynomial maps from R 5 to R 2? 26.
Determine
the
smooth
2-knots
such
K
that
M(K)
admits
a
4-
dimensional geometric structure. 27. Let G
be an extension of Z
by
the fundamental group of a closed
orient able 3-manifold which is hyperbolic or Haken. Is Wb(G) = O?
28. Compute
Wb(rr)
and L~(rr) when rr is an extension of Z
by a finite
normal subgroup with cohomological period 4. 29. If G
is a finite group with cohomological period 4 there is a (finite)
orientable PD 4-complex with fundamental grOup GXZ and Euler characteristic O. When is there such a closed 4-manifold? (Such a manifold can fibre over SI only if G is a 3-manifold group).
30. Determine the finite homology 4-sphere groups. In particular, is I-Xlone such?
150 References We shall abbreviate "Lecture Notes in Mathematics (-), SpringerVerlag, Berlin-Heidelberg-New York" as "LN (-t. [AR 1984]
Aitchison, I.R. and Rubinstein, J.H. involutions on homotopy spheres,
Fibered knots and in [GK], 1-74.
[AS 1988]
Aitchison, I.R. and Silver, D.S. On certain fibred ribbon disc pairs, Trans. Amer. Math. Soc. 306 (1988), 529-551.
[AF 1967]
Armitage, J.V. and Frohlich, A. Class numbers and unit signatures, Mathematika 14 (1967), 94-98.
[Ar 1925]
Artin, E. Znr Isotopie Zwei-dimensionaler Flichen im R 4' Abh. Math. Sem. Univ. Hamburg 4 (1925), 174-177.
[AY 1981]
Asano, K. and Yoshikawa, K. On polynomial invariants of fibred 2-knots, Pacific J. Math. 97 , 358-383. Cappell, S.E. Mayer-Vietoris sequences in Hermitean K -theory, in Hermitean K - Theory and Geometric Applications (edited by H.Bass), LN 343 (1975), 478-512. Cappell, S.E. A splitting theorem for manifolds, Inventiones Math. 33 (1976), 69-170. Cappell, S.E. and Shaneson, J.L. There exist inequivalent knots with the same complement, Ann. Math. 103 (1976), 349-353. Cappell, S.E. and Shaneson, J.L. Ann. Math. 104 (1976), 61-72.
Some new four-manifolds,
Cappell, S.E. and Shaneson, J.L. On 4-dimensional s -cobordisms, J. Diff. Geometry 22 (1985), 97 -115. Cart an, H. and Eilenberg, S.
Homological Algebra,
152
References
Princeton University Press, Princeton (195:3). [Co 1983)
Cochran, T. Ribbon knots in S4, J. London Math. Soc. 28 (1983), 563-576.
[C)
Cohn, P.M. Free Rings and their Relations, Academic Press, New York-London (1971).
[CR 1977)
Conner, P.E. and Raymond, F. Deforming homotopy equivalences to homeomorphisms in aspherical manifolds, Bull. Amer. Math. Soc. 83 (1977), 36-85.
[Da 1983)
Davis, M.W. Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. Ma tho 117 (1983), 293-324.
[Du 1983)
Dunbar, W. Geometric orbifolds, preprin t, Rice Universi ty (1983).
[Du 1985)
Dunwoody, M.J. The accessibility of finitely presented groups, Inventiones Math. 81 (1985), 449-457.
[OF 1987)
Dunwoody, M.J. and Fenn, R.A. On the finiteness of higher knot sums, Topology 26 (1987), 337-343.
[DV 1973)
Dyer, E. and Vasquez, A.T. The sphericity of higher dimensional knots, Canadian J. Math. 25 (1973), 1132-1136.
[Dy 1987)
Dyer, M.N. Localization of group rings and applications to 2-complexes, Commentarii Math. Helvetici 62 (1987), 1-17.
[Dy 1987')
Dyer, M.N. Euler characteristics of groups, Quarterly J. Math. Oxford 38 (1987), 35-44.
[Ec 1976)
Eckmann, B. Aspherical manifolds and higher-dimensional knots, Commentarii Math. Helvetici 51 (1976), 93-98.
[Ec 1986)
Eckmann, B. conjecture,
Cyclic homology of groups and the Bass Commentarii Math. Helvetici 61 (1986), 193-202.
[EM 1980)
Eckmann, B. and Miiller, H. Poincare duality groups of dimension two, Commentarii Math. Helvetici 55 (1980), 510-520.
[EM 1982)
Eckmann, B. and Miiller, H. Plane motion groups and virtnal Poincare duality groups of dimension two, Inventiones Math. 69 (1982), 293-310.
[Fa 1977)
Farber, M.Sh. Duality in an infinite cyclic covering and even-dimensional knots, Math. USSR Izvestija 11 (1977), 749-781.
[Fa 1983)
Farber, M.Sh. The classification of simple knots, Russian Math. Surveys 38:5 (1983), 63-117.
References
153
Farrell, F.T. The obstruction to fibering a manifold over a circle, in Actes, Congres internationale des Mathematiciens, Nice (1970), tome 2, 69-72. Farrell, F.T. and Hsiang, W.C. The Whitehead group of poly(finite or cyclic) groups, J. London Math. Soc. 24 (1981), 308-324. Farrell, F.T. and Hsiang, W.C. Topological characterization of flat and almost flat riemannian manifolds M n (n ... 3,4), Amer. J. Math. 105 (1983), 641-672. Farrell, F.T and Jones, L.E. Topological rigidity for hyperbolic manifolds, Bull. Amer. Math. Soc. 19 (1988), 277-282. Finashin, S.M., Kreck, M. and Viro, O.Ya. Exotic knottings of surfaces in the 4-sphere, Bull. Amer. Ma tho Soc. 17 (1987), 287 -290. Fort, M.K., Jr. (editor) Topology of 3-Manifolds and Related Topics, Prentice-Hall, Englewood Cliffs, N.J. (1962). Fox, R.H. A qnick trip through knot theory, 120-167.
in [Fl,
Fox, R.H. Rolling, Bull. Amer. Math. Soc. 72 (1966), 162-164. Freedman, M.H. Automorphisms of circle bundles over surfaces, in Geometric Topology (edited by L.C.Glaser and T.B.Rushing), LN 438 (1975), 212-214. Freedman, M.H. The topology of four-dimensional manifolds, J. Diff. Geome try 17 (1982), 357 -453. Freedman, M.H. The disk theorem for four-dimensional manifolds, in Proceedings of the International Congress of Mathematicians, Warsaw (1983), vol. I, 647-663. Freedman, M.H. and Quinn, F. Topology of 4-Manifolds, Princeton University Press, Princeton (in preparation). Gabai, D. Foliations and the topology of 3-manifolds. III, J. Dif£. Geometry 26 (1987), 479-536. Geoghegan, R. and Mihalik, M.L. A note on the vanishing of HnCO;Z[GD, J. Pure Appl. Alg. 39 (1986), 301-304. Gerstenhaber, M. and Rothaus, o. The solution of sets of equa tions in gronps, Proc. Nat. Acad. Sci. USA 48 (1962), 1531-1533. Gildenhuys, D. Classifica tion of solnble groups of cohomological dimension two, Math. Z. 166 (1979), 21-25.
154
References
[GS 19811
Gildenhuys, D. and Strebel, R. dimension of soluble groups, (1981), 385-392.
[GI 1962)
Gluck, H. The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308-333.
On the cohomological Canadian Math. Bull. 24
[GK 1978)
Goldsmith, D.L. and Kauffmann, L.H. Twist spinning revisited, Trans. Amer. Math. Soc. 239 (1978), 229-251.
[Go 1988)
Gompf, R.E. On Cappell-Shaneson 4-spheres, preprint, University of Texas at Austin (1988).
[GM 1978)
Gonzalez-Acuna, F. and Montesinos, J.M. groups, Ann. Math. 108 (1978), 91-96.
Ends of knot
[Go 1976)
Gordon, C.McA. Knots in the 4-sphere, Commentarii Math. Helvetici 51 (1976), 585-596.
[Go 1981)
Gordon, C.McA. Ribbon concordance of knots in the 3-sphere, Math. Ann. 257 (1981), 157-170.
[GK)
Gordon, C.McA. and Kirby, R.C. (editors) Four-Manifold Theory, CONM 35, American Mathematical Society, Providence (1984).
[GL 1984)
Gordon, C.McA. and Litherland, R.A. Incompressible surfaces in branched coverings, in The Smith Conjecture (edited by J.Morgan and H.Bass), Academic Press, New York-London (1984), 139-152.
[GL 1988)
Gordon, C.McA. and Luecke, J. Knots are determined by their complements, preprint, University of Texas at Austin (1988).
[Go 1965)
Gottlieb, D.H. A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840-856.
[Go 1979)
Gottlieb, D.H. Poincare duality and fibrations, Proc. Amer. Math. Soc. 76 (1979), 148-150.
[Gu 1971)
Gutierrez, M.A. Homology of knot groups: I. Groups with deficiency one, Bol. Soc. Mat. Mexico 16 (1970, 58-63.
[Gu 1972)
Gutierrez, M.A. Boundary links and an unlinking theorem, Trans. Amer. Math. Soc. 171 (1972), 491-499.
[Gu 1978)
Gutierrez, M.A. On the Seifert surface of a 2-knot, Trans. Amer. Math. Soc. 240 (1978), 287-294.
[Gu 1979)
Gutierrez, M.A. Homology of knot groups: III. Knots in S4, Proc. London Math. Soc. 39 (1979), 469-487.
[HK 1988)
Hambleton, I. and Kreck, M.
On the classification of
References
155
topological 4-manifolds with finite fundamental group, Math. Ann. 280 (988), 85-104. Hausmann, J-C. and Kervaire, M. Sous-groupes derives des groupes de noeuds, L'Enseignement Math. 24 (978), 111-123. Hausmann, J-C. and Kervaire, M. Sur Ie centre des groupes de noeuds multidimensionels, C.R. Acad. Sci. Paris 287 (1978), 699-702. Hausmann, J-C. and Weinberger, S. Caracteristiques d'Euler et groupes fondamentaux des varietes de dimension 4, Commentarii Math. Helvetici 60 (985), 139-144. Hendriks, H. Obstruction theory in 3-dimensional topology: an extension theorem, J. London Math. Soc. 16 (977), 160-164. Corrigendum ibid. 18 (978), 192. Hillman, J.A.
Alexander Ideals of Links, LN 895 (981).
Hillman, J.A. High dimensional knot groups which are not two-knot groups, Bull. Austral. Math. Soc. 16 (1977), 449-462. Hillman, J.A. Orientability, asphericity and two-knots, Houston J. Math. 6 (1980), 67-76. Hillman, J.A. Aspherical four-manifolds and the centres of two-knot groups, Commentarii Math. Helvetici 56 0980, 465-473. Corrigendum, ibid. 58 (983), 166. Hillman, J.A. Polynomials determining Dedekind domains, Bull. Austral. Math. Soc. 29 (984), 167-175. Hillman, J.A. Seifert fibre spaces and Poincare duality groups, Math. Z. 190 (985), 365-369. Hillman, J.A. Abelian normal subgroups of two-knot groups, Commentarii Math. Helvetici 61 (986), 122-148. Hillman, J.A. The kernel of integral cup product, J. Austral. Math. Soc. 43 (987), 10-15. Hillman, J.A. Two-knot groups with torsion free abelian normal subgroups of rank two, Commentarii Math. Helvetici 63 (988), 664-671. Hillman, J.A. A homotopy fibration theorem in dimension four, Topology Appl., to appear. Hillman, J.A. The algebraic characterization of the exteriors of certain 2-knots, Inventiones Math., to appear. Hillman, J.A. and Plotnick, S.P.
Geome trically fibred
156
References two-knots, preprint, Macquarie University and the State University of New York at Albany (1988).
[Hi 1979]
Hitt, L.R. Examples of higher-dimensional slice knots which are not ribbon knots, Proc. Amer. Math. Soc. 77 (1979), 291-297.
[Ho 1942]
Hopf, H. Fundamentalgruppe und zweite Bettische Gruppe, Commentarii Math. Helvetici 14 (1941/2), 257-309.
[Ho 1982]
Howie, J. On locally indicable groups, Math. Z. 180 (1982), 445-461.
[Ho 1985}
Howie, J. On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 289 (1985), 281-302.
[Ka 1980]
Kanenobu, T. 2-knot groups with elements of finite order, Math. Sem. Notes Kobe University 8 (1980), 557-560.
[Ka 1983]
Kanenobu, T. Groups of higher dimensional satellite knots, J. Pure Appl. Alg. 28 (1983), 179-188.
[Ka 1983']
Kanenobu, T. Fox's 2-spheres are twist spun knots, Mem. Fac. Sci. Kyushu University (Ser. A) 37 (1983), 81-86.
[Ka 1988]
Kanenobu, T. Deforming twist spun 2-bridge knots of genus one, Proc. Japan Acad. (Ser. A) 64 (1988), 98-101.
[K]
Kaplansky, I. Fields and Rings, Chicago Lectures in Mathematics, Chicago University Press, Chicago-London (1969).
[Ka 1969]
Kato, M. A concordance classification of PL homeomorphisms of SPXS q , Topology 8 (1969), 371-383.
[Ke 1975]
Kearton, C. Blanchfield duality and simple knots, Trans. Amer. Math. Soc. 202 (1975), 141-160.
[Ke 1965]
Kervaire, M.A. Les noeuds de dimensions superieures, Bull. Soc. Math. France 93 (1965), 225-271.
[Ke 1965']
Kervaire, M.A. On higher dimensional knots, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (edited by S.S.Cairns), Princeton University Press, Princeton (1965), 105-109.
[Ke 1969]
Kervaire, M.A. Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67-72.
[KS 1975]
Kirby, R.C. and Siebenmann, L.C. Normal bundles for codimension 2 locally flat imbeddings, in Geometric Topology (edited by L.C.Glaser and T.B.Rnshing), LN 438 (1975), 310-324.
References
157
Kirby, R.C. and Siebenmann, L.C. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Mathematics Study 88, Princeton University Press, Princeton (1977). Kojima, S. De termining knots by branched covers, in Low-dimensionsl Topology and KJeinian Groups (edited by D.B.A.Epstein), London Mathematical Society Lecture Notes Series 112, Cambridge University Press (1986), 193-207. Kropholler, P.H.
