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 non-degenerate
me.p? g say,
in any triangulatirm each simplex
is
many simplexes
with
where n0n-degene:r'a"tf::: llloa.n8
respect
to ,r'hich
mapped non-degenerqtely 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 continuous-map
"continuous-map"
thesis
Then the
part
For example suppose
us there
our normal usage
eM.
points
qlre8-dy 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
self-intersections
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
non-degener3.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 continuous-map 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 O-dimensional 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 4-space, 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'···'Yi-1)
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 n-simplex
y E
We say
definition
it
Define
y.
1
y.
1
i-1 )
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
counter-ex3,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+y-m.
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(X-XO)
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 low-dimensional
about
to
for
will
definitely
Y.
The intuitive
high-dimensional feeler
fl Xo
attached
in
engulfing
not be in
blocks, to
being
idea and
Xo by its
is
to X- Xo
-8Xo ~ X-Xo.
frontier isotope
the fOGler keeping
interior
of the feeler
frontier
ffi3Ynot).
flxo
in gener81
Corollarv - =~ 1 fXO
The theorem
meets
position,
If
flxo
f(X-XO)'
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
t-shifts?
choices
piecewise
structures)
and choices
The t-shift
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
non-degenerate
(it
an embedding). Let
A be
t-simplex
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
(t-1)-skeletons
of
and which we construct
line'"")r structure
of points
M, Y with
use a sequence
0
L.
st(a,K') where
t-sirJ.plex
:q
Let
=
3,
Q
of
K', L'
•
If
is an (m-t-1 )-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?
A-t Band
Q ? and thereforo
p-t
~AP in the boundqry The idea
that?grees isotopic
to
is
BQ of the
to construct
with
f .nA
f,.
keeping
another
ill-ball
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 t-shift 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'
t-simplexes
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 (m-t-1 )-sphere, Q,
Q.
Therefore,
an (m.-t-1)
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 (m-t) -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
t-shift
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+y-n
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 t-simplex
frontier
local
shift
from
Xc>.
Since
g
f
to
agreAA
we h8,ve
LP, o
dim(g(X-XO) Therefore with
it
the
•
bBQ
suffices
interior
Since only
if
(\ Y f\BQ)
to examine
of the
Y is
dim(f(X-XO)
~
ball
intersection
~ x+y-m 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(X-XO) 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,
oy-t-1
r o.-t-1
dim
Y
under
° f y- t -
1 •
-14Reg"'crd[) as ealbedded in subspC1.cespanned by C.
Case (<:1)
J
[C) span
(v 1c1
and
[6] ,
li.Y1pn.r
There '1re two possibtli ties, .,)cc()l~ding
[C]
and
(c1)
[c11
span
[r]
as to whether or not
(C] denote the
Em.-t, and let
[r) sp,!':..n
.
Therefore
[vc]
and
3.nd so
dim.(V1C1 (\ vc) 5 dim v1C1 + dim vC - dim.6= (x-t)
+ (y-t)
- (m-t)
.
Therefore
[CJ
nnd
sp':"in 3. proper subsp3.ce of
[C1)
contain
and
[C1J
Case (bt
do not span
[r]. Therefor8
[6]?
which does not
v 1 ' by our choice of v 1 in, and the definition
general position in
6
with respect to
6
t
(vC] of,
•
(Note th'1t this applic'1tion W'1S the reason for our definition of the gen0r~1 position of generqted by the vertices
'1
of
point off the lqttice of subspaces ~t
'1more complic'lted 3.pplic~1tion
of the same kind occurs in Lemma 34 below.)
which does not meet the interior of ~ nSAumption
that it did.
Therefore
Therefore
, contradicting
our
C'1se (b) does not apply, and
the proof of p~rt (i) of Lcmm'1 30 is complete.
-15(ii)
are
~.1iJe
given
~nd h~ve to show that Again it
suffices
f(X - Xo) (\ Y
the
t-shift
=
this
dinen8ion
shift.
Since
drops
to ex~mine the local
is
(\ Y)
dim(f(X-Xu)
cont'?.ined
in the
.
bBQ (\ (t-skeleton
of
t-skeleton L)
t > x+y-m
of
L,
,
by one.
'3.nd since
= B
we have
•• g (X - XO) " Y "BQ
which is for
of dio.ension
some J3,
show
2nd
If
B ¢Y
'1
Moreover
dim(g(X - Xo)
80
C'1se
nd so
?
does .not meet the
this
Agqin there In
t - 1 •
dim(g(X - XO) " Y) ~ t - 1
g( X - XO) "Y
f (X - XO) " Y "BQ
=
is
qre two cases,
f(X - Xo) f\ Y ;:) B
r.
Y) ~ t - 1
it
suffices
int erior
tri vi "llly true, and,
c. B
Conversely,
0
to
to show that • bBQ, for "lny B.
of
'l.nd so '1ssume
B
=
as ~bove? only case
fA C Y • (n)
~pplies.
(';.)?
v 1C1 () vC
not meet the
is
interior
eJ"lpty.
Therefo re . bBQ, snd the
of
g(X - Xo) "Y proof
does
of Lemma 30 is
complete. Proof q,nd
Given
of Theorem 15
~ssut:1e
Perform
s"
t-shifts
x+y for
by Lemma 30 (ii) intersection in gener'J.l
8?
knocks
1 , until with
respect
leI?
the
let
s = dim(f(X ....XO)('\y)
theoreo
,x+y-m+1,
8-1".0
each t-shift
down by position
=
->
otherwise
ill,
t
f ~X
the
'~le ~re left to
Y.
is
tri vir'.l.
in that
dimension with
order:
of the
'1n embedding
E':1ch t-shift,
"md
-16therefore
~lso
e,rbi trnrily The pro0f
their
small
composition,
'1mbient isotopy
of Theorem 15 is
x+ y
embeddings
=
m9
Y
Therefore
rJI,
Given
if,
for
Let
fX" Y
s t (V 1 fX), is
to
M and
is
"'. finite f
be m'-:.nifolds such
Y C 1',1 be proper
position set
is
with
of points
tr~nsversal
there
is
3,
it
interior Y
§.1
to to V
of
homeomorphism
We
respectively.
onto
Y if
respect
to
any) triangul~tion
s t (V , Y)
tr"1.nsversal
:fixed.,
III
is
tr::;,nsversal
"it e3.ch point
fX (\ Y
Ex:m:ple
Let
J:f1 be a 4-bnll
be two loc3lly
unknotted
to two unknotted f
9
-+
in general
"Ne say
as vertex
fXO
X9 Y, ill
is
some (and hence for
throwing
of
X
f:
f
v E fX " Y
cCJnt~,ining v
sew f
"'ondlet
such that
keeping
by an
. U
complete.
O-DIEENSION.AL TRANSVERSLLITY thD,t
c~n be reGl;Red
g
X C M is
trmsversal.
curves
in general
with boundary
disks in
S3
in
o.nd let
11 formed by joining
th'1t
posi tion
S3,
X, Y the centre
linlc more th8:n once.
with
respect
to
Y
9
Then
but not
-17Theorem 16 and let
Let
X, Y, M be manifolds
f' ~ X ~ M and
can ambient
isotope
isotopy
such that
Proof
First
perform
a O--shift
f
ambient
interior
=0
t
applies,
and the
of a finite y-simplex
into
with
).
f
respect
Therefore
to
Y
small
general
position?
Since
fX () Y
0
it
suffices
to both; to
and then does not
=
points
of Lemma30, only case
where an x-simplex
transversal
ambient
to examine
gA ~ AP ~ BQ (A,B
of the two cones in
interior
Then we
Y •
shift
proof
intersection
is
g A
into
shift
1\.s in the
point
to
of e~ch local
number of points at
embeddings.
g by an arbitrarily
isotope
of one local
because
M be proper
C
g is transversal
meet the bound~ries the
Y
x+y = m ,
such that
~x+y
crosses
such a crossing
BQ Ii Y,
and so
(a) consists
a is
g is
transversal. tr~nsver$al
to Y. We now
SINGULARSETS introduction, Let it
pASS
0nto situqtion
to the homotoping
(2) of the
of oaps into
f::X: ~ M be a map between polyhedra is
not necessary
The singular S(f) Then
S(f)
The branch Br(f)
set =
that
of
f
set
Br(f)
=
{x E. X;
is
and only if of
defined
f-1fx';'
closuretX~X; if
(for
this
position. definition
Iv1 be a manifold).
S(f)
= ¢
general
f
f
is
xJ
by~ •
an embedding.
is a subset
no neighbourhood
of of
S(f) x is
defined
by:
embedded by f
J.
-18We deduce that
:Br(f)
is
closed,
only if
g is an immersion.
The rth
singular
---------
set
=
t
Sr(f)
=
closure
triple
is
To prove the last
'1
-Tx,
which tends
i'x
= fy,
is
not embedded.
TIr 3 The singular reference K, L of
x
that
¢
-T
3 (f)
:;)
r
points
J .
S3(f)
the
s2 - S2'
C Br
c
~ Br ( f ) to show
x n
I.
sequence
32'.
f. y n ?
is a sequence
X is
because
Y
{Ynl?
comp8.ct.
Consequently for
some n,
32 -3'<:. 2
- 3'2 ¢
?
in general
and
Br
r>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
f-1 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 non-degenerately. (iii) There is a subcomplex L
I L\ =
that Proof
Br(f)
dim (L - Koo) <. dim K2
and
Let A be a p-simplex of K
(i)
o
K2
7
S rI(f) then
=' L :> Koo , ~ 0
We shall show thqt
0
-1
o
0
A C. S rI(f) For f f.i is the disjoint union of open simplexes of K, and must contain either
if A meets
a simplex of dimension > p, or at least r simplexes of o
dimension p.
In either case, each point of A is identified
under f with at least Therefore closure
r- 1
other points, and so
Sr(f)
is a subcomplex , K
r
say
•
0
Let n be the number of simplexes of K then A is identified with at least
rs
r
S rI(f) is the union of open simplexes, and so the
o
so
• ACS'(f)
n
Therefore
Kr = K
00
for
If A € Kroo' -K r - 1 other simplexes, and
r> n
0
0
Define s to be the
least r such that (ii)
If a simplex is n~pped degenerately then a continuum
is shrunk to a point and
S~(f)
I¢
Conversely if every
simplex is mapped non-degenerately then at most n points can be identified, and so
Soo(f) = ¢ .
-20-
(iii)
If
degenerately
.A E K"" , then
and so
A C Br(f).
mapped non-degenerately, according Br(f)
subcomplex
AE L-
'XAe K2•
so
or not
of closed
NON-DEGENERACY
Define
-
given
::: ¢. that
Xo C X
Proof is
M, with
is
is
i;S
or
1..•C. Br(f)
embedded.
Therefore
and is
therefore
a
f ~X
for
D1§:.E
f ~ XX
of the
the
definition
to the
, and
is
if Lemma 31 (ii)
more general
definition
rJl:m froill a polyhedron
-t
tlten ,,"Nt can homatope
is the
already
X
to ~
to a non-
f
homotopY.
If,
further,,
we can keeE
Xo
homotoBl_
let of
small
non-degener3te?
Choose triangulations
trL:mgulation
M to be non-degenerate
-t
equivalent
x ~ rn,
and
during
is
x € S2(f\lk(K,K»
a vertex
g :!2l... an arbitrarily
m~
simplicial?
stars
A C X - Br(f)
s t (A, K)
2.
Givan a
degenerate
fixed
this
in Ch3pter
manifold
the n
o
lk(A7K)
The justification
Lemma 32
mapped
dim A < dim K2 ' and so
Therefore dim K2 •
We recall
A Jl. Koo'
simplexes,
there
K"",?
<
dim (L - K",,)
Soo( f)
some simplex
,
L say.
7
If
union
If
and either
as to whether
is the
A faces
X,
or a half-open
in the
interior
X,M with
K be a first and let
triangulation
m-cell
of
B1".'
of
m-cell,
or boundary
derived
Th1
,Et
(each
according of
WI).
respect
to which
f
complex of the denote
Bs
is
the
either
to whether
The set
{Bs
open vertex an open vertex
1
is
lies
an open
•
-21covering (p)
of
for
TIT,
each
sufficiently
and A E K,
close
Now order order
has the
f
of locally
c some Bs.
f(st(A,K))
to the
property:
f
also
satisfies
simplexes
increasing
Any map
(p).
A1' ••• ,fir
dimension
of
(i.e.
K in some if
A. < A.
VA ..
K. =
starting
with
(i)
(ii)
f.~ 1
f. 1
f.
satisfies
f.IK. 1
1.
is
The end of the
a sequence
I
Xo = Kj'
,
X ~ M such that
homotopy keeping
K.1- 1
non-degenerate.* g=f
case when'
Xo C X
we choose the
original
80C1e
fi:
on i,
(P),
Xo as a subcomplex,
contain
by induction
of maps small
induction
For the degenerate,
,
by an ~rbitrarily
1-
1
(iii)
fO = f
We cons truct,
J
1
1
then
J
1
i
i ~ j ) , and let
X ....•~Il
j,
r
proves
the
and
fl Xo
is already
triangulation
choose the
and then
lemma.
start
the
non-
so as to
ordering induction
so that at
with
j,
f. = f • J
Wo must now prove defined,
and let
the
such that
Bs
h ~ B ....•~ the
the
L=.A.
,
1
f_ L i 1
onto a simplex
inductive
step.
L=st(A,K), C B
and let
fi-1
B denote
Choose a homeomorphism
0
and define
diagram
Assume
g
= hf. 1-1
;
in other
words
f.1- 1 L ----)
B
~z< I:::>
is
*
commutative. \7e do not clai~J. that
th':tt
f;
f. ~K.....• 1
1
e:.1beds end sim.plex of
v!:
Ki
is in
si~plicial, M, but
nor do we claim.
only th':'ct Soo(f.lrc. 1
1.
)=¢.
-22Choose subdivisions g ~ L' -+,6'
l' 7~'
of
L,A
such that
is SiUlplicial, and such that l' has at least one o
vertex in A
1et
0
denote the vertices of l'
G
xp+1"
contained in 1~, and let vertices of l'
.
0
0
,xq
denote the remaining
Choose a sequence of points
(Y1"
arbi trarily close to
fx n ' 1~n~p • Define g' to be the linear map determined by the vertex map
•
Then
P" o
so
gl A
dim
J~ <
f.1.- 1
I ~; == l·h,
• gill.
,Yp) c. 6
4' , such that Yn
in general position with respect to
g'x n
o.
g
is
L' -+(;>
1sn~p
==
which is non-degenerate by induction, and
is non-degenerate 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 h-1 g' -- h-1g 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 Ki-1
-23homotopy is sufficiently small. because
=
I(.
~
Ki-1 U A
and
t
f.1~
Thirdly
f.
K.
~
is non-degenl">rai;c
is non-degenorate 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 r-fold point dimension 0.
r
Define
g
g
X
-t
M:
x - (r-1)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 non-degenerate
and
dim Br(f) < 0.2 . The first follows from Lemma 31 (ii) , because we are assuming x < m , and so
dr <. 0
from Lemo9, 31 (iii) .
for r large; the second then follows
-24Rems.rk2 for
We shall
simplicity.
oursolvGs
The engulfing
the boundary. be treated
confine
theorems are
In applications
independently
to 'the. interior
of' mg,nifo1.dS
especi~lly
tricky
at
the boundary problem can generqlly
and more cl santly
by the Addendum to
Theorem 12. Re [11:') rk 4
In applications
situation of
of wanting
X which already
often
e8bedded).
introduce
happens
a relative
]2.ositism for
Xo eX.
:Ji0mark 1
g\XO
Xo
if If
is
.
xo == x
:But if
Remark 2
instead
sight
it
of werely
in
S (g) r
we
-position.
in general then
~ then
xO< x
9
position. diGJ,(Sr(g) f) Xo)
(i)
implies
then
(iii)
(i)
is,
1£ dr -
1 •
(ii),
g i;:lplies
eJ.ch r.
and (iii) general
is vacuo\ls,
position
for
or (ii1). possible.
we ought to be able
to make
But if
of
dr'
~ the ['est
X is not a r.19,nifold~ then
X may C3use certain
independent
<
does not i::J.ply (ii)
surprisingly
would see~ th~t
the non-hoillogenei ty of to lie
the Qain theoreo
g: X ~ M to be in gen~
pos i tioD of
ConditioD
At first
st9ting
(.1nd is
(X,XO) if
<: x,
and so then genera.l (X,Xo)
Define
g is in general
(ii) (iii)
to be in gener9.1 position before
Xo
on a subpolyhedron
definition.
the pa.ir
(i)
hqve the relative
to keep a map fixed
Therefore
Suppose
frequently
WG
g.
It
points
of
X always
is not the r-fold
points
-25S~(g)
theGselves~
cause
but the
li~it
points
in the branch
set
that
the trouble.
:Ex3mple
Let
polyhedron X
cannot
Xo
C Br(f)
X be thG join
not embeddable be locally
of a p-simplex
in 2n-space,
Xo to an n-dirc.ensional
and let
embedded at any point
,for all
g.
If
g is
m:= 2n+ 1 •
of
in general
Then
XO' and so position
for
(X,XO)
then diffi S2(g)
:= d2 := p + 1
=
dim (S2(g)nXO) but
d2 - (x - xO)
<
the whole singul~.r wa:1t to "pipe S(f)
set
n large. we shall
S(f)
awaylt the
~ and in order
interiors should
for
In applications
Remark 3
of
0
.
XO'
it
consist
point
concerned
But in critical
to do this
the triple
subpolyhedron
be priillarily
cliddle:s of the
of such simplexes avoid
p
set,
1Ne summarise
cases
siG.1plexes
be important
of pure double the branch
this
'1Neshall
top dimensional
will
set,
inforr.1ation
with
that
points,
the and
and a certqin in a useful
form: Theorem 17 (X,Xo),
f :X
Let
M be in gelleral ~~ote
'INhere
Lc?t K bo
11Y.
(1:= 2x -
Xo
8 S
for
some tri.8ngul'.1tion
G:l
,
a su'Jcomplex,
(i) of
o
-t
Then the
K of di::lens ion
[1.
f
g
K
-t
i tion
for
point X,
M is
the pair
dioension
th7 t contains SiLlpJ.icial
m.
singul.:,rities ~ d •
the double
trian,2;111s.tion ..of
and such th? t of,
DOS
S(f)
of
f
foro. a subcomplex
-26(ii)
If
one other denote in
A is
d-siillpl(lx
the
X - Xo
d-sioplex
'=l
A* of
open st':trs
of
of
8 (f) 11.1
A* in
K,
other
points
Proof of
fU
(i)
8.re em.bedding§..~
••
no
By Lemma 31 9.nd the definition
of gener'3.1 pas i tion
f. o
and so ~
contained
fA = fA* , and cont8.in
in
u*
,")f fX.
(ii)
A
fU.* intersect
1
r?
If
U1 U* are
f I u*
flU,
the:reJ1L~Xa--2-~lx
fA = L'J.*.
then
ell
the. imqges
then
t
such that
and the res1;rictions
1
S(f)
.Ii
1- 83(f)
for Nowl\.
U ==
Also
Therefore
"
'!lords
of general
•
Ii
consists
exactly
¢.
2
for
flu
if
a contradiction.
the
simplex
u -I- u* , then
fU,
fU* meet only in
pair
The rest
(X1XO). Xo is
for
U*. o = .1.1 ,
u€S(f)"U
map can be moved into
1
by defini tio:p.
Therefore
a subcomplex.
Since
f
is
Finally
then if
A C Br(f)
u E U?
fu
,
= fu*
Therefore
qnd so
($
and contain
fA = fA*
of Theorem 17 is of the
snd so
A*.
was not an embedding,
o
The proof
points,
Q
Sim.ilarly
and
because
dim.(S(f) (\ XO) <. d
because
position,
by Lealma 31 and so
of eX'"'lctly double
st(A,X:) C. X-XO ' because
non-degenerate,
fX.
3 ' (f)
one other
Xo'
position
<.
Ii
of general
<: d
dim Br(f)
0
¢ Br(f).
In other
< d by definition
Dim S3(f)
chapter
is
no other
points
com.plete. devoted
gener8.1 position.
to showing that
any
of
-27Theorem 18 interior
Let
f ~X
'1
mg,p from a polyhedron x<
of a ffi'3.nifold ill, where
gener9.1
position
where
by an 3rbi tr3rily is
o
M be
-t
in general
(Notice
Xo is
small
position
th8.t the
a sll.bpolyhedron
hoootopy for
theoren
the
is
keeping p'lir
of
X.
Xo fixed,
is
in
Then
f -- g
such that
g
(X,XO).
trivial
if
X=ffi).
1
t,ny ill2:l?
X -t M is homotopic
First
homotope
X into
With the
hypothesis
Coroll9.ry
f I Xo
Suppose
ill.
X to the
to a mg,p in general
position. Proof in the
2
a finite
family
(X.,X.)
J
1
Choose a tri'3.ngulation
put
Xo == ¢
{xil
be
position
and extend theorem.
adjacent XiCKn,
the
K of
X containing
Kn
homotopy from
position
begins keeping
of skeletons
and so
glXi
for
all
Kn be the
(Kn ,Kn-1)
pair
we have a map g, pair
each
is
1;0
with
is
cont'J.ining in general
every
K by the
the
I
f Kn
Kn-1
keeping
At the
into
f I Kn
end of the
position
for
XO.
n == x.
then
position,.
o:e
fixed,
in general If
X.1
homotopy extension
n = Xo ' by moving
Xo fixed. that
i.
n-skeleton
use Theorem 18 to homotope
The induction
general
induction
on n, for
the
ex,
j=O).
------
by Theorem 1, '3.nd let
:By induction
general
Xo c:.Xi
X.::::>X. (including
-----
as subcomplexes,
that
g .§.SLas to be in gener9.1 position
forwhich
J
1
Proof
into
Qnd then
of Theorem 18, let
of polyhedr·4..~~ch
Then we C8.n choose
K.
interior,
theorem.
Corollary
]ag
the
1
If
each
Xi:;' Xj then
-28oi ther
x. == x.
general
position
1
3nd the
J
for
the
( X. ~X .) C (Kn ~1(n -1 )
because
Rem.ark
In Coroll,'3ry
J
to the
pair
2, if
of the
far
(iii)
gen0ral
of
t o 9:enerELl·.J-·pOS1iJ10n
such that
g
Hewing c10ved f
we then
use Theorem 15 ~ by induction
Stnd Sr( g) ==
Theorem 15, and uses
general
Xo C
p08iticn~
and so by
to
LOU,L1:1
Xo
LT.13p keeping
-+
derived
L
is
K of
equal
X ~ V'le
a generBlisqtion
position
s pg.ce.
then 32 we fixed.
si8plici91.
f:
gSr+1 (g)
nf the
X -~ iiI
first
Therefore K, KO Let
complexos modulo the
of
with
like
t-shift
such tll'lt
flxo
1,
r
general '-'
fixed.
of Theorem 18 is
in pqrticular CQl1
starting
~Sr (~) ,-, "~C M into
isotope
Y keeping
X and
g by Corol11ry
on r?
a generalisation
Choose triangulations f : Ie
and the family
e'3.ch r,
general
The proof
of R ~ap
Given
into
0 ~ 'll:lbient
\I"ith respect
The t-shift
•
K in euclidc3n
for
F'roof
position
for
Yc III, we can hOLJotope f
CorollEl?:'Ll
l'3rge
(K•.n ? Kn-1)
in nqnifolds,
position
9.nd
~
satisfied
of a triQngul'=ltion
position
pri::.1iti ve gencro.l
is
Xo == ¢
put
'I've
x. > IrJ"
or else
J
1
implies
()f
(X.?X.);
. sa t·18 f'1e d f or 1S
it
f'1r'.lily of skoletons
recover,
position
so condition
J (X. ,X.) 1 1
gener"ll
of
as follows.
flxo
is
that
is
in
non-degenerate~
homotope
f
into
8,ssune
f
non-degenerate
<J.
non-degener3. te 0
X, Xo g,nd 1J of M such th8..t K', L I
denote
(t-1)-skeletons
the bSl,rycentric of
K?L.
-29Then
f ,K'
-+
L'
rem'1ins siwplicial
IJ,;t B be a t-sicpl?x f::ji[]plexes of since
f
is
l'-;,st?
CO:J.8
l,. ,B
of
n-:n~-d.egenerate.
=
f.
dE~notes the
1
then
f.
1
-t
1 1 1
is
i:'-n be the
so that
those
Xo
in
such th'."lt
q
be the barycentres
i
=
st(b,LI)
bBQ
restriction • bBw
-
th0 join of the three The loc'}l
a
A's
each
a.A.P. 111
9
'Nhich aro ':\,11 t-siuplexGs
9
is an integer
Then for
1
1
f,
there
fa. ;.::b ).
f . ~ a. L. P.
of
Ord8r the
is non-degene·rate L1' •••
ai,b
st(a.,K') 1 :-:md if
B
and only if (with
1
\lords
f
fIC, and let
K th:J.t qre m3pped onto
in othur
if
of
bcc8,use
shift,
w?PS
a. -+b
,
1
given below,
determines
nl3W C3p
that equ21s f.1 if A.P. 1 1
frontier vary, that
Xo
fixed
th(: local
Local
shift
before, centre
if
G.'i.aps
is houotopic fixed.
i > q ,
f:r. Cl
to
f
and is h0Dotopic
except
ThE~local
03p
that
instead
ench cone is
-1
g
letting
keeping i
the
'ind
B
a £~lobD,lc.a.p .~ s8all
a t-shift
shift
fi
is
L'lUeh
homotopy keeping k(~e'Ping Xo the
fixed.
S'1rneas
of moving one cone away froD the
we h~ve to Clove several
Llthough
into
by 9.n grbi trl.rily
We c'1.11 the Clove f of a
Therefore,
i ~q •
c00bine
to
cones away froo
not in general
each other.
ei.:.1beddcd,the Qovement of
•.
-30e~ch cone c~n be reqlised shift
of an eobedding
but
j
of the cones
is
by qn a~bient of course
isotopy
the
choose
a hOrJ.eoc:orphis2
r
Extend
h; bQ -', 6 by o·J.pping b
6,
Stnd joining
simplicial. (v 1 ? V 29
t:J: '. on
6.
each
•••
,
i
vq)C ki:
possible
Subdivide
into
v in general
the
the
by hypothes is
x <: m ).
to the barycontre
so that
q <: i S n,
vi = v, near
,t).-t
since
(1 onto
-t
h ~ (bQ)
and choose position
be the homcoC1orphiso joining
v -t
V •• l
In pQ,rticu12r
m~ps
f ~A.
k.
l
=
1
D,,'
-t
I
If
then
i;> q
isotopic
of the ,2;.
"l
= f.l
(and therefore isotopy
frontier
Proof
is
respect
the
to
identity
i > q.
For
keeping "
A.P.). l
l
,
B
-t
l
9nd if
i ~q
homotopic)
to
'1nd
h
then f.
cowpletes
k . hf ~ a. P. l
g.
is
l
l
l
31
In the local
shift
ThE: singul-' r S9ts A t-shift
S (g.) r l
are unaltered
preserves
-t
the
.
r
small
BQ (~nd therefore
definition
== S
bQ
g,mbient
by an arbitrarily
l
fix8c] the bound!lry This
-1
of the
shift. LeCl~lla
of
sequence
Il,
with
6.
v
define
9
join
the
[)
to the map
to be the
a~bient
is
linearly"
Dcofine
Let
.
(which
.
h ~Q
y. fP i
bound'1ry of an (m-t) -s ir.1plex, throwing (m-t-1 ) -fqce
local
illovenent of the union
only a hoootopy .
As b afore,
h to
as in the
(f.)
l
•
by ambient
non-degeneracy.
isotopy.
local
-31LeW:"18,
34
t-~hift
With the hypothesis keeping,
fixed.
Xo
of' Theorer:l 18_-1.et f ~ g
If
then the s 9rJ8 is true
for
CO!.Qll§!J:
preserves
s
A t-shift
sufficiently
die Sr(f)
~ dr
geneT"'!.l posi tio.Q (for
r ~Q
choose
lqrge). d > dr'
Suppose not:
suppose
d == dLJ. S (g)
r< s.
g agrees
and
r
Since
of the 10c8.l shifts,
interior
fO~8,1J-,-
6.
Proof of Lemos 34
frontiers
be a
of SODelocql dir.'1
r
f
on the
sOL1ething D.ust go wrong in the
shift.
S (g I
with
where
Therefore
n
.•
u (a. L . P . i==1 l l l
-;,
. P. ))
l
==
d ,
l
and n
dim S.jgl U (a.P. .•.
Therefore " I D with
respect
hg
to which D E
1\ .,
l 'N8
l
- P.)) l
of at least
1:!
i~ q,
hg is
r
(U'l.P.) l
and,
and suppose
I
1
Sitjlplicial,
si::J.plexes. D,
d- t
==
in the interior
u ,
sir:.lplexes r:J.appingonto a.P.,
l
if "'.Ie ch'Jose subdivisions
(d-t)-siDplex under
1
rO lie
of
Ua.P.
then there
is a
Select
g,
set
suppose n U a.P ..
in
q+1
1
1 1
,
6., thq,t is the ioage
of
of these,
,6,11
l
of exactly r.
lie
1
1"1
in
Therefore
l
h~}ve q
r == ')ITs
shall
r. r. v'"
each
l
now choose cert'"J.in sL.lplexes
i == 0,1 , •.• ,q,
vJith the
,
i.
.....
C. C. l
and
properties D. J D l
dim D.l ~ diD 6 -r.c • l (Reca11
C ==
c()di:J.ensinD. ::::r:l
- X
a)
D. C l
A,
for
-32Firstly
if
r.l == 0,
triviqlly
(gla.P.)
ri
bqsG
C. == rand
D. ==
l
l
IJ. ,
satisfied. Secondly
hgS
choose
l
hf S
is
l
r.
f. 0
a subc0ne
of
supposo
(f I P.).
ri
l
and
hg(a.P.) .
Therefore
l
i ~1 l
there
:By
0
33
LewDl'?l
with vertex
l
is
v.
J.
C. E.6,'
a siGlplex
l
and such
that C.
C
hfS
D.
=
v. C.
l
l
l
ri
I
( f P. )
:>
l
l
D
"
Therefore dio C. :;: di:n S
r-i.•.
l
==
(flp.)
.
l
diu S (fla.L.P.) ri l l
~ dr.
- t - 1
l
-
t
-
1
by hypothesis
1
since
l
=
rJ. -
r.l c - t - 1
;"1
ric - t
9
'NhGre c
=
codi8Gnsion
Therefore dim D.l ~
.~-
::: diLl 6 - r. c l
verifying
property
(*) • rO f. O.
Fin'111y suppose
:By construction
n as
f
on
U a. P. ,
q+1
l
f-3.nd
so thero
is
Go
l
n Co C hfSr DO
==
o
(fl
vCO =:J D
U
q+1
P.) l
SiGlplex
g is
Co E 6,'
the
saGle
such that
~
-33We verify
(*)
in the previous
9.S
Sr (f\
o
_\8
ro -
spr.J.ce, and denote by etc.Q
ls before
of
Q
[ciJ
1 1 1
t - 1 ,
Le~J.t:13
P. ) - t - 1
a.',.
q+1
d
in the proof
q+1 n
SrO ( flu
::: diG <.
case: n U Pi)
sin ce
r 0 ~ r < s,
'3. nd
so
6.. in euclidean
30, eobed
subsp3.ce spanned by Ci' are two possibilities, each leading to a
there
the linear
contrqdiction. j -1 C':lS6
For
(s. )
6Qch
j
~I ••• < j ~q ,
9
(\ [D.] o
9.nd
[Dj]
span
1
CSt.se
(b)
Not (a)
In C'1.se (a)
we defrQce dio
Summing
for
0
()j [ D.1-' 1 +
o
j::: 1 ,2, .
0
•
,q
q
dim "[D.) o
diC:1 6.
:::
. ~ jn -1
dhl
= 1 :::
q , dir:l D. t'(j 1 dim
6
(l:J.-t) :::
lJ. -
:::
d
<
d-t
r
to
+ -
t - rc - t
+
dim
we h:lve
3.nd cancelling,
9
. o CD 1·l
6-
q dim (dit:l
- diC1~)
1
q
L r.c,
o
D.
1
by (*)
D 1.
[6J
.
-34contradicting
jno [D.]
the fact
In C'-lse (b),
1
so
J
r.
exists
J proper subspace,
¢
vj
position
TI ,
point
the definition
sufficient
t)
'completes Ler~lCl'3.
T7
>
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 t-shifts 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.
Dd-t 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
-35?roof of LeDm~ 35 9.
Let
d::: dim Ss (g)
•
By LenDa 31
t-dir.lensionalsubcoo.plex of the tri'1.ngulation K
in the t-shift
f~ g •
By construction
S.s (f)
is
of X used
the t-shift keeps the
(t-1 )-skeleton of K fixe d, and so S s (g) .:> S s (f) ioplying th'1t
«t-1 )-skeleton of K)
f\
d ~ t-1 . d -;,t-1
;
d :> 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 d-t 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 t-skeleton
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.)
" (t-skeleton
::::
(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
t-sim.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
t-shift
Z D:1pped by
g to
J.
and so
~)s(g\ Z) C Z (\ (t-skeleton
of
K)
n
==
1) 'Vi <:'>
q+1
SinC8'lq+1
, ••• ,an
v , and since Ss(glz)
==¢
q,re the
there
,and
are so
less
rO< s.
only than
points s
of
of theGl, we deduce
The proof
of Leooa 35 is
cOQplete.
-37Lemo'), 2.§. both t-shift
XOCX
flxo
and
f
(i)
Given
are
keeping
xo<
9
,
X
in general
f :X -, IiI , supp.Q.se tl1ll
and
Xo fixe3.0
If
diw.(Sr(f)
tru0
for
g.
(ii)
If?
further,
(\ Xo) < dr
for
9
.1hen
t==d s
all
1"<
s , then
A t-shift
preserves
For use the
Corollq,ry
to Lec.:o3,34, :1nd Lewwa 36(i)
ill
of Lem~:lFl 36
have
'dio.( Sr( g) " XO) == d 1" ,
Lemma 34 Coroll,~lry, of .'1 locql
dimension
d1" - t
general
Suppose not.
Proof
shift,
8.nd
, in the
proof find
position
Then for
bec'3.use
As in the
the
is
S8.ITl8
d i [1( S s (g)" XO) < d s •
Coroll~£l
interior
be a
f-+g
Let
position.
din
for
pairs"
with
sorJ.8
1"
9
s
large,
< s
9
S 1"(g) ~ d 1" by
of LeoDa 34 we exaoine o
a sill1plex
Dc 69
siuplexes
i~aage of
we
the
of
of
n
V
Z ==
q+1 D C
fXO'
Therefore glZ
=
flZ
a.P.
'3.nd r.
D
is
Xo () a i Pi in the
of
a.P.
,
1 ~ i~ q.
1.1so
1 1
¢ for
==
im3.ge of
1~
i ~ q , :1nd SOl"O
SrO(g I z)
n
XO'
I0
But
, and
~nd so in the verific~tion
of LOOQ'134)
Therefore
sit:.lplexes
-L
But
by hypothesis,
proof
9
1 1
in case
we gain
(a)
one diuension;
we have
of
(*)
(~s in the
,
-38d -t
contradicting In case
the const1"Uc-tion
(b) the contradiction
(ii) p~rt
D r
The proof
(i)?
except
for
of
in the proof
and so
r.]. <:
should
h[-lve s
contradicting
included
the oodification
siCiplexes
in the general
position
of
,
1"'0J 0
bec8,use
D C fX
O
1"'0 <: s ? ()thervvis e we
Ll9.ppedonto
x::>xo
the saDe as for
= i ~q
Fin'J.lly Xo
of
0
Firstly
1 ~ i ~ q.
is
of h~ving to show
for
the hypothesis
Leo~a 36 is
].
36 p':'lrt (ii)
1J8:.1::'1"1
of LGD.rJ8, 35. for
S
f't D.•
is unch~nged.
r.]. <: s , 8.8
C
B,
irn.plying
and the
condition
flxo.
The proof
of
complete. to
Proof
of Theorem 18
genar'll
position,
for
pair
the
shows that
f
x:::::Xo
of
g iClplies
c'~n be moved into
general
reGlains to 2chieve
induction s ::::1 •
Xo fixed.
pas i tion
condition
36 shows this
on s
putting
The general
Lemma 34 Corollary.
(iii)
t
position
=
ds' of
pg,ir.
for
gener'll using
and starting f
in
position
using
the general
the
can be also
f! Xo
general
position
because for
into
with
Ler.1!J.3. 35 Corollary
general
we are then finished,
L0208.
f ~X ~ lvl
':lnd we h:;.ve to Dove f
(X,XO) keeping
If
pair.
We are given
If
t-shifts.
position
x::>xo ' there
position t-shifts,
of the by
trivially
Gleanwhile is preserved
with by
INSTITUT DES HJ~UTES ETUDES SCIENTIFI~UES
:1..9.ll 1..Fevised-12~21 Seminar
on
Combinctorial
by
E.C. ZEEH.AN.
Topology
The University Coventry.
of Warwick,
VII INSTITUT
DES
HAUTES
SeminE'tr on
ETUDES
Combinatorial
The idea of' 8:,1 enGulfing homotopy
stateme~t
step in passing
into a geometrical
from algebra
map (2)
a homotopy
X
X
aoout
Obviously
is k-connected~
in
is homotopic
is contained X,
it is a key
subspace
M
in the interior
two statements
theorem.
about
that is to say the
to a constant.
in an m-ball
in
M.
Tho first is
and the second is a geometrical
the sGcond implies the first, because
The converse
our first engulfing
for
:E
c
X
is contractiblG.
liI
statement:
be a compact
is inessentJ-&
statement
statement.
the ')rem is to convert a
M, and COIlsider the following
(1 ) inclusion
Topology
to geometry.
For exam::::Jle le t of a manifold
SCIEr~TIFIQImS
is not so obvious~
a ball
and is given by
For the thecrem He shall assume that
that is to say the homotopy
groups 1t.(I\1) l
vanish
i ~ k.
£illd....-~,_~
ffi_
+ k .::'_.1..:.. ..JhelL_X __ is
111.8 S sen tJ~.. L.~tn
it-+_S3_.~cp!,-';.!.~ill9.9-_ in _1L.~11 ~1L~}1y __;tn.!9rior_of"JvI~~
11 if' ~d
onlY-if.
X.
VII
- 2 We shall prove a gelJ.cralisatioY.l in Theorem ~rom which the above result ~ollows at once. proof of Theorem 21 is long9
illustrate
the main idea of an engulfing
19,
called "pipingl:,
theorem.
The proof
48, involves a more complicated. technique
and we ~Qostpoll.e this UIJ.tillater in the chapter
in order not to interrupt
our main flow of thought.
the piping lemma is only concerned with winning
=
which will
four lemmas, the first three of which are straightforward.
The last one, Lemma
x
eince the
s~ecial techniques for the
we give a shorter proof 07: Theorem
boundary,
requires
involving
However
21 below,
In effect
the last dimension
m - 3. If
L~n'Y
o
in..~!t__ .ill:}d.-:i_.s;._B ._tt~DLe._.~c.ll11~2.illJ? i ~n tis
0
t. 0"0_L....lL1l..t..:Ll
o
Y c B, for if not
First we may assume that
o
isotop
B
onto a regular lleighbourhood of itself in
Theorem 8(3). to
X
The proof is easy to visualise
we push
other bits of was necessary
B B
along with it.
out of the way as we
to have
B
M
so tllat X9 Yare
we can ensure that
y .11..
g09
: triangulate
subcomplexes. collapses
on the number of elementary
Y
expanl!is
which explains why it M. a ~1oighbourhood of
By subdividing
simplicially
simplicial
becausE.:as
that we may have to push
in the interior of
Now for the details in
Notice
lvI, by
to
collapses,
X
if necessary
By induction it suffices to
VII
- 3 consider the case vvhen X ~
Y
is an elementary
(Notice that we do not say anything about the triangulation,
for otherwise
because during the induction
aF
w
the link of
L
F.
f:i."om the face
of
gets pushed around.)
that
across the simplex
X~Y
F
Let
denote the barycentre
F
in
b.
be a simplex of' dimension
M, which
of
F,
is a sphere because
0
c X
F
being a subcomplex
,\
, = ~. and
collapse.
this would violate the induction,
B
Suppose, therefore,
B
simplicial
c M.
Let
1 + dim L, and
1\
choose a homeomorphism
h : L ~~.
Map
F
to tho barycentre
1\
6
1\
of
6, and extend linearly
to a homeomorphism
h : FL ~ b..
0
Since
Y
c
sF
. + we can choose a pOlnv
b
c
B,
in the 1\
interior
of the segment 0
that
abF c B.
Let f
the homeomorphism
.
aF such b.
defined by
1\
•
mapping hb ~ b., composition h-1
keeping fh
composition
st(F,
M)
therefore
b.
fixed,
and joining linearly.
throws on
F
b
onto
to a homeomorphism keeping
fixed, because
Y
g : M ~ M
M - st(F, M) fixed.
does not meet
F.
Join this
to give a homeomorphism
onto itself, ·which l{eeps the boundary extends
The
1\
: ~L ~ ~L
to the identity
to the identity, Y
6. be
->
fixed,
of
and which
ambient
isotopic
The isotopy keeps
step, 1\1), and moves
ab F
onto A.
o
Since
B ~ Y u abF,
o
gB ~ Y u A
=
X.
a more delicate
the isotopy moves
The proof of Lemma version
37
of this result
B
to gB where
is complete.
We shall prove
in Lemma 42 below,
replacing
-4 the manifold
B
by an arbitrary
VII
subspace.
1)1en. fX _':~.1. fY~ Let
Prool'.: f : K ~ L tha t
Y
K, L
is simplicial. J. S
be triangulations
Let
a subcomplex,
K'
a:ad K'
o~
X, Z
be a subdivision
of
such that K
collal!ses simplicially
such to
Y.
Let
=
Ie I
denote
the sequenco
for each '{oJe
fA.
1
across fK
i-1
"'.,
-
fK.
from the face
that
,
.J;.".
1
non-degenerately fB. a face.
o
0
1
Ai u Bi face
of
the simplex
C
X -
Also
Y
collapses, A.
1
and suppose,
from the face
collapse
across
B .• 1
the
1
in general
X - S(f).
to
fB .•
A
1
0 -r11 f\ .L \ .••••
y C
there is no subdi vi sian
i U
Sef).
B.) l'
n
L'
The reason for our claim is
into some simplex
A.
!
,
and
=
n
fE. (notice that we do not claim it is a
linearly
because
K
simplicial
f ~ K' ~ L' is simplicial). mags
f
'::!
is an elsmentary
1
sim.Pli£.-t~1. collapse becau5c such that
K1 '::,\ •••
of elementary
i,
claim that
ball
Ko ''-~
of
Therefore fK. 1
Therefore
=¢
L, and fA.
is a ball
becaus:
fA.1 n fK.1
is the complementary
1
The sequenc;: of eleme';l tar;y-collapses
gives
fX
~;fY.
- 5 The proof' is
x
==
y
Z.
==
result,
v lal if'
>
:It
ill-
3, for th en
-
assume x ~ m - 3.
Therefore
namely
tl'l
VII
the same statement
ChOOB0
We r~~o~ p~ove a weaker
cxcept
that Z is one dimension
higher. Proof of thc.yve"gJcerresulLCg; Let C be the cone on X.
~_2x - m t.31. ..
Sinco X is inessential,
o
simplicial keeping
approximation
fIx fixed.
position
kee~ing
fix fixed.
By the relative
Therefore
linear,
l' into general
the singular
set S(f) of
~ 2(x + 1) - m. of C through
S(f); tlmt is to say D
of all rays of C that meet S(1') in some point
than the vel~tex of the cone. fC, Z --fD.
Them dim D ~ 2x - m + 3.
Since a cone collapses
Y--~·..••Z, and the proof
of the weaker
38.
Since fX
of the piping
lemma
As in the weaker
=
X, we have
+-.?J~~
(Lemma 48) below.
lemma is long we postpone
J,et
r(~sult is complete.
Eroof .9..£:....:tl1LSltronge..r_xes1;1J."t .£ z ~ 2x..- m For this we need the piping
other
to any suocone we have C -.... :,D,
and since D ~ S(f) we have Y'~Z by Lemma
x c
M.
theorem we can make l' piecewise
Let D be the subcone
y ==
1':0 ~
By Theore;m 18 we can homotop
l'will be of dimension
is the union
Since the proof
it until later in the chapter.
caso, let C be the con.:;on X, and
o
0
f:C ~ M a (piecewise; linear) Triangulate of X.
extension
X and let Co be the subcone
By Theorem
the
0
X c M to a continuous-map
inclusion
we can extend
18 we can homotop
of the inclusion
X c M.
on the (x - I)-skeleton
l' into general
position
for
VII
- 6.....•
the :pair
Co
-.;r
c Cx+1
X
keeping
Co
U
.JI..
'--'9
:fIX
The triple
f'ixed.
(in tho senS0 of the piping lemma)
is cylinder-like
and so by the :piping lemma we can hornotop 01 c C
and choos8 a subs:pacG
S(f)
n
dim(Co
Let
D
00 U 01 O~,D by
":..\ D U
01 because
Lemma 38~ bcc;::iuSC
fixed,
n
2x - m + 2
~
dim D ~ 2x - m + 2.
01;
-...''"':::\ D and
y == fO, Z == feD
Define
U 01 •
Co
00
f Ix
lweping
C1
< dim 01
be subcone through
f
such that
C
01)
XX9
T11erefol'e
and the result follows
01)
U
D U C:1.:::> 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 in-JsscEtial 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
Qorq-llarx ,2•. a »Qm
immediately
0 tC2.123~ Il1- slllle r e...iJ,L ;::5-f~j;.t~.g
J~f_.1 s ..:t-op 0 1o,g,i ca 111L..JL~me omor-phic
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 Mazur-Brown.
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 x-skeleton
VTr
- 8 -
of
x... .,-
M~ and
the dual
largest
subcomplex
meeting
X).
by Corollary
x1j-I-skeleton (which is defined
of the barycentric
Now a homotopy
1 both X~ X*
m-sphere
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.]jLck-Schir~e~man catc~ory
8
ProoL~_
Let
M
be k-connected.
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 ~azur-Brown
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 (m-l)-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 = m-2.
3--manifold
OpOj.'l
in the
and 2x ~ m+k-2 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.
m-3
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 q-sirnplex
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 non-compact
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 simply-connected
For a description
In particular is
4-lIlanifold
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 (m-2)-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,
m-manifold~ m ~ 3, is a sphere,
is contained
missing
for k .., =: . __m-2.~
--=::;:-.-
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+k-2 is nocessf:'.ry in TheoreD 19. let
is closed manifold
the hypothesis 1
Let M =: S
linking
x
Sm, n ~ 2, and
two little
n-spheres
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~m-3) 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 k-core 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 1-dimE.: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 1-diD.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:.
k-corc.
thclJ.
a point,
or a ball,
- 14 ~~ampl~J.
VII
The k-skeleton
of a triangulation
of M is
The k-sk8leton
of a triangulation
of a
a k-core.
3.
~~e
k-core is another k-core. Exa~~~ __4.
A regular neighbourhood
~~qlAQ~~
If D'~ C then C is a k-core
of a k-core is another
k-core. if ~~d only if
D is a k-core. Exam~le
6. If p ~ q then
Defini~la~~2f
SPx point
cng£1fin£.
Let X, C be compact subspaces ~_X
is a (q-1)-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 (D-C) ~ 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 D-C, 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 non-compact, 22 below.
So far collapsing
and we extend
the definition
as for example
in the proof
has only been defined to non-compact
where
{ D-O -~
for compact
spaces,
spaces as follows.
Define
D-O n C
the right hand side is compact
non-compact
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 (D-O) ~ 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 D-C such that
triangulation
~A:B n
B.
Whenever
we shall say
9J
~xcisio~.
sides is X':B
D-C
D-C 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 non-compact
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 m-cel11 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 ~
L';Lt If[ b~",§:",l~aniJ'_Ql_Sl wi tll.ou t""£0llnCll!P1l-'l-..
.9 onIQ.Q.S~ t .8.}l.b s.J?§.~.()~~ ~L e t~.Q_J)8....§l.,"£ 1 0
Sl,§§ ~-l:2..s.pa co ~1l2..t
n~...£e ssar ily c:ompact L..
- 17 -
prgo? regular
I? C is compact
n0ighbourhood
and umbient
° in U,
o~
neighbourhood
Mo of D"":C tn
C n Mo, Do == D n Mo·
of Co, Do in
SOElC
thE::interior
of
to M by keeping
and another
Thon
of) Theorem
the rest of M fixed.
fine, then U ~ N
case when X u U = M. engulfings?
-p
hU
C
because
the amoeba
is th& tit
is contained
For example apyroach
eacl",;. neYI engulfing
in
was over X.
consider
the
is no good for
may mess up what
The advant.age of' the feeler approach successive
colla:t;)sibilitycondition
q-collalJsibili ty below).
FJ~:HC
neighbourhoods
If the triaDg~lation
price that we havo to pay for this advantage a certain
Let
and so U will be isotoped
tha t the core C stays f'ixe':~ while
satisfy
to a regular
isoto}? NC into ND keeping 8(3). Extend the isotopy
U in IJe:nmaL~1.
Therefore
has a.lready beE.m engul?od.
° fixed.
Let NC' I-jD be second derived
and so we can anbisnt
MO?
X, by Lenmla 40,
containing
which will be compact.
tr ia~J.gula tion of 110'
In general
successive
is easy, for choose one
confine attention
]\/1,
Co u Mo fixed, by (the proof
sufficiently
the proof
isotop one onto the other kee~ing
If C is non-compact,
°0 ==
VII
feelers
are addecL
is The
is that the core must (see the definition
of
ii f\lrther aclv8Dtagc of the' f'eGler approach
can handle bounoJ3.ryppoblcE1s which pre sen t certain
- 18 -
difficulties. consider
Before
the special
the general
case9
case of a collapsible
core.
19~ we shall be able to deduce
of Theorom
from the general proof
handling
VII
Theorem
21, but again
however,
we
As in the case
the following
it is worth
Theorem
giving
20
a short
separately. Th_~~.m
~O.~ ~eJ: l~~mb.2_8.k7.9.QP.nE,).c~2fl. _manif.9ld...l_L~m-3~ ~~
L
both J-11
Le.1-J2J?e Sl;_9Q:J-lgJ2.§Aple~ s~bsl!ac...sL_sll£l X >.§_ COElPEl.9"t..§..1£1?.m2~
the int oct.o_r-_Q.f....~,l._~~If~LLt.J_~~t]1.iUl-1LEL.9_~p._e~.Kll)J'J."t11 a_£Q.1J.D.;gsi ble .1:rq£ll1Ja.c!L1?jl.?-~lli~~=inj~ .. erior of_Ji.L_tha.1..j.s t
Proof· so that
Triangulate
C is simplicially
simplicial
collapses
claim that
it is possible
the complex There
C u Xy
is no trouble
C u X, and subdivide
collapsible.
of C in order
during
the elementary
of decreasing
to perform
collapsing
Order
dimension.
of dimension>
cannot g...,t in the way.
There
with those
x+1, for COIlsider a collapse
of dimension
from the face EX. principal
looks as though
It is possible
'iiJe
all those of dimension>
it to Xo say, dim Xo
collapses
if necessary
=
x on
x, as follows.
x+1, because
X
there might be trouble AX+1
across
that F c X, but since F is
in X, F is still a free face of ii, and so the collapse
is
valid. o
Now Xo is contained Therefore
in a ball
in M by Theorem
19 Corollary
by LGlJma 37, C u X is also contained. in a ball,
1. o
B say, in M.
o
We may assume
C u X c B by taking
D.
regular
neighbourhood
if necessary.
- 19 Now a ball interior,
is a regular by Theorem
neighbourhood the proof feeler
of Theorem
neighbourhood
8 Corollary
B of C.
By Lemma
of any collapsible
1.
40 vIe can engulf X.
by one in special
This completes of the
cases.
If there exists X
Let
set in its
X lies in the regular
Therefore
We now show how the dimension
20.
can be improved
VII
x
.
such that W ...•... "J X,
and X n C = W n C, then we say W can be furled. to X relative __
or, marc briefly, where
Cis
W can be f~rled.
•••
t~~
••
~_
~
'4
_.
__
••••
to C
_' .. ~
~,
The term comes from sailing,
a ship, and the 2-dimensional
the i-dimensional
._
x < w,
sails
Yif
can be furled
to
masts X.
o
111.,1.1 ~.~--1.f... w call1?..L._fill:l~.J~.](_
~ h. theJ~lL..£P...a1l:JL_ 'N. <;. j)
=...Q
o
1n._l'L....~~1h.ql?-_th8;"t_lliLlP=C2!i ..J'-~ Notice
that there is no restriction
and that the feeler has the same dimension corollary
as W.
To provo
we need a lemma, vfliichis a more delicate
We take the opportunity version,
on the d.imension of W,
while proving
sharpel'"the.n.is needed
h0:C'C,
Lot X"': .••Y in tho manifold
version
the of Lemma
37.
this lenmla, to prove a sharpened \';hich
'.iill
M. INrito
be l-~s(:f'ul later fo:.'"
VII
- 20 0
if
X-y c
M
X~Y ~, X~ ~-.Y
if
X-Y c
1~11
if
X "-~~(X n M) u y~y.
The ambient manifold understood. and""~
0
X~Y
0
M is not included
in the notation
We call ~--; an interior _c,.Q.llajLs£" ~
but is always
a bou,n.dar¥_co~,;L.ill?~
an ~!li.§..~tp:t~~c21]'...§.p~ . .At the end of this chapter
define j.,nili[§..rd,g., co~llo,psing '-~ admissible.
Notice transitivity
shall
was introduced
by Irwin~ and inwards
both f'or t:ngulf'ing spaces that meet the boundary.
of 0, ~ is obvious,
int8rch~ngod,
\1'8
is in a sense o:9posite to
tho.t all three relations
tl~t two elementary
simplicial
because
o since it lies in M.
are transitiv.:.;. The
and that of a depends u~on the fact
collapsGs
. 0
_ R
K~ ~
K2~
the froe face of the second rGDains
free in K~,
. -X._~ gJ.ven ~ Y ~ .~ Z, tricmgula to and
Therefore
push all the interior collapses
which
Adnj.ssible collapGing
collo.:psingby Hirsch,
collapses
to the front,
leaving
all the boundary
z.
at tho end, X-~
If N is a derived then N
.
neighbourhood
of X in M
-..g,--;. X. If a ball B in M meets
B is admissibly
collapsiblo, If X ~
X in M is a ball ~e8ting Exanmle _
admissibly
. ."..
••
;.,j,ijI.
4.
collapsible.
in M in a f'ace, then
B~O. 0, then a derived
neighbourhood
of
M in a face.
A ball properly
embedded
in a manifold
is not
VIr
- 21 -
.;LLX':'-a,~ Y
c.
f.~~ed such
that
LW.l!I,L
then
Z i
a,
X u Z* -'-:4
s
ambiE:}~ ... t=i130t..opic
to _Z..:.: •..}cG~,EJ~g
I
~~:'
Remarlcs. -1.
The spaces
. 2.
are
not
nocessarily
compact •
X-Z"A.•. C X-Y •
3.
dim(X-Z~)
4.
The lemma is
~ dim(X-Y), true
if
by 2. a, is
replaced
by 0 or
~, again
by 2. El?oof. Triangulate if
necessary,
triangulate __ ....• r ,or .1'..-1.
A from
tho
face
x=Y in
M, and by subdividing
collapses
of Le::lr'la 37, by induction
proof
sir:rplicial
when X ~ Y is
the
of
..••.0 -.........\ ...•.
As in the elementary
a neighbourhood
collapses,
an elementary F,
say.
it
sufficos
sim:Qlicial
There
are
to
collapse,
on t he number
consider across
two CD.ses according
the the as
of
case simplex
to whe ther
VII
- 22 -
x
Y or
'~:;1.
X
---f3~ Y.
case A~ F c M. was to exclude
st
In the first case A, F
(The purpose
M, and in the second
of the admissibility
the third possibility
.
st
A
.. M~ F
in the hypothesis
eM) .
Firs.t caScLL-.:b-J!..,--i_1!,. As in Lemma 37
'IiVG
to be a subcomple:;~ of the triangula tien, because isotoped
around during
Thorefore
Z,..• '."
the induction~
we may expect A
n
until
do not assume Z
Z is going to be
it reachos
the position
Z to be an arbi trary subpolyhcdron
of i>.. 1\
F
a
1\
Lot F be the baryccntl'o of P~ and let A' be a subcUvision
of A
1\
containing
aF and A
n
Z as subcoIDplexcs.
Choose a point b in the
A
interior
of the segment
aP, and sufficiently
for there to be no vertices
of
i~'
in
cloSG to the . abF other than those
We claim th3.t
--oi o
abF
abF n Z.
point a in aF.
VII
- 23 To prove this, recall that Z ~ Y ~ aF, and so the idea is to collapse
abF onto aF, but leave sticking up those bits lying in Z. let P be a simplex of AI meeting bF and not contained
More precisely, in Z.
=
Let Pi
=
P n abF, P2
P n bF.
Then Pi is a conveA linear
cell with face P2, and so we can collapse Pi from P2• o
collapse
is interior,
because
0
Moreover
this
0
Pi U P2 cAe
for all such P, in order of decreasing reQuired
0
M.
Perform
dimension,
cullapsos
and this gives the
By excision
collapse.
•
0
abF U Z "'-::,\ Z.
i
Since F
M, the link of F is a sphere, and so we can define the
homeomorphism abF onto A.
in the proof of Lemma 37, throwing
g:M ~ M described Let
Z~;:
=
gZ;
then Z is ambient
isotopic
to Z* keeping 0
The inage under g of the colll;;lpse abF u Z~
y fixed.
Z
is a collapse
0
A U Z:::l:
But A U Z>;-. = X u Z .... -., because
Z, .,...•
X
=
A u Y and Z* :J Y. o
Therefore:; we have the
required
collapse
X u Z~;:~
Z*.
S iq£ olld._c .... a_s_e_:_l:~ F c 1\1. In this case,
but
case
the definition
L = lk(F,M) of dimension
is
no longer
the
construction
of g is
different
a sphere
r
1 + dim L, and let
of abF is because
but a ball.
as in the
first
the link
Let 6 be a simplex
be a top dimensional
face
of 6,
1\
with barycentre 1\
F
1\
-+
r,
r.
Let h be a homeomorphism given
0
L
-+ tJ. -
r,
by map~ing 0
and. joining
linearly'.
Then hb E
r,
because
A c M.
1\
There is
a homeomorphism of
6. detel"T'lined by r,mp9ing hb
-+
r,
keeping
o
tJ. -
r
fixed,
and joining
linearly.
As before
this
h08eonorphism
- 24 determines
a homeomorphism
of
~,!=,
isotopic to the identity
abF onto A and keeping Y fixed. as the first case.
VII
The rest of the proof is the same
The collapses
this time are all boundary
The proof of Le~~a 42 is complete. reason for avoiding the third case ll.. the required
throwing
g
Notice .. i M, F c 111is that
collapses.
that the otherwise
isotopy would have had to push stuff off the boundary
of M, which is impossible.
We have to show that if W can be furled, then it can be engulfed with a feeler of the samc dimension.
Recall that furling and X n C
means there exists Xx, x < w, such tfilltW~X
=
W n C.
u O.
This implies that W u C~X
Since x :::; k, we can engulf X c E-·":..!C, dim(E - 0) ~ x + 1 :::; w, by Theorem 20. ",
WuO
such that
Apply Lemma 42 to the situation
0
..... "'>'XuOcE
(the collapse being interior because all subspaces are interior to M), we can ambient isotop E to E* keeping X u C fixed, such that ('Nu C) u E~>~ E*. E*~'O
because
W c D~O.
Let D = (w u 0) u E*.
Then W c D',"",Eojd
the pair (E~;l' 0) is homeomorphic
to (E, 0).
Finally we have to check the dimension
~*)
:::; aim[(W =:
u 0) - (X u C)]
dime-IV - X)
:::;w.
Therefore
of the feeler:
by the Remark &fter Lerr~a 42,
dim(D -
and
VII
- 25 dim(E~ - C)
=
dim(E - C)
~ x + 1
:::;; w. Therefore
dim(D - C) ~ w, and the :proof of the Corollary We have now completed
that
\~c
shall
the proofs
need in the ensuing
lemma (L0mma48).
For tho r0st
prove the generali
sation
that
of the engl11fing theol'ems
chapters,
ot' this allows
apart
chapter
from the pi~ing shall
W8
go on to
for more complicclted cor0s,
permi ts both C and X to meet the boundary of generalisation
is complete.
To
'fl.
stnte
and
the
we need two definitions.
pe f i.~lJ.j..2!';;..,~'lt.~<1=.2.9J-1£J?1l.t1JJ~1.itY... We
descr'ibe
tho collCtpsibili
ty condition
was mcnt,ioned in tho Remark aftc-:r I;emma1.-1-1. of ;JI, not :c.eces sarily
compact.
Let C bG a closed
th9.t subspace
Define C to be .a•.~cq]J-...? ... Jl.,flLP1..e~ __ ~p....11 if .•.••• C
is a subspDCG ':j such thE).t C ""',~ ~~and dim(
theru
on the core,
Q
n
0
~ q.
iK)
If dim C ~ <1 th8n C is q-collapsible. .~~. co 11apSlG~e 1"\:n-;1 closed
If C is neighbourhooc1 of C, of C
n
:'11, is also iDl
BUDspace
subsrace
of :'1: is
O-collapsible
Q.-collapsible
compact and q-collapsiblc9 that
• for all
q.
then any regular
meets flI in a reguln1'" neighbou:r-hood in M
q-collapsible.
arc properly
embedded in a 3-ball
Let C, X be subspaccG
i~ the inclusion
o
of ~ is
of M •
map X c M is homotopic
is not O-collapsible.
---._
We call X .•..... C-inessential ~.-~~
in
~9
keeping
in M
....•~:.;:""-~
X n C fixed,
VII
- 26 to a map X
C.
-4
£lJCam1?.J.:!L~If C is a point in M.
the dei'inition reduces
to X inessential
'rherefore the cO~1cept is a generalisation. If C is a k-core
for if X is triangulated no obstruction boundary
hemisphere
so that X n C is a subcomplex~
to deforming
fixed,
and dim X ~ k then X is C-inessential;
into C each simplex
in order of increasing
then there is
of X - Cl
keeping
its
dime:;'1sion.
If C is the northern hemisphere and X the southern n of S ~ then X is ll9~ C-inessential • .~theOI'~TIL?J ..:.
r....Q.t C~coll
ElllsibJ..lLl~o~Qf
j;h.£.
9 ..£..Jll::.2..:-1&j~_XX be ..£qmp~~:t ancl C-:iIl~~~li~ntJ~L...._illl-.L.~'tll?J?02e
(1)
.
d:Lm (~:LLJ. ..JIL..:S...J<. ~'ld
(2)
..
p__nnJYL==_J..~.~Q.L.D.J4• Notice
that the converse
C-inessential,
because
is trivial~ tile collapse
if we can engulf
X then X is
D-~C gives a deformation
r'etrs.ction of: D onto C, vifhichhomo tops X into C II l~eeping X n C fixed.
- 27 -
VII
Before pl 0ving Theorem 21 we give some examples and corollaries. 1
Let 1.1 be k-cormected, X
o
le t C oe a point
in M~and le t
o
be inessential
in M. Then w'e C2~1e~1.gulfX in a collapsible
A reg"ular neighbourhood
of' D is
8.
bc,ll,
set D.
arn so we deduce Theorem 19. o
Let Mbe x-connected. If C is a collapsible set in M, x 0 then C is an x-core. If X c M, there is no obstruction to deforming
X into C, keeping X n 0 fixed, \7e can engulf ~9JIrqJSL-3~
X~ alld
Theorem 19.
X is
X
illustrate
Yle have l,jI ::: Six
Then 0 is a i-collapsible can be engulfed;
contained
be easily
example~ vv.hichwas described
['.round the S1.
in M by be i11g self-linked
us that
Therefore
so deduce TheoI'em 20.
Consider' Irwin's
Example 3 after
81 x 82n•
and so X is C-inessential.
in a regular
abo\,-e in
82n $ and X ::: S11embedded
Ohoose C = 81xpoint
(2n-1)-core.
in
'l:heorc;m21 tells
by I,emma39 this
is equivalent to saying ne ighbourhood of 81• Of cou:..'se this can
seen by eleme:"1tary methods - but the purpose was to how Theorem 21 can be applied
in situations
where Theorem
19 fails. ~§i~J2l~
We give an example to show that
the hypothesis
(l+X~ m+k-2 in Theorem 21 is the be st possible. Let M = E2n-1 ::: En x En-1, n > 3,0 ::: 81 x En-1, X::: sn-1 x 09 where 8n-1 cont.ains
81 in its
interior .........
---......
.
1'-..--,---,: C
,.--'--' i
!i---~
I
I
I -''''---
J
I
I
~"-.t-~ I -- -.-------I
!
i
!I
...•.....
~j
\
- 28 Then C is an n-collapsible
i-core o~ M (it is not a 2-core
z).
=
~2(M,C) ~ ~1(C)
because
VII
=
Putting x = n - 1, q
n, m = 2n + 1, k
=
1 we see that
q + x ~ m + k - 2 by one, but all the other hypotheses X is trivially
C-inessential
because
are satisfied.
it can be shrunk to a point of C.
we could engulf' X.
SUPT)OS0
Let A be a regular neighbourhood is an open set containing h moving •
X lies lnnbnll by Theorem
1
n in E.
0
Then A x E
Let N
i:a IT by TheOl~em 19.
=
h(A x En-i).
Spun
-r
Ji.
Since ~ (N) n
••
,nth a dlBlc D
to C
a fundamental
class in H~(C
l;
E
Hi (0) be the image of thj.s class under the inclusion
o
II
~ D· c C.
disle
..
l;n thJ..sball
N keeping X = oDn fixed (which is possible since X Then C II Dn is i-dimensional, oriented by orientations
of 0, X, and therefore possesses
1"
0,
II
does not TIleot0).
C ,.lI.v
Tl
=
9. By Theorom 15, isotope nn in N into general position
with respect
.p O.L
n-1
so by Lemma 41 there is a homeomorphism
0, and
this open set over X.
of' S
Then l; is independent
n E,2n-1. ' We verify
in En x
o.
But ~
=
of D
that l;
J
n
because
II
Dn).
Let
homomorphism
it is the linking class
0 by spanning X with the unique
0 from the commutative
diagram of inclusion
homomorphisms
~
···_'::"~H1(N) = The contradiction
z.
shows that X cannot be engulfed.
E~.. €PW..9"~_
We give an example to show that the hypothosis 2x ~ m + k - 3 is the best possible in the case that X c M. Let M = 81 x B2n+1, n ~ 2,
and in the boundary
M
=
S1 x S2n let X
=
Let C be a point, which is a O-collapsible
Sn be as in Irwin's example. O-core.
Then all
VII
- 29 the hypothGSGS 2x
i
arc satisfied,
m+k-3 by one dimension.
We cannot engulf X, for if we were able
to, then X would be contained ·a disk in M.
except that
in ball by Lemma 39, and so would span
This disk would lift to a couiltable set of disjoint
1 , - cover l:\. "B2n ne un1versal x +
't' d'1S k S In
d' 1Sk s wou 1d therefore construction
0f
The boundaries
bv v
bOlli1dary spheres are linked.
The cont:padiction shows that X cannot be engulfed, hypothesis
of these
l' 1 d'1n R un~ln~e ~ x s2n, but
be homologically
we know that adjacent
',rM.
8.l1.d that the
2x ~ m+k-3 is best possible. I cl0 not know whether
that X c M.
x ~ m-4 is best possible
It would be the best possible
if the following
in the case conjecture
were true.
90I.lJ..~~9>:t1IT~ .. L~,U1~~J2~~ ~a__c:t;l}l,.t~",",m§:n.~ fg 12-~ ..?~.Sk
4. let.- =""'-'"""81~_.~,,~-.., be essential _"""~_ ,__ ,,,,,,,,,.~"._,.,.;;:-.:.-=- in •.__ ""'="' •... _.,_.••-...~
M
Observe
Then 81 .••...does bOlmd a disk in M4 . ....• "'._.~_ •.••••... ::w_.~....."._ __ ~ •. -'" _.~.not ..'~ ...-.. ;
~
•.•••
~.-"<--
that S1 does bouncl a P..iDgy.~_~ disle because
A good candidate
for this conjecture
is Mazur's
.__ ~,. __"'._.~~..".,~
tvi4 is contractible.
manifold
" M4 (see
, S1 x pOln . t c (S1 x B3)' , eh ap t er 3 , page 10) , an d lor t' 11e curve cnoose ..P
0
drawn so as to avoid the attached
2-ha:1.o.le.This curve bounds
with one self intersection,
seems impossible
Before proving we state and prove Corollary
Theorem
which
21, which will require
a disk
to remove. several lemmas,
two corollaries.
1 to Theorem
~"""'''*'"'~-'-~'~'~'~'.-~.'''''''''''.'"""",4.~~.-._~-~~,--
21. .•,....,.,_
~Jt t._Q.h :x ..b EL a 8,. Jn ~.Jl''".<::~?l'£l;'L.?1.• _~.Lf_~~V~cpn .2~~L~("i9.Il]".ts~ il?&~tlJ,r 1 ~,:l.J2 _~, then we can engulf' W with a feeler
~.~~
•••.•.••--.'~"
....
, ...•••.,'._"'~~""'-'
.""'-'---.,.-Z"'_'-'a...- •••••••,.:
of ."'_'*...-....~· the same dimension. .-._o......, •.•• _~••.-;.·.•..•.••••.••..••.... _.....
;:.:a~._"".• ,,,,.-..,..._ ..~.-,,,-_~"---.••••.c.=·
The proof is the same as that of the Corollary
*_',.:.:.._o_.~
to Theorem
209 with the
VII
- 30 proviso that it is nacessary to verify admissibility C.£.r:.2J~l,q£L_?__~2~...T1J.PB1~~1!L,,2.:1.DrY£.1nJ
at each stage.
.
J.e t.J!.~ J:_€L~~Llc.. -.2..,9-I?-l1.~L9.!.esl_"_m.ap: i t.o1.q,P... Jc_ £..ID=.3.La.lli1"1 ej~ Q~~
h-connected ~'_~-<"
.•.". ~,
submanifold of 1ft
...-oa.~''''''_'''''''~_''''~'''''' __ '._'~'_'"''''''''''
__
"'O-!
__
m-4.
11 ~
~_''~''~'~'~~''''''''''''
Let C ,·-9:,.;0 in M __
•••••
"'-.......-.,...
••
_--'_
••
¥''''-'''"-'"~''''-'''','''''''
"' __~"""'&~a.
o
~,!lll
and
.'"
0
£LD_.JL~('..~.:...1t§ll ..... 1?..~.9.9nrQ~.U_j.. A."Jv~.,_JS. ~ lSt,.~QL1~L+ Y-.,t )Ln_.Lc;...;,8. wi th
9-llItlllJLlU
_~j}.:-._.TJl;.~ll ..w ~ ~~~~l).Ji1lU' ..X._c:.1L~~_~__ pu 911." t l}.al
..
£li11!.lD-:-c) "i.xz1 al14.Jl..D_1L;::.jl~ ....1L2.LjJ_Ni_~Con.s8Quel'Lill X is q,Qntaip.!2,.cl o
in a ball that meets M in a face in ~.
_
.•.. --"
..••• .;,,~,---,-'
•.•••.. ,,~-"'_••. --...::..-~~.~
.--.~-~ .., ~-_.__.. -.......:..••.=. -.••..•. ,~ ••..•..•••••.••
o
Apply Theorem 21 to X n M, C n M in Q and engulf •
X n M
0
E--~C n M, where dim(E-C) ~ dim(X n M) +1 and E
c
o
E u C --y
R
c
Q.
Then
Now apply Theorem 21 to X, E u C in M to engulf
E ~O.
X c D~--..iB u C, where
u 0))
dim(D-(E
~ x+1 aDd D n h
Because of' the last remar1:::, the colla:9se D~:8
=
.
(X u E u C) n M ::::E.
u C is interior.
Therefore D .,0..;:} C. Therefore D
a.,
o~ and so a derived neighbourhood of D is a
----j.,
o
ball meeting M in a face in Q. Vie
now pI'oceed to the Drool' of Theorem
2"1.
The last statement
of the the 81 s is talcen care of b;)rthe following lemma •
.
ctJ2gS.§:;~J?._so.~]}lttJ2 ...J1.Jii
.
=._lX
lL..9J..,it}l·
The iden is to isoto!) D inwards a little, keeping X u C fixed.
Let Y :::: X u O.
We can assume M compact by confining attention to ~,
a regular neighbourhood
....
,-
of D-Y.
Let c : M x I ~ M, c(x,O)
be a collar given by Lemma 24 Corollary. triangulation
of M
x I
=
x,
X E
Choose a cylindrical
(that is one such that the projection onto M
M
- 31 is simplicial)
VII
1y is a suhcomplex.
such that
Define an isotopy
C-
of M x I in itself as follows.
For each vertex v E M n Y keep
v x I fixed.
For each vertex v E M - Y isoto}? v x 0 along v x I and stop before hitting (v x 1) u C-1y; extend to an isotopy of
v x I in itself keeping v x 1 and tho intersection
with c-1y fixed.
Now extend the isotopy cylinderwise
A x I, A E M
in order of increasing Wl.t'n
d c -1y...,.. J.lxe.
to each prism
d.imension, keeping A x 1 and the intersection
The image under c extends to an isotopy of M
keeping Y fixed and moving M - Y into the interior.
The restriction
to D gives what we want.
We introduce useful
a notion duo to Moe Hirsch, which will be
in the proof of Theorem
21.
Let s denote the simplicial
collapse j7"
•••
The symbol s includes collapsGs.
~L..§.
"-:.'J.\.
'~.•••••
-:--
T
1 ">-\K
n11 the given ordering
Let W be a subcomplex
as follows, by induction in~uctiv0~trail+
of' K.
=.u.
of elementary
We define the E..~t...1of W If n
on n.
simplicial
=
0, define trail
W has been defined for the collapse
s
W = W.
t:
v
Ko ~
and
sunpose K n-·1~K .1:'_
K:t. '::, ••••.
-..,;. Kn-1 •
n is across
the simplex A from the face F •
Define trajl . s W ==
,trailt W,
F ~ trail
.<
~A u trailt W,
t
W
F e trailt W.
VII
- 32 ~~en there
is no confusion we shall
Geometrically
the trail
the deformation retraction K~. L.
drop the suffix
is the track
K 4 L associated
left
s.
by Wduring
with the collapse
We leave the reader
to verify
the elementary properties:
(i)
trail
K = K~
trail
L.
(ii)
trail
(Wi
(iii)
trail
(trail
Therefore
"trail!!
u
W2)
w)
is a closure
L =
= trail
Wi
= trail
~\!•
operator
subcomplexes of K, and the trails fibering
u trail
W2•
on the set of' all
form a sort
of combinatorial
of the collapse. Remark
Notice that
the trail
depends upon the triangulation
the order of the elementary collapser. collapses K',::,L
are re-ordered
then the trails
are different.
For example a~~issibility is an admissible
triangulate
so that
in the boundary.
simplicial
collapse
a simplicial
Therefore
collapse
the trail
is not
invariant.
However trails
X-yY
to give a different
are used to define
then the trails
a piecewise linear
If the same elemen.tary
turn out to be the same, but if different
elementary collapses K~L
and
can be related
is a piecewise collapse
the trail
linear
invariant.
invariants. If
(in a manifold now) then we can
of anything in the boundary remains
Therefore admissible
to ilboundary preserving"
to piecewise linear
collapsing
in terms of' trails.
is equivalent
For Lemma53 below
'. -,.)
we shall introduce inwards
-
73
VII
another piecewise
collapsing~
which
linear
is equivalent
invariant
to "interior
called preserving"
in terms o~ trails. We;
now prove tvvo important
Lemma
propertie s of trails.
dim (tra,11...1V) _:(, dim (L ('Lt~il Let s denote K
=
Ko ~.'
K:1.---.; •••.....•.... ">,
starting
K
=
n
L.
and dim (K1'1_1n T) T then trail
results
hold for s.
Therefore
dim A ~
Let t be the collapse
trailt'fJ.
By induction
on n~ Ko--""
K. 1 11-
dim T ~ w + 1~
.l.
S
If F E T then F E K 11-1 n T, and so dim F :(, w. 1, and so dim (trails
Also L n trail s W:= L n (A
u
w) :=
u
T) ~ w
+
1.
~
~ w.
For each elementary
collapse~
either both A~ F are in the trail~ or neither ordcr~ all those elementary in the trail, giving K
dim (A
T)
A u (Kn_'}.l. n m)
which is of' dimension
Proof
is by induction
Suppose K '--"K n is across A from F. n-1 TT := m n K W == T~ and T n ,no and so both n -,..., ~ n
w +
C
..:LJ.
vv.
:(,
l:-
then
Jil._~VI.
The proof
=
Vi
c K9
'N
the collupse
with n = O.
trivially
of' length n - 1~ and let T
If' F
W
It.K-::,.1is a.simpli,£ial coll.m~d
)+~
"'",,>\
collapses
L u trail W.
across A from F~ say~ are.
Perform~
in
for which neither A, Fare These elementary
collap ses
VII
- 34 are valid, because A principal
if A principal
in Ki u trail W;
and F ~ trail W, then perform,
F
Proof
\l"v2
=
trail W, then
also if F a free face of A in Ki,
is a free face of A in A.1 u trail W.
Let s denote the collapse K--~L. s
trails W2 because W2
C
collapse.
Let t denote
L given by the lemma.
Wi -~
Wi.
Now
collapses, giving
to Lel.lli'TIa 45. If K--~ L is a simplicial
the induced collapse L u trail trailt
1
in order, the rest of the elementary
Corollary
i
in K., and A
Then
Now apply the lemma to t
to obtain L u trails 1N1.--"-"; L u trailt W2• s for t in tho right hand side gives what we want.
Substituting
Corresponding inessentiality to generaliso X
-t
C.
to the genoralisation
to C-inossentiality,
in the theorems from
it is nucessary
from the cone on X to tho mapping
cylinder of
It was for a similar purpose that Whitehead
the mapping
cylinder in 1939.
introduced
We need a relative version
mapping
cylinder here because X n
pointed
out in Chapter 2, the mapping
rather than a piecewise
in the proofs
C is kept fixed.
of the
As it was
cylinder is a simplicial
linear construction:
it is a tool rather
than an end product. Let K, L be complexes meeting
in the subcomplex K n L,
- 35 and let f:K ~ 1 be a simplicial The relative_~apping as follows.
cYlJnger
one of sufficiently
high dimension
¢.
of K uLand
=
1.
~K u L defined
inductively
B E
=
each other).
are
If A E K n L
.
a(~A u A u fA)
AS.
dimGnsion~
where
Define
~K = Ui~A; The relative
choose
so that the new vertices
in order of increasing
..
space9
a say.
If A ,-K - L~ define
~A
= Ui~B;
of f is a complex
always lie in some Euclidean
independent
define ~A =
~A
map such that fiR n L
For each simplex A E K - L choose a new vertex~
(Since our complexes
linearly
VII
ma~:9ing cylinder
K u L as a subcomplex.
A E
Kl.
is ~K u L.
In particular
it contains
VII
- 36 b
c
d
Let K be the 2-simplex~ diagram
abc, and L be the 1-sirnplcx, ad.
shows the relative
~~K ~ L given by ~a
= ~ =
mapping a, ~c
=
The
cyli~deI' o~ the simplicial d.
map
VII
- 37 I&mma_~~
.CP
(NewmfUl)
For this proof
in the complex ~K.
view we replace formulae write
a
~A
=
The symbol ~A will
the statements
0, fA
=
Therefore
o~ = ~o chains.
inductively
If B is a ball,
and so by the ambiguity a general
~A
To empbasise
=
¢,
~A
We replace
=
for a
the chain point
flA is degenerate
2
of
by th9
u by +, and
a(~oA + A + fA).
We ded~Ge
+ 1 + f
on simplexes,
and extending
additively
then oB is the chain of the complex of our notation,
.
oB
=
B.
to
• e,
(Of COl~se for
chain 0, 00 is defined but 0 is not).
The proof is by a double following
stand ambiguously
0, respectively.
for boundary.
by verifying
K - L t,hsm_vA is an
only we shall use chains modulo
and an (n + i)-chain.
subcomplex
E
A.
+ 1)-ball 'y!i th face p~
If An
induction.
Let 1, 2 denote
the
statements'
1(n):
if An E K - L then ~A is an (n + i)-ball.
2(n):
if AB E K, A E K - L, dim AB
=
n, 0 ~ dim A < n,
then p,(AoB) is an n-ball. Notice
that the lemma follows
non-zero
coefficient
and A is a vertex for dimensions
in
o~A.
from 1(n) because
A occurs with
Both 1, 2 are trivial when n = 0,
(A ~ fA because
AiL).
We shall assume
1, 2
< n, and prove first 2(n) and then 1(n).
The proo~ o~ 2(n) is by induction
on dim B.
of 2(n) when dim B = q.
denote
the statement
begins
with 2(n, 0) trivially
Let 2(n, q)
The q-induction
implied by 1(n - 1).
We shall prove
VII
- 38 2(n, q - 1) => 2(n, q), 0 < q < n. Given An-q-1Bq E K, write B
=
xOq-1•
Then
lJ.AoB= !lAo(xC) = !lAC + j.lAxoC
= X where X, Yare
+ Y, say
n-balls by 2(n, 0), 2(n, q - i),
respectively.
Then X + Y is an n-ball provided that X, Y meet in a common face. Now X n Y
=
oX n Y, because X
=
= (fJ.(oA)C+ fJ-AoC+ AC
=
aoX~ a i Y + fAC) n fJ-AxoC
!lAoC + (fAC n fAxoC)
coY.
=
Therefore X n Y
oX n oY.
There are four cases.
= o.
(i)
fAC
(ii)
fAxC ~ o.
(iii)
fAG
~ 0,
fx
fA.
Therefore fAxoC
= o.
(iv)
fAC
~ 0,
f'x E fO.
Therefore fAxoG
=
Therefore 1-'ACn fAxoC E
In each of the first three cases X n Y (n - 1) ball by 2(n - 1).
=
=
fAoC.
fAC.
fJ-AoG, which is an
In the last case X n Y
=
fJ.AoC+ fAG,
which is the union of two (n - 1) balls meeting in the common face fAoe (because fAG ~ 0), and consequently X n Y is a ball. Therefore X, Y meet in a common face, so that X + Y is an n-ball~ and 2(n, q) is proved. l'T r,e
write A
=
now
h ave t 0
prove
XBn-1, x E K - L.
2 ()n
') G1-vel" An E K - L., => 1 \.n •....
Again there arc four cases.
VII
- 39 (i)
B E K.
(ii)
B
i
K9 fA
(iii.) B I=. K (iv)
9
J O.
fA = 0
B ~ K9 fA
=
9
fB
fB f O.
=
O.
In the first case we prove, by a separate induction on n9 that ~A
=
(xyB),9 where y = fX9 and the dash means take a first
derived complex modulo xB + yB.
For if this is true for
dimensions < n, then ~=~xB ==
a(~xoB + xB + yB), since ~oB = 0
~ a«xyoB)' ==
+ xB + yB), by induction
a(o(xyB))'
~ (xyB)'.
In case (ii) ~B is an n-ball by 1(n - 1). ~B + fA is an n-bal19 because ~B n fA face.
Also ~xoB
+
(~B
+
==
Therefore
fB, which is a common
fA) is an n-ball, because ~xoB is an
n-ball by 2(n7 n-1) and ~xoB n (~B + fA)
==
~oB + fXoB
~ ~oB + fB, by a homeomorphism, ==
o~B + B,
which is (n - 1) ball, because it is the complementary face to B of the n-ball ~B9 by 1(n - 1). We have shown that
~oA + fA
==
~xoB + (~B + fA)
VII
- 40 is an n-ball.
=
But o(l-LoA+ fA)
is an n-sphere9
and
joining
aA.
Therefore
to the point
l-LaA+ A + fA
a gives
~
nn
(n + 1 )-ball. Case
(iii) is simpler
than case
l-LaA= I-LB+ I-LxaB9which
Thererore are n-balls
meeting
is an n-ball
in the common
since fXoB
= (n - i)-ball and ~
Case (iv) is yet simpler
because
fA
=
O.
I-LB9I-LxoB
face
~B n I-LxoB= !-LoB+ fB9
Then !-LoA+ A is an n-sphere9
(ii) because
=
fB9
as above. an (n + i)-ball9
because
as before.
this time
!-LBn p,xoB = !-LoBy
= = The proof
o!-LB+ By since fB (n - i)-ball
as before.
Q9ro:l;lar"y_ to Lemma Lt6..:.,
l-hILu
Proof
~
Collapse
across
A E K - L9 in order of decreasing
hkILuL
of' f:K
L can be obtained
x
= x
x
=
Eroot'
=
09
X
from A9 for each
simplex
dimension.
x x 19 fx
=
x
X
t,2
We construct
cylinder9
cylinder
from K u K x I u L by identif'yin.z X E
K
X
E•.K
x
E
a continuous
cp : K u K x I u L
onto the mapping
L~~.:..
1...o'Dological:j..y_ th_e rela..!Jve mapping
-+
fx
09
and L8mma 46 is complete.
of 1(n)9
J;&l@1a4L
=
K n Lx t E I. map
.-.-.--, ~ u L
that realises
the identifications.
We
VII
- 41 emphasise
that the proor
and not piecewise
(or this lemma only) is topological
linear.
Dorine
~IA
and ror A E K - L construct order or increasing
step~ lot Sn n
•
=
(A x I). n
h :8 t
~or the moment
that this pseudo-isotopy
~lsn has already been constructed
Sn
.
L..
~ (~)
"-"
hi
~
//
the identi~ication~
.
is a homeomorphism.
the pseudo-isotopy
enables
/
/
1jJ
8n/
~, 1jJ
By induction
~
"
where
the identi~ication.
exists.
so as to realise
and so this map can be ractored
We shall show
such that ho = 1~ ht
~ 8
ror 0 ~ t < 1~ and h1 realises
is a homeomorphism
in
as rollows •
that thero is a pseudo-isotopy
Assume
u L to be the inclusion,
x I : A x I ~ ~A by induction
dimension~
For the inductive
~IK
Using Lemma
46
~ to be extended
or A x I onto a collar or~.
By filling
that ~A is a ball~ to a map of a collar
in the complementary
balls~ we obtain a map of Ax
I onto ~A, such that the interiors
are mapped
This is the required
homeomorphically.
map
~IA
x I,
a-
because
un.der the identification
no point
of (A x I)
is identified
wi th any other point. We now construct spanning
the verticos
each vertex
the pseudo-isotopy.
of A n L (possibly
B
Let B be the simplex
= ~).
of B~ and so
fB, = B
=
A n L (~ Aj beeauso A ~
t).
Then fb
=
b for
VII
- 42 We first construct
the pseudo-isotopy ht:B x I ~ B
on B x I by defining
[t, 1]
X
by mapping the segment x x I linea~ly onto
to be given
x x [t, 1], each x E B.
ht keeps B x 1 fixed, ho
=
1,
for 0 ~ t < 1, and h~ is the required
ht is a homeomorphism identification
Therefore
B x I ~ B x 1.
Next we construct
the pseudo-isotopy
on A x 1, as follows. Lift A ~ fA; that is to say choose a simplicial embedding g:fA ~ A such that fg and t E I, define at with vertices
= 1, and B c gfA. Given = (1 - t)a + t(gfa). Let
a vertex a E A, At be the simplex
~at; a E A~, and define tIle simplicial map ht:A x 1 ~ At x 1.
Therefore
ht keeps gfAx
1 fixed, ho
=
1, ht is a
homeomorphism
for 0 ~ t < 1, and hi is the required
A x 1 ~ gfAx
1.
identification
Moreover ht is compatible with pseudo-isotopy
already defined on B x I, because both keep fixed the intersection B x I n A x 1
=
B x 1.
We can show by elementary
geometry,
that
given 8 > 0, there exists 0(8) > 0, such that given 0 ~ s < t < 1 such that t - s < 0, then tIle isotopy from hs to ht of B x I u A x 1 can be extended
to an 8-isotopy of Sn, keeping fixed
of h s (B x I u A xi). Choose strictly monotonic sequences 8. ~ 0 and t. ~ 1
outside any chosen neighbourhood
2
such that t.
2+
1 - t.
l
< 6(8.). l
Choose a sequence of neighbourhoods
V. such that nV. = B x I u A x 1. 2
2
l
Suppose inductively
have chosen the isotopy ht of Sn for 0 ~ t ~ tie
that we
Now extend the
- 43 -
VII
isotopy ht of B x I u A x 1 for ti ~ t ~ t. 1 to an e.-isotopy ~+ ~ of Sn keeping fixed outside h~ (V.). Therefore [ht.l is a lJ. ~ 1
Cauchy sequence
u
of homeomorphisms
limit map hi exists? making
~
of S ? and consequently
[ht; 0 ~ t ~
11
the
a pseudo-isotopy.
Any point of Sn outside B x I u A x 1 has a neighbourhood
outside
Vi? for some i? which is kept fixed for ti ~ t ~ 1. Therefore tho point is not identified with any other point under hi. By construction B x I u A
X
hi makes the required
1? therefore
the pseudo-isotopy,
on Su.
•
(V x I) u (V
the construction
of
of cylinderlike
Let V be a manifold?
=
Therefore
on
and the proof of Lemma 47? are complete.
p~fin~~ion
W
identification
X
and V x I the cylinder
on V.
Let
i)? the walls and base of the cylinder
(perversely we regard V x 0 as the top and V x 1 as the base of the cylinder). We call a triad xx? J~ if there exists a manifold
C
JX+1 of compact spaces QllJpderlike
V and a map ~:V x I ~ J such that
~(V x 0) c X
and ~ maps (V x I)-W homeomorphically V x 0 1" --.I
•.• --
.- .•-
i
w 1.
'-'.,
!
!
I
I
.".'-
V x I
I
~ 1
~
-.--.-)
onto J - Jo•
VII
- 44 Example
1.
Let XA- be a complex, subcone
on the (x
cylinderlikc.
A.1
~ by mapping
to a homeomorphism
for each i.
Then X, Jo c J is
For let V be the disjoint
Define
extending
of X.
i)-skeleton
fAil in 1-1 correspondence
x-simplexes of X.
J x+1 the cone on X, and Jox the
-:Jc have
union
with the x-simplexes
0 isomorphically
x
of a set of fBiJ
onto B., and 1
on Bi, in the proo1~ of
of' Ai x I onto the subcone
alread.y used
thi s example
39 above.
Lemma
of g to the (x - i)-skeleton
Let go be the restriction
let J~ be the subma:pping cylinder cylinderlilw.
For- choose
x-sirn.plexes {A.l
1
of X-C.
Define
homeomOFphically
~ by mapping onto I1.Bi'
to example
Then X, Jo c J is
to be tho disjoint
in 1-1 correspondence
In the special 2 reduces
V
of go'
Bi
of X, and
union
of a set of
\frlththe x-simplexes
fB.}
1
the pair A. x I, A. x 0 1
(by Lemma
1
46) for each i.
case that C is a point,
not in X, example
1.
1emm[L~..lThc_.P1J2.ip.,gJ2mmaJ be a man5.f.9J2:.2.. an4_l£J.
L~tM:
:~..9_J~.,.£..Jx+1
be..£Y.J.i.12§c£like,
o ~2
J£"..iL-J 0 an.£.S1:Lch that ..£( J
f.:L=J
-t
]1.,. homgtopi c
t}L f
-=...J0)
eM. _
Th~2} thqre
exi ~i.s.§... IQ,aJ2,
k..£.9 .. EJng X j,LJo-±:ixed..Jancl.. a sut? .§J?.fLco_J:l. . c;: J".2.
- 45 -
VIr
such th~t o
11(J 7_3.02
.§lf1) S!ll!L31
c Li. c
~ 2x - I!l.. + 2
dim J..J0 ..D.~J1)
3 ~
~ 2x -
m---i"-1
3 0...!..L.31 -.:...30
The meat of the lemfJa is the combination condition J~Jo
of the collapsing
u 31 together with the dimension
dim J~ ~ 2x - m + 2.
In order to achieve this the homotopy
f ~ f1 has to be global~
rather than local liko tho homotopies
of simplicial
approximation
and general position. Before proving the lemma we introduce notation.
The proof
will then follow~ and involve three sublemmas.
Let V be compact.
9X.±',ll.td.r:.ts&
of V x I
if the suo cylinder through each simplex is a sub complex •
Given any triangulation9 Theorem
We call a triangulation
we can find a cylindrical
1, by merely making tho projection
Given a cylindrical
triangulation
subdivision by
~:V x I ~ V simplicial.
we can ohoos0 a cylindrical
derived complex by Lemma 5. In a cylindrical
triangulation
simplex~ call A ~~~Etal call A ~~tical
if ~A = ~A.
if ~IA:A
~
there are two types of ~A is a homeomorphism,
In an arbitrary
triangulation
and of V x I
VIr
- 46 it is possible vcrtica19
to havG simrlexes
but in a cylindrical
thercof9
that are n0ither horizontal
triangulation~
every simplex is either horizontal
Q;z1.jlld..Q£VYJ...~J.§.~j.Ylg
nor
or any subdivision or vertical.
•
Let K be a cylindrical triangulation of V x I. For each a simplex A E ~K, the subcylinder A x I consists of horizontal a-simplexes
and verti~al
(a + 1)-simplexes, alternately. interiors
arranged
Removing
the
of the top horizontal
and the top vertical an elementary Proceeding a co 11a p seA
simplex is
simplicial
collaps~.
in this way we obtain
.
x I '~,; (A x I)
U
(A x -1).
Do this for all simplexes A E ~, dimension,
and
W0
in some order of decreasing
have a simplicial
collapse V x I ~V
Let P be a sub complex of a cylindrical V x I. of P.
Call P sQl.L
x 1.
We
triangulation
of
every simplex beneath
a simplex
P is solid if P = trail P under a cylinderwise
collapse. A subcyJ.indel'is solid. Examnle....,.~ 2. -~ .
The intersection
and union of solids are solid.
If V is a manifold then W is solid.
and W the walls and base of V x 19
- 47 -
VII
The result follows immediately p
=
from Corollary
trail p~ Q = trail Q under a cylind~rwise
1 because
collaps~.
~Nithout loss of' generali ty we can assume J.l compact~ for
otherwiso replace M by a regular neighbourhood z
=
2x - m + 2.
hyDothesis~
of fJ in M.
Let
Let ~:V x I ~ J be the map given by the cylinderlike
and let ~
=
V x I
"
f~. .----.
_CJ2.
.. __ > M
/ ~, J /f ~I
Let Z = ~-1 S~1"·. I
)
Th~n Z has the properties
( 1)
dim Z :;:; z.
(2)
dim [Z n (V x I)" ]
These properties position~
~ z - 1.
follow from three facts: firstly f is in general
implying dim S(f) :;:; z~ and dim (S(f) n (X u Jo))
~I(v x r)
seconilly~ is non-degclierate because and thirdly ~-1(X u Jo)
=
(V x
~ z - 1;
o
is a homeomorphism;
r)·.
If we could now find a subspace Q ~ Z such that dim(Q n W) < dim Q ~ z V x I ---../N u
I~ ---.,
'[if
then the proof would. be finished by defining f1 In particular
the collapses J---;Jo
u J1~JO
=
f and J1
=
~Q.
would follow from
VII
- 48 Lemma 38 because
W ~ S(~).
However in general
VVhatwe have to do is to homotop f to 2:1.
== ~-1S(~1)'
so that
there
~1~
does exist
a
no such Q exists.
and replace
Z by
con"taining
,';:1,
Z1 and with
the above properties. Digr'cssion. =_._~-~,. obstruction
We digress
to finding
intui ti ve idea behind Then
1CZ x
I is
for
a moment to explain
a Q containing the proof.
the subcylinder
Z~ and to describe
Let 1C:V x I through
the the
V be the :9rojection.
-+
Z and wu can collapse
cylini:lervJ'i se
It
is no good }JHtt.Lng Q, ==
high.
f.,nd the trouble
cylinderwise horizontal the is
tc try
j,s that
if
V'/C
this
start
is one dimension
collapsing
stuff
in these
underneath
them.
Bimplexl;s in order
of a map (wh~ch is essentially we alter
Z minus the punch-holes.
x
f
too
I
thG dim...;nsion by one 1 then the
How the only way to "punch holesOi
Roughly spea~ing
1CZ
of Z ~or'm an obst1"'uction to collapsing
(z + 1 )-dLnel1sional
underneath,
I because
X
and reduce
z-simolexes
to punch holes
1CZ
what Z is) to
f:1.
a homotopy froEl f to f1 keeping
to release
(V x I
r
V\'O
to alter
shall
fixed~
X u Jo fixed,
the idea the stuff
in the singular
and Z to Z1 so that
More prcc:isely
homotopy ~rJm
is
'I'hercfore
away
the map.
Z1 equals
describe
which will because
set
a determine
~ I (V x I)
o
is a homeOIflorphism. The way to punch holes
in a simplex AZ E Z is as follows.
VII
- L~9 Since A is a top--climellsiom-11 siL1plex of a singular from where interior
two sheets of ~(V x r) cross one another.
point
sheet containing sheot.
of A, and a neighbourhood A.
Then pipe
The "free encl."moans
dividing
set, it arises
N into a central
1'1"
Choose an
N 01' this point
in the
over the free end of the othel'
~(V x
0).
The
d.isk N' surrounded
pipingll
i1
is done by
by an annu:::'us Nil, and
re:ple.cing ~N by
U
-Plhere~:1.NH is a long pipe running
along a path in the seconcl sheet
to the fr0c eno.9 and
docs not meet the rest or ~(V x I).
had to have
w)
in the interior
The following
show how the riping want.
pictures
illustrate
enables us to perform
Of co~~se the pictures
that we
of M was to make room for
tho pj.pc and the c~rl?over the e-11d. Define V x I.
The reason
~i
:;:
the idea when z :;:1, and the collapse
are inaccurate
tllat we
in that x ~
ill -
3.
VII
- 50 -
!J$FORE fl"ee end
---------t 21
1
path:
\....~l . ,
,'.., N .... , 1-0;-'
-
·1
I
..-;.----. ."
'\ Z
v
I
I
/'
x I
obstructed by Z
.....,;
.. of
,
pt.
... 7.•••••. t.~
I
~.::.'_:
----7
'
".~~~""'''''':.
~,/':
.L:-
~
I/
J.-...-
1--
.----"'/'.'~
f
i --
iY
I
,
\ .1
!-..--
~._
'
'---
._.~/
.
VII
- 51 -
A tecru~ical di~ficulty it is impossible with respect
in the proo~ is that in general
to ~ind a cylindrical
to which
o~ V x I
triangulation
the map ~:V x I ~ M is simplicial.
reason is that we cannot make both ~, ~ simplicial, by Examplo
1 after Theorem
sometimes
want 9 sioplicial,
necessary
to switch back and forth.
1.
A.S
The
as is shown
the pI'oof progresses
we shall
and other times ~, and so it will be
Q.()A..tJ...nuing th.£..J2£.2~f, of Lem~L48. Let K, L trian~~late successive
subdivisions
denote the induced subdivision
of K.
the pair V x I, Z.
K1, K2?
subdivisions
•••
of K, and L1?
of L.
(for a suitGble
L2,
•••
will
First let K1 be a cylindrical
Next let K2 be a subdivision
~:K2 ~ M is simplicial
We shall construct
of K1 such that
triangulation
of
M).
Then
L2 has the properties:
(1) dim L2
~
(2)
The z-simplexes
(3)
9 identifies
identifies
interiors
of Z above.
of L2 are interior
the z-simplexGs
of L2
to K2• in pairs~ and
of those simplexes with no other points.
The properties
Theorem
z.
Property
(1), (2) (3)
follow
(1)? (2)
from the properties
comes from the general position
and
17. Su 121crnm.§LJ..:..
WL®1l.-qp.oos_~~~,JC
.L41
80
thi?t~La .E?.!L"ti sfi_e~§.. t.h.e __:further prQJ2.~:
If A.J s a ll<2!i,&ontaJ._2!,-..EimJ21.ex oLL2~
.
~b._n
q~.~~.L7Sl::..
- 52 Although Ki
Eroof. general satisfy
(4) because
is cylindrical~
two horizontal
lie in the same subcylinