1 INSTITUT DES HAUTES ETUDE~ SCIENTIFIQUES
Seminar on Combinatorial Topology by E. C. ZEEMAN.  .......•..•. .........
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1 INSTITUT DES HAUTES ETUDE~ SCIENTIFIQUES
Seminar on Combinatorial Topology by E. C. ZEEMAN.  .......•..•. ............
The University of Warwick, Coventry.
l
J
INSTITU1'
HAHT}i:S
nes
Seminar
ETUD3S
SCIETITIFIQU.15'"
on Combinatorial
By
Topology
E.C. ZEEMAN 
Chqpter General
position
from polyhedra homogeneity
into
of the
intersections. polyhedra will .r'8
6: GEtmRALPOSITION is
~ technique
manifolds. manifold
this
and ~il a compact
shall
denote
the
is
dimensions
of
to (poly)
m~ps
to use the
the
ch~pter
manifold.
x ,y < m.
assume
The ide~ to minimise
Throughout
always
applied
dimension
X, Y will The sillsll
of
denote
letters
x, y,
X, Y, M respectively,
In particular
tackle
1N8
the
m
apd
followillg
two situations. Situation
(1)
subpolyhedron move f
There
of
$,
Situation embedding.
(2) First
Let
g such that
x + y  m. .f
into
general
f:
into
is
the
position
Y
move with
pos i tion
f + g
f
by
respect
to
Xo of
0
X ~ M be a map, not necessarily
we show in Lemma 32 that
to
of minimal
a subpolyhedron
general
be a
possible
gX" Y is
We describe
such as keeping f I X  Xo
and moving
and let
In Theorem 15 we show it
isotope
arr:;l refinements
fixed,
X ~ M be an embedding
embedding
namely
ambient
f:
M.
to another
dimension, s3ying
Let
is homotopic
an to
X
Y.
2a nondegenerate
me.p? g say,
in any triangulatirm each simplex
is
many simplexes
with
where n0ndegene:r'a"tf::: llloa.n8
respect
to ,r'hich
mapped nondegenerqtely of
h say,
dimension, 9.lso the
for
X may be mg,pped onto
namely sets
we describe
the
composite
into
keeping
Xo fixed
position,
a
finite
if
unless
x::: X.l
We observe
)
tbat
from algebra
algebraic X~ M
hypothesis (illie
confusion
situation
X
(2)
but
by saying such as
to be in general
tells
is
maps, and is
(notice
posit1on
that
the
position
of
of a general
an essential
exists
step
map
=
the
essential
in that
an
a continuousmap
"continuousmap"
thesis
Then the
part
For example suppose
us there
our normal usage
eM.
points
qlre8dy to be in general
flxi
to geometry.
to deduce as 9. geometrical embedding
of minimal
refinements
of subpolyhedra
use the hyphena tad with
double
to a
0
progr9.mme of "improving" passing
are
does not imply the general
f
M).
we ,nish to make minimal.
There are
for
of
homotopic
f ~ g ~ h
happens
also
tXi)
of
0'
homotopy
f' Xo
family
position
etc
pasi tiona
qnd arranging
gene ral fl ,X.l
general
g is
Not only the
points,
of course
one simplex
selfintersections
S 2x  m.
of triple
move f
for
which the
simplicial
(qlthough
Next we show in Theorem 17 th'l t map,
g is
that
to avoid
polymap),
and that
existence
of a homotopic
steps
are:
we want
3
continuous
m:=tp
1
simp} icil=tl
J
general position
n.P1ll'oxim!;l:tif'rt
map 10c9.1 homotopy
nondegener3.te map
l
I
Chapter 6 general
\II
map in general position
global homotopy
e~g':llfing} Chapters 7 and 8.
pJ.pJ.ng
{embeJng
Remark on homotopy
Tho general programme is to investigate
criteria for (1)
an arbitrary continuousmap to be homotopic to a polyembedding, ~nd
(2) for two polyembeddings to be polyisotopico
Therefore although
we are very careful to make our tsotopie~ piecewise linear (in situation (1))
~j\le
are not particularly interested in
making our homotopies piecewise linear (in situation (2)). We reg'1rd isoto py
'3.8
geometric, and homotopy as algebraic
topological. Invariant definition
If
JYI
is Euclidean sInce 9 then gener1l
position is easy because of linearity~
it suffices to move
the vertices of some triangulation of X into "general position", and then the simplexes automatically intersect
4minimally.
However in a manifold we only have piecewise
linearity, and the problem is complicated by the fact that the positions of the vertices do not uniquely determine the m~ps of the simplexes;
therefore the moving of the vertices into
"general position" does not gu~rantee that the simplexes intersect minimally.
In fact defining general position in
terms of a particular triangull tion of X le:lds to difficulties. Notice that the definitions of general position we have given above depend only on dimension, and so are invariant in tho sense that they do not depend upon any particular triangula tion of X or M.
The advqntages of an invariant
definition are considerable in pr'1ctice. For example, h9ving moved f
'3.
map f into general position, ~}\!e can then triangulqte
so that f is bo th Simplici3l and in general pasition (a
convenient state of '1ffairs that was not pOSSible in the more naive Euclidean space approach).
The closures of the sets of
dr;uble points, triple points, etc. will then turn out to be a descending sequence of subcomplexes. Transversality
In differential theory the corresponding
transversality theorems of ~~itney ~nd Thom
serve a different
purpos e, because they 9,ssume X, Y to be manifolds. in our theory it is essential that polyhedra than manifolds.
'Whereas
X? Y be more genersl
For goneral polyhedra the concept
of "tr8.nsversality" is not defined, and so our theorems !:j,im at minimisip~ dimension rather than achieving transversality.
5~~en
X9Y
are manifolds then transversality is well defined in
combin~torial theory~ but the general position techniques given below are not sufficiently delicate to achieve transversality~ except in Theorem 16 for the special case of Odimensional intersections (x + y When follows.
= m)
x + y > rJ.
•
the difficulty can be pinpointed as
The basic idea of the techniques below is to reduce the
intersection dimension of two cones in euclidean space by m0ving their vertices slightly apart.
However this is no good
for transversalitY9 because if two spheres cut combinatorially transversally in En, then the two cones on them in En+1 , with vertices in general position~ do not in general cut transversally: there is trouble at the boundary. The us€ of cones is a pri~itive tool compared with the function space tecbniques used in differential topology, but is sufficient for our purposes because the problGms are finite.
It
might be more elegant, but probably no easier, to work in the combinatorial function space. WIld embeddings
Without any condition of local niceness, such
as piecewise lineqrity or differentiability,
then it is not
possible to appeal to general position to reduce the dimension of intersections.
For consider the following example. pOSsible to embed an arc and a disk in E4 (and also in
It
is
n ~ 4) intersecting at one point in the interior of each, and to choose
f.:> 0 ~ such that is is impossible to €shift the disk off
5the arc (although it is possible to shift the arc off the disk) • The construction is as follo'ws~ Let A be a wild arc in E3
,
and let D be a disk cutting A once 3.t an interior point of o8ch, such that :b is essential in E3  A. If we shrink A to is 4space, a point x , and then multiply by a line, the result ( E 3/ A) X R := E4 (by a theorem of Andrews and Curtis) • If D' of D denotes the i:':1'1ge in one point
x)(
Therefore if t
0,
in
2nd
E3/.~ .1:1
D')(;
0
,
then
D' X 0
:~leets xxR is essenti'J.lin E4  (x XR)
.
is less thqn the dist~nce bet~8cn dD' X 0
x X R , it is i::::;.possi ble to !shift the disk
D' .x 0
qnd
off the '1rcxx R.
We restrict ourselves to the case when X is a
Compactness
polyhedron and therefore compact.
Consequently we can assume
th":t t M is also COGlpact, for, if not, replace M by a regular ne1.ghbourhood N of
fX
in
j\,~
0
Then N is a comp3.ct manifold
of the sq,me dimension, and moving f into general position in N
a fortiori moves f into general position in M.
General position of ~oin~3 in Euclidean spac~ move maps into general position we need
'1
Before we can
precise definition of
the general posi tioD of a point in Euclidean sp'3.ceEn with respect to other points, as follows. (finite or denumer'lble) subset of En
Let X be a countable 0
Each point is, trivially,
a linear subspace of En, and the set X generates a countable sublattice, En •
L(X) say, of the lattice of all linear subspaces of
Let flex)
in L(X) •
be the set union of ,'3.11 proper linear subspaces
Since L(X)
is countable, the complement
En n(X)
6is
everywhere
dense.
wi th r.e.spect to
X if
D
Now let in if
6.
the same is true being
subdivision
of
an ordered respect position
~
if,
Eosition
Proof
v.1 7
v.,
1
to
B.emark 2
This is
previously
all
(the
Let
~
say.
be a We define
position
with
is in general
(x1,···,xr,
(not
sequence
~,
Y1'···'Yi1)
necessarily
•
distin~t)2
(y 1 ' ••• ,y s) C A
such that
Yi
th£a
in general
is arbitrarily
complement t:>. n(x1,···,y
the
7
that
us to choose all
the
the first
our theory
would sufficG
1
X
.
enabling
Notice
~cEn
to
t:;.I of 6 and a sequence
of ~ 9.
y.
of points
with respect
••• ,xr
Si ~ s ,
1
to the set
1~ifS
Remark 1
x1,x2,
each i,
to choose
Inductively
dense at
now it
for
position
set
(y 1 ' ••• ,ys) c D. to be in general
with respect
to
posi tion embedding
vertices
of vertices
possible
and X a finite
of the embedding).
G~ven a subdivision
(V11 ••• ,vS)
close
6 ,with
to be in general
.
some line~r
independent
with respect
~emma 29
is
,
t !l(X)
6 is in generfll for
sequence
to
Y
y! En
be an nsimplex
y E
We say
definition
it
Define
y.
1
y.
1
i1 )
arbitrarily
near
h'1ve to be interior
time we hqve used the
field,
like
v .• 1
to
6 .
reals:
would work over the rationals,
to use smaller
is
and even
the slgebraic
number field. Remark 3
There is an intrinsic
a sequence
of points
being
inelegance
in general
in our definition
position,
because
if
of the
7order
is changed
counterex3,mple contains
all
they :E~
choose
9
r:::ttionals
of this
property
?
in
complex numbers), rid
may no longer
on the
'1nd then
ineleg3nce9
4 points
rr
add
(regarding
Iff
9
and ~t the
To construct X such that
rel11 axis
would be ruol'e trouble
we need is
be so.
..Q.(X)
]2
as the
in ths. t order.
same time preserve
than
a
it
is worth ~
To get the lattice
because
to make Lem.f.:1?s 30 and 34 work.
some gqdget
ISOTOPING~.~BSDDINGS INTO GENERAL POSITION ~~ consider (1)
of the
let
lVI
Let
introduction.
posi tion
7
K
be
with
Let map.
8.
resEcct
Given
Theorem 15
by an arbitrsrily of
Xo fixed~ Y
Remark 1
In the
general in the general think
as a little
"y)
~ x+ym.
m == dim M•
in general
•
lnd an embedding
and
then we can ambient ~mbiGnt isotopy
gl X  Xo
is
f
isotope
keeping
f
.X ~
into
~: and the
in gener:J.l position
th(lOrem we say nothing In fQct
two chapters,
position of
is
~nd
M
g
im~g,
with
0
position. next
y = dim Y,
glX  Xo
 YCM,
XOCX
such that
resnect to _..'
be pOlyhedra,
r'1" ~ ,I,'t
x = dim X  XO'
(g(XXO)
small
situation
Y if
.: f(X  XO) C NI,
such thqt
Y
9
U
we say that
to
dim
v .L\.n C v A
Let
be a manifold.
g ~X
all
with
in many applic:::ttions flxo
respect
Xo 9.nd Y as large lowdimensional
about
to
for
will
definitely
Y.
The intuitive
highdimensional feeler
fl Xo
attached
in
engulfing
not be in
blocks, to
being
idea and
Xo by its
is
to X Xo
8Xo ~ XXo.
frontier isotope
the fOGler keeping
interior
of the feeler
frontier
ffi3Ynot).
flxo
in gener81
Corollarv  =~ 1 fXO
The theorem
meets
position,
If
flxo
f(XXO)'
Proof
Apply the theorem Jnd ambient to
is
may prevent overlap
c:
coroll~rjes.
position,
or if
Theorem 15 is true
to the embedding of the
isotope
fX into
fXO
ne cessqry,
fixed.
ge neral (Notice
otherNise
for
maps
im8ge
position the
of
with
extra
h°'ving to keep
us from moving 'lwkward pieces
2
(Interiur
Case)
M then we can ambient
'Nith respect· For put
three
have
fXO
X  Xo
fixed
th'J.t
X,.,). \j
Corollary Y
its
we may already
in general
then
the
0
Y Iceeping
hypothesis
so th9t
in the following
is alreqdy
as embeddings
fixed
applicqtions qS
as .vell
respect
frontier
Y miniffi3.11y (9.1though
In other
does not meet
fX eM,
its
says we can ~mbient
3
Corolll.ry Xo= f
g
such iha t
in Corollary
M,
YO = Y f'l M.
gl Xo
is
'J
f
o
map
f ~X ~ rK
into
general
and position
M fixed,
(Bounded C'?se)
1 ••
let
isotope
t.:? Y keeping
Xo = sO
Given
1
0
Given a ml.p f ~X
?
Then we C"l,nqmbient
in general
position
in
1'i[
9nd
isotope
1!I with
Y eM, f
to
respect
o
to
YO'
to Y
g,nd glX Xo
in gener"l
position
in
r.•l with
respect
0
Proof
First
ambient
isotopy
Coroll:rv
1.
apply of
Corollary M to
2 to the boundary,
and extend
M by Theorem 12 Addendum;
then
the
apply
9For the proof
of Theorem 15 we sh~ll
special
moves which
below.
The parameter
The construction
call
1Ne
t
concerns
involves
(i. e. replRcing
the
tshifts?
choices
piecewise
structures)
and choices
The tshift
of an embedding
X, Xo
of
simplici
and Let
':11.
K' ? L'
denote
dimension f ; K' is
of loc~l
coordinate
systems
by local
in genernl
to which
linear
position.
f ~X
tri 'J.ngula tions
derived
K? L (obt~ined
remeins
0 ~t ~x •
is
~,:
of
X, M.
Let
co ..~plexes modulo the
by stQrring
of decre9.sing
sir.lplicial
7
beC'1use f
all
simplexes
dimension). is
of
Then
nondegenerate
(it
an embedding). Let
A be
tsimplex
of
= b)
Then
fa
the
j
~ t ? in some order
L'
7
respect
K L denote
of
dir.J.ension, with
By Theorem 1 choose triangulqtions
the b8.rycentric
(t1)skeletons
of
and which we construct
line'"")r structure
of points
M, Y with
use a sequence
0
L.
st(a,K') where
tsirJ.plex
:q
Let
=
3,
Q
of
K', L'
•
If
is an (mt1 )sphere
the of
=
st(b,L')
aAP
and
B = fA
?,na
b be the b~,rycentres
P? Q nre subcomplexes
dim P S x  t  1?
of K,
img,ge A? B (with
bTIQ A
t¢
XO'
because
then o
fA eM.
Let fA ~ a AP
denote
the
restriction
of
f.
7
b BQ
Then
fA
is the
join
of three
10
m8.ps a
t
AP of
b?
At Band
Q ? and thereforo
pt
~AP in the boundqry The idea
that?grees isotopic
to
is
BQ of the
to construct
with
f .nA
f,.
keeping
another
illball
bBQ
embedding
on the frontier
AP,
the boundary
and is ambient
BQ fixGd.
We shall
""i.
the move
f~ "i
ion below.
t
gA
a local
shift,
and give
From tho construction
it
can be chosen to be arbitrarily isotopy
be made arbitrarily Now IotA
A C X  Xo A C Xo
define
wi th
f,
fA
The closed
only in their
clnd therefore
f.
Also since
boundqries latter
with
isotopy
from
supported
the st3.rs
to
g.
'Jnd so in particular
fixed.
gA
{st(b,L')}
f
Notice
fXO t
g
thg.t
overl:lp
isotopies
each cover tgA~
the
close
only in their
global
the
ambient
isotopy
of
agree
{gAl
keep fixed,
small
is
f(X  XO)
in
L' ,
fixed.
a tshift Y
e8.ch
M arbi tr9.rily
t
neighbourhood
keeps
lYe C911 the move keepiQ,g Xo
Therefore
g: X
. U tv!
for
on which the
Tlloreovor the ambient
by the simplicial
for
{st(a,K')3
stars
to give an arbitrarily f
K;
gA ,,:,nd
each other.
which the 10c'1l ambient
c0~bine
that
and the ambient
of
t
frontiers?
combine to give a glob'".l embedding to
fA'
tsimplexes
a 10c8.l shift
gA = fA.
X and overlap
construct
be app~rent
to
call
small.
run overall
construct
the explicit
will
close
the frontier
cmbed:3
entered
with respect into
the
to
Y
construction
11
'Jllhen choosing
the tri9ngulqtion
L of
M so
aG
to have
Y a
subcomplex. Local
shift
We are
of 8n embedding f : aAP
which is
the
join
of the
.
three
given
sim:plicial
3.
erabeddjng
b13Q
!:laps
a
+
b,
A. 13 and
P . Q ,
and ",e want to construct g : aAP
(We drop the
subscript
Now Q is a subco:nplex lower with
of
from
A
and
ff'
_1
an (mt1 )sphere, Q,
Q.
Therefore,
an (m.t1)
fqce
r ,
we can choose
6,
and extend
and joining
h : Q
linearly.
simplicial.
respect of
,
to
~.
Choose
,
to be the
to
+ ~
Then in p':1rticular
Define
the homeomorphisGl
of the
identity k
to be the
join
of the
Let
h: bQ
h
1 k h 1
of
an (mt) sim.plex
+ ~
+ ~
v be the barycentr~ by Glapping
A in general
in
v1.J. v
.
on 6. to the
because
position v
c:r.l9.pv'v1
+
bQ
0
with
is a vertex
.
on 13 to the homeomorphisrJ. : bQ
b + v
such that
: bBQ + bBQ
identity
are
A'
+
v 1 neg,r v
6.
'3.nd fP
is
a hOweoC10rphism
Choose subdivisions
6,.
join
is
r.
the f'lce
k1 :
Y (\ Q
6
Q .
h : (bQ)' is
if
.
into
.)
Y f\ Q
6
than
fP V (Y f\ Q)
li
both
dimension
throwing
gA
and by construction
'1nd by hypothesis
h
of
b13Q •
+
Define
12Then
k
18
~
h~~oQorphis~
bound'1ry fixGd.
of the b~ll
aAP
g
g is al'!lbientisotopic
15.
Lemma
We Cl.n m.ake g
completes
to
the
the definition
.
~
bBQ
f keeping
arbi tr~rily
sm3ll? by choosing
arbitrarily
kOGping
Dt:fino g = kf
Then
bBQ
the boundary
near
f,
fixed by
and the isotopy
v 1 sufficiently
near
v.
This
of the local shift.
h
a
~
~1
~p?
r
f(aP) !lre joins, it follows
Remr:..rk Since
f,k
is the join of
g : 9.P ~ bQ' to
is not a join with respect as the diagram structure
induced
Lemma 30
tshift
shows, on
However
but is a join with respect bQ
from
with res:pect to Y f
.
f:A~B
is in general
g ~ aAP ~ bBQ g
to the sim.plicial structure
Given the hypothesis
(i) If
that
A L..)
by
h 1
position
Xo
of bQ,
to the linear
a
of Theoreo
keepin~
aP ~ bQ
g
15, let
be 
f~g
a
fixed.
with respect
to
Y,
then
so is g. (ii) On the other h'1ud if f
dLl(f(A:Xo)
ray)
is not in genernl
:: t "x+yn
then
position,
di~(g(X
 Xrv)
and if
"y) ::t
 1
13Pro0f
(i) ••
g ~ ?AP wi th
+
f
It
suffices
bBQ,
for
on the
to eX":lmine the ~\ c X 
a tsimplex
frontier
local
shift
from
Xc>.
Since
g
f
to
agreAA
we h8,ve
LP, o
dim(g(XXO) Therefore with
it
the
•
bBQ
suffices
interior
Since only
if
(\ Y f\BQ)
to examine
of the
Y is
dim(f(XXO)
~
ball
intersection
~ x+ym of
•
g(X  XO)
n
Y
bBQ.
a subcomplex
Bey,
the
ny)
of
L,
Y
meets
and so we "lssume this
the
interior
to be the
of
case.
Therefore
and so diCl(g(XXO) where the of
6'
m3ximum is
such that
v101 ('\ vO
meets
Since hi' , h
f\
• Y f\ int(bBQ)) t~ken
01C the
~ll
t + m'l.Xdim(v101 "vC)
pairs
of simplexes
° c h(Qf\Y)
hfP,
interior
B01, BO are
respectively,
over
=
of in the
we h?ve
01,0
'1nd such that
0. . im'1ges of
x  Xo '
dim 01 ~ x  t  1,
oyt1
r o.t1
dim
Y
under
° f y t 
1 •
14Reg"'crd[) as ealbedded in subspC1.cespanned by C.
Case (2
for
have been defined
such that
Br
f
g
inv3riantly?
without
Now choose triangulations K
T
L
is
simplicig,l.
Therefore any
and so it
x E Br •
to any triangulation. X? WI
GO.
Then there
x
although
r  S r' sets
n
contains
S2
points?
suffices
because
Hence
Notice
and
if
by~
at least
with a disjoint Y
x=y
of
¢
00
x E 32  32'.
to a limit
neighbourhood
GO
S2 • (f)
it
identified
and so
::>
::) 32(f)
statement
suppose that
contains
of the double
32 ( f ) =
=
defined
:::=
~e deduce
S1 (f)
S(f )
is
Er( f)
3r'(f)
etc ..
x =
f
f1 fx
x f Xj
the closure
points,
therefore
of
r
Sr' (f)
Thus 32(f)
X
3 (f)
and that
is
19Lemma 31
(i) There is an integer s , and a decreasing
seguence of subcomplexes K  K1 ~ K2 :) such that 1Krl = S r (f) •
0
•
=
=
Ks+1
Koo
0
SoJ f) = ¢
(ii)
=
:>K s
if and only if f maps every
simplex of K nondegenerately. (iii) There is a subcomplex L
I L\ =
that Proof
Br(f)
dim (L  Koo)
1sn~p
==
which is nondegenerate by induction, and
is nondegenerate by our choice of y's because dim 6
Now define
outside 1, and on 1
f.: X 1
+
111
so that f. agrees with 1
the diagram f.
1
1.)
B
~A~ is commutative.
Having defined f.1 we must verify the three
inductive properties. Firstly g' ItV g by straight line paths in 6., keeping the frontier Fr(1,K) fixed. Therefore h1 g'  h1g can be extended to a homotopy
f. 1 'V f. 1 1
supported by L
0
By the
choice of ordering of A's, K.1 1 C K  1 , and so the hoootopy: fixed. Secondly fi satisfies (p) prOVided the keeps Ki1
23homotopy is sufficiently small. because
=
I(.
~
Ki1 U A
and
t
f.1~
Thirdly
f.
K.
~
is nondegenl">rai;c
is nondegenorate on
~
K.1 1
by
The proof of Lemma 32 is
induction and on A by construction. complete. GEN:ER\1 X
< m.
of XX into Mm, where
Consider ~ps
POSITION OF r:rAPS
Define the codim.ension c
= m.  x
Define the double point dimension 0.
:::
0.
2
::: x 
C
:::
2x

m •
More generally define the rfold point dimension 0.
r
Define
g
g
X
t
M:
x  (r1)c
==
•
to be in general position if dim S (g) ~ d
r
r
each r.
Our principal aim is now to show that any map is homotopic to a map in general position. Rem.ark 1
The dimensions are the best possible, as can be seen
from linear intersections in euclidean space. Remark 2
If f
is in general position then f is nondegenerate
and
dim Br(f) < 0.2 . The first follows from Lemma 31 (ii) , because we are assuming x < m , and so
dr
general
its
and
J
d
~nd our choice
J J
of
D. J
I
J
in
th'1t
position
of a
to cover this
o. c J
r
in the interior
D
v.
q,nd so
J
:::
of
6 ensures
to
operations
s •
of
of
6.
This
LemLj/l
34 suppose that
Then
By incr2'1sing each
kee12in,g Xo
by tshifts induction s,
sets
S1'" (f),
fixEJi.
on s , starting
by decre'3.sing induction
t::: dLl S (f) , we can reduce s correct diwlension by Lo:J.::J."1 35,
singul'lr
J ~,
D.
of Lemoa 34.
posi tion
s = 1 , q,nd, for with
Therefore
VITi th thE: hYEothes is of Theore[.l 18, vie C9.n m.ove f
001"'011'11"';:[
Proof
1 S j $ q , such that
v.O. ::: O. ,
With the hypothesis t
into
f\
en"
1.
our choice
the proof
35
•
1.
of the general
lattice
q D C ("\ D.
co~tr3.dicting
Ii ,
of
Therefore
eventuality.
A0 [D.]
C 1.
[6].
f:. with respect
in
because
involved
o
SODe j ,
do not span
and D I S ~:lre 9.11 vertices general
"D.
j;.1 [D.] D.=v.O .• ~lso I \ 9nd [ OJ' J span a o 1. J J J of the IT Sqy, of [6]. Now the vertices
and
0
there
[D.J ]
and
1.
Ddt C
th~t
1'"
< s,
trivially on t,
89,ch singul::,tr set at the
by Lerr.:.o8 34.
\}IIi th starting
S (f)
to s S'1o.e time keeping d s , and wi th one the proof is exsctly the same as that of L8083. 341
Suppose Dodification
then
That is to SRY, we exarJine the interior dt C of a loc~l shift, and find D tha t is the image of r0 1 c.6,
substi tuting
s for r. n
si::J.plexes in
U
q+1 S
9,. P. J. J.
, and
to
:::
r.].."sL.mlexos
in a.J.J.' P ..
where
r.
]. for
O~i~q
The ffin,dificationthat we need to prove is for
1'.< S J.
in order to be able to verify contradiction est3.blish
(*), and therefore achieve a
in each of the tino ca.ses 0
The contrqdictions
d::: t  1 .
There remains to prove the Qodificqtion, use two pieces of hypothesis dim Ss(f) :::t position.
and
flxo
'1nd for this we
that we h'1ve not yet used, that
is qlready given to be in general
36Using Lemoa 33 and that tskeleton
of
K,
Ss(f)
we h1.ve for
s s (g\
a.P.) ~ ~
eaeh
is
s ~2
?
in the
i,
S (fl a.P.) s ~ ~
==
(a. P.)
" (tskeleton
::::
(a.P.)
('\!i.
=
a ..
J. J.
Now trivL'llly
cont~ined
J. J.
()f K)
J.
).
because
if
s==1
then
and
Ss(f)==X
dim. S (f) > d. And d ==x 'J.nd so we could not have s s s mapped by g to v .• the only point of a.P. Therefore ~ ~
=0,
S (gla.P.) s J. J.
re:':lains
le9.st
s
are
at
the
tsim.TJlex
contradicting n  q < s. g
agrees
with
r.<s
for
J.
tho
case
sic1plexes
"~ 1" q+.
the
hypothesis flxo n Z = U a. P .• q+1
on
Z,
0
9
0
If /i
n
n  q ~ s,
of
~
then
Xo mapped by
dir.1 Ss (f I XO) ~ t " ds'
ir.lplying
f
1!:i::q.
i = O.
B,
Let
is
J.
J.
and so
There
a.
in general
f
into
and
position.
By definition
there
Therefore
of the
tshift
Z D:1pped by
g to
J.
and so
~)s(g\ Z) C Z (\ (tskeleton
of
K)
n
==
1) 'Vi S(f).
Co n 01 cD.
Then
Tho proof of L0~~a 39 is complete. c>
We havo
x
is contained
inessential
in a ball in
stai,ting trivially wi th dimensions
less than B;y L.:nnma
in
M9
and have to show that
The proof is by indnction
x == 
1.
Assume
the re suItis
on
x,
true f'or
x.
39 choose
11
Y~ Z c
such that
X c Y
''>J
ZZ ~
where Z
~
.2x  m
~ k, Th8refore
Z
x ~ m  3.
+ 2,
j.
39
by the l~pothesis
is inJsscEtial in (Thi s
by Lemma
s one
OJ
Iii.
2x ~ m + k  2.
But z < x b;l the hypothesis
the place s wher
0
codimensi on
;>:
3 is
o
crucial).
By
Lenmla
Therefore
37
so is
Y.
Z
ic contained
in a ball in
M
by induction.
'rherefore we };',9.ve put a ball round
X, and
 7 
the proof
19 is complete.
of Theorem Corollary
VII
We deduce
some corollaries.
1.
,then al'}x, subsp~ce, o:C dimel'l§.i.QI,J. ~ l{ is ...Q,Qntainedin a.bali.. The corollary
follows
to
from Theorem
19.
, " (,~veaK
Qorqllarx ,2•. a »Qm
immediately
0 tC2.123~ Il1 slllle r e...iJ,L ;::5f~j;.t~.g
J~f_.1 s ..:top 0 1o,g,i ca 111L..JL~me omorphic
Sm • ..We call this the w.:.;alc Poi:i'lcaroConjecture
although
the hypothesis
structure
(we always
assumes
assum0
this)9
homeomorphisffi9 not a polyhedral the proof
Schonflies
proof9
which
upon
does not depend
the stronger
result
that
The stronger
result
for
it in these notes9
In Chapter handlebody
Theorem
9
and which sphere,
upon
9
theory, gives m ;::6.
is also true 9 but we shall not give
from differential
theory,
upon including
r4 :;::o. Let
Then since of
is that
depends
the only lalown proof depends
smoothing 9 and deep results ==
which
is in fact a polyhedral
m == 5
because
proof9
combinatorial
the Schonflies M
The reason
of MazurBrown.
using
manifold
gives only a topological
homeomorphism.
Theorem
we shall give Smale's
has a polyhedral
the thesis
that we give here is Stallings'
the topological
e5
that M
because
x
==
m ~ 5
[m/2]
x.~ = m  x . 1.
and
'P
we have both
M9 and call this complex
M
x, x~ ~ m  3. also.
Let
X
Choose
a triangulation
be the xskeleton
VTr
 8 
of
x... .,
M~ and
the dual
largest
subcomplex
meeting
X).
by Corollary
x1jIskeleton (which is defined
of the barycentric
Now a homotopy
1 both X~ X*
msphere
x.
that
and if they dontt already
X9
X*
IVl= N u N*. ambient ball, ball
Noyv picl{ a regular
isotope A
it onto
say, whose
A*, whose
N.
interior
in the interior
of
the topological
Schonflies
ball.
Therefore
contains
A •.• '.'
=
Theorem
A u C
sewn along their boundaries;
o'~ M.
Then o
..I.
in
carries
E
into another
Similarly
Therefore
H
to Lemma
o
=
A
C
M
is
0
'.
Therefore
!vi
t.§..Jheunion of
r
balls.
1ustcr.]jLckSchir~e~man catc~ory
8
ProoL~_
Let
M
be kconnected.
balls
topological
Q.onse.s..ucntl",Y M_is
~ r~
by
is a topological
sphere.
then
a
u A.,.
of two topplogical
in other words
and
(m _i)sphere
24).
of ~azurBrown
B9
construct
is a collaled
is the union
illltilthey
of
N.
Then
M,
noi ?;hbourhoods of
complex
A) (by the Corollary
M
them a little
neighbourhood
contains
C = IvI 
Now let
derived
of
if necessary).
of the two ba:.ls to cover
The isotopy
interior
Therefore
B, B*, say.
of' B, B*
then we stretch
in the second barycentric
M~ not
are in the interiors
''''
N, N .....be the simplicial '
Let
do 9 as follows.
i~ ballE
X.,.
(by taking regular neighbourhoods We nmv want th.:;interiors
of
is (ml)co~Lected.
are contained
We can also assume balls
first derived
t.obe the
of
 9 
Now
~ k rr/r
[m/r]
VII
< k + 1
m < r(k + 1) m + 1 ~ r (k + 1 ) • Thereforo set G., 1.
{o, i =
the
condi tion
1,
•••
1,
• • • 1
9
[m/r]
~ k is
nl
can be partitioned
r,
each
disjoint
all
1.
if
ClEG
simplexes,
all
t he role
that
play
of the preceding of K
•
1.
each
A~bient
.•
Let
~ k + 1 intogers
Divide
thcver~ices
of
M'
K.1. be
1.
isotope
9
hypotheses
possiblo.
suppose
proof'
ncighbourhood
1.
1.
for
1.
B. by Corollary N..
of
then
By construction,
1.
M
Then M = UA., as 1.
of showing by examples
that best
x = m2.
3manifold
OpOj.'l
in the
and 2x ~ m+k2 in Theorem 19 arC) the
In 19':57 Whitehead contactible
played
Them 1\1= UN .•
Cluestion
of
1.
simplicial
in a ball
r
The K.' s will
1.
A~~ containing
a ball
1.
First
•
1.
m3
in J ..
1.
and so K. lies
B. onto
x ~
lie
into
M' consisting
subcoffinlex of •
N. be the
:'!c nail' tlH'n to the the
the
the
of a qsirnplex
compleii1ontary skeletons
corollary.
dim K. ~ k
• denote
thE: barycentre
the
subsets
M'
of whose vertices the
disjoint
t.hat
1'1, and let
by putting
1.
saying
of
th. lJ becond derived
I" 1," It 1."••L ' '>
i,
complex. J.,
subsets
J.1.
into
derived
to
r
into
contai:aing
Choose a triangulation barycentric
equivalent.
produced.
the
follmving
example
~
M' (opon moans noncompact
of a
without
boundary
A
The manifold in:;edontial
is
rCT'1arl{abl·,;;in
(since
L~3is
that
contractible)
it
contains but
is
a curve not
S I thE'e t is
contained
in a ball
VII
 10 
iu M~.
The mani~old
is constructed
as follows.
Inside a solid torus
T~ in 83 draw a smaller solid torus T2~ linked as shown; T2 draw T3 similarly
D e f·lnc"
7.: ·J.·,"1':';
..
j,
'',,)
links T~~ then S
I:)
,3 •
a baJ.1 lIi M exam:plu
linked, and so on.
7
_
Ho
then inside
Cf.li t the proof'~ bccau8c
1
is ~ot contained
t..l1G proof of the next
s SiElp18r.
P00naru
(1960) and Mazur (1961) produced examples of a
in
VII
 11 compact bounded boundary. page
contractible
inesscmtial,
a regular
example
M4has
bu~ not
r2
of B.
L;t
as
spine
contained
were
neighbow~'h')od
interior
Let
of 1illzur1s
it+
3,
see Chapter
10.
Suppose
is
wi th non simplyconnected
For a description
In particular is
4lIlanifold
contained
for
in a b&ll
the
B.
neighbourhood 2 D fixed
of B, M, under
in the
2
B.
in
There
3, Theorem 8).
(Chapter
this
B by
lies
of D
2
reason.
By replacing
nQceSSar~T we may assume D
a homeomorphis~n M + Ivl1 keeping
Then D
following
2
if
B1, M2 be tht!, images
2 D •
dunce hat
in a ball
be a regular
~1
the
homeomorphism.
Therefore
we have
By the
regular
2 and 3),
neigrbourhood
annulus
jll in

B
B:l
Iv1
M2
tho
top
arrow
1\:1(M) ~ O.
is
0:_'
83 x I
'"
'"
M
cOI!lnutative •
the manif'olds
induced
by inclusions
(M:\. )
2
Therefore
x I
triangle
an isomorphism,
R3marl;:~ one
'"
0
0 M
!Vl1
1\:1
the
(Theorem 8, Corollaries
we have 0
Therefore
theorem
D It is
is
is
not
and the bottom contained
significant
open,
an,d the
that other
group
zero,
contradictin!
in a ball. L1. the is
two examples
bounded.
It
is
above
 12 
conjectured
that
no similar
VII
example exists
~or ~ose~
manifolds.
More precisely: Con,ject\l}~e..:. Corollary ~
Observe ~lat
this
the Poinc8I'c Conjecture m
=:
is
3,4 the conj3cture still
unsolved.
an (m2)connected
__
for m ~ 5, because
In the missing
conjecture
true,
if the Poincare
dimensions In
in a ball.
th0n tll.e proof =:
dimensions
to the Poincare" Conjecture, Conjecture
is true,
mmanifold~ m ~ 3, is a sphere,
is contained
missing
for k .., =: . __m2.~
=::;:.
is true
for m ~ 5.
is equivalent
For,
true
••;:a~",,,;'
conj0cturu
is true
subpolyhcdron is
1 is
~
then
and so any proper
Conversely
of Corollary
which
if'
the above
2 v\I"Orl\:s for
the
3,4, because there are complementary sl{eletons
of cod.imer~ s i on ~ 2. Bing has shown that
in dimension 3 a tiGre delicate
result
7..
will
suffice:
he has proY.:;.dthat
every simple closed
Wi;;;
curve lies
if 11
next give an exam:glo to show that
X ::;:Sn, embedded in Mb;y f'irst
locally,
and then connecting
in which
then M3=: S3.
in a bal19
2x ~ m+k2 is nocessf:'.ry in TheoreD 19. let
is closed manifold
the hypothesis 1
Let M =: S
linking
x
Sm, n ~ 2, and
two little
nspheres
them by a pipe running around the
\
st.
VII
 13 
Notic0
that
2n+1~ x ::: n~ k ::: 0, and so the
III :::
hypothesis
:rails
by
one dimension
I, Ia
2x Next observe acrOSf:l
tho
n to a poin.t by pulling
that
we can hOlilOtope 8
other
and back
Therei'ore be can ta ined ball In
9
(by Theorem
countable
set
theref'orG
link.
spher'cs,
i'or
since
otherwi se we could
n~m3) and span it
of disjoiJt
disl:s,
There w'hich suggests it
in
is
its
be contained
if
"vo cannot
sn
homotopy gr'oups 0qui'vDlcnt
to 'iC
i
(M, C) is 1C
i
(£:1,0)
saying
(F) .il
,
tbat
disk
obstruction
would lift
This
in the
";IT!bed X in a ball,
to a
could set
of
contradiction
i'or
that
i ~ k.
thB inclusion
II
\VG
lS;3t example, migl1t try
to
of 1\1.
5'~be:pace 0 to be a kcore is This
to
say the
condi tion
C c M induces
I'olati
ve
is
isomorphisms
i < .':., an.a. 8n epimorphism ~_(O) c.
Ii' M isk··coflJlectud, 01' ar~:T collapsible
an (n+1)disk.
two l1oighbours.
t:connected;
vanish
it
to a countable
some BOl"'tof 1dimE.:1"lSional II core
the pair
in thi s
with
lifts
r.1ore ~p'(:cisGl~l de:f:'in8 a closed of I'd if
unkIlot
in a ball.
a 1diD.GllSional
that
8n cannot
none of whose bourldaries
But by construction
8n cannot
otherhand
On the
R x 82n of' 81 x fJ2n the
COYer'
one end
81 .
the
inclssi:mtial.
any one oi' ·'Iihich links
shows that
engulf'
around
S::.1 is
in a ball,
the universal
+ k  2.
set
in IvI is
C:.
kcorc.
thclJ.
a point,
or a ball,
 14 ~~ampl~J.
VII
The kskeleton
of a triangulation
of M is
The ksk8leton
of a triangulation
of a
a kcore.
3.
~~e
kcore is another kcore. Exa~~~ __4.
A regular neighbourhood
~~qlAQ~~
If D'~ C then C is a kcore
of a kcore is another
kcore. if ~~d only if
D is a kcore. Exam~le
6. If p ~ q then
Defini~la~~2f
SPx point
cng£1fin£.
Let X, C be compact subspaces ~_X
is a (q1)core of
of M.
We say that we can
b;L..ll.\ll>hin/L.0llt Ta• .f.9.e1c£=fr:.qm t.h~_Q1'..!L.9., if
thel~e Gxis ts
D,
such that XcD~C
din (DC) ~ x.+1. More briefly
vie dGscribe
or ::.m£1l.lf' ..;X:Jn ~. applications
this by saying ~np;ulf'J.,or ~:n,g\l.lf X ::from ..Q.,
The fesler is DC, and it is important
that it be of dinension
special cases of the same dincnoion chapter we shall engulf singularitics may introduce
new singularities,
for
only one more than X (and in as X).
For exawple in the next
of maps, and the feeler itself
but these will be of lower dimension
th.an the ones we started with, and so can be absorbed by successive engulfing.
Rewriting
Theorem
19 from this point of view, the core C
would be a point, and X would be engulfed
in a collapsible
set.
 15 The proof
that this statement
by the following
VTI
is equivalent
to Theorem
lem~a.
LeD}11::'llt0. Let
C, X
9.e~pact
subspac~J? of M.
"Q..e_"Wgulfed:'r.£l11 C if. 8d~n..kY _if..JL_ilLcOR~ned
Proof...:..If X can be engulfed regular
neighbourhood
of Oy because
N of D, which
N ~ D "'",C.
simplicially
elemental"Y collapses
is also a regular
dimension,
by Lemma
of dimension
Performing
11.
collapses
all those
D, say.
Then
we have only removed
simplexes
of
the rest of the elementary
collapses
gives
N01l=.90lTIPact_ collapsing
and~~ci
sion.
We shall always aSSUDe X compact,. but it is sometimes to have the core 0 noncompact, 22 below.
So far collapsing
and we extend
the definition
as for example
in the proof
has only been defined to noncompact
where
{ DO ~
for compact
spaces,
spaces as follows.
Define
DO n C
the right hand side is compact
noncompact
the definition
useful
of Theorem
D_C compacty and D '"~.~O if
N
so that N
simplicial
Perform
';;:: x+2, leaving
in
neighbourhood
if necessary
Order the elementary
dim (DO) ~ x+1, and D ~ X because dimension,;;::x+l.
in _~ re.R~
in Dy then it is contained
and subdivide
to C.
in order of decreasing
Then X can
Oonver'sely given X c N :)0, triangulate
so that X, 0 are subcomplexes, collapses
19 is given
collapsing.
If C, Dare
is nCVlr; if they are compact
then the
 16 
de~inition
agrees
hand
we can triangulate
side1
elementary
with conpact
simplicial
collapsing1 and perform
collapses
face of any elementary
VII given the right
1
the same sequence
o~
since C does not meet the free
on D1
collapse.
because
imm.]diate consequence
Jill
of the
defini tion is the C:i;SJ.J2Jon property A·~
because
the condition
use this property Given D~C,
n
A
B
1'01"both
in either
u
1.'....
B '..... :;.,B
direction
when we say triangulate of DC such that
triangulation
~A:B n
B.
Whenever
we shall say
9J
~xcisio~.
sides is X':B
DC
DC n C? and choose a particular
the collapse collapses
sequence
we mean
simplicially
of elcoentary
we
choose
a
to simplicial
collapses. The definition remains
of engulfing
from a noncompact
the same, with the new interpretation
given
core C
to the symbol ~
~.2msn:K· Stallings He envisaged swallowed
introduced
a dif'1'erent point
an open set of M moving1
up X.
Rewriting
amoeba
of the ball containing
the connectlon point
between
like, until
19 from this point
Theorem
open set would be a small open mcel11 the interior
of vi ew of engulfing.
our definition
X.
and we could
of view, the
isotope
The following of engulfing
it had
lemma
this onto illustrates
and Stallings'
of viow. kemmCLlI:1 ~
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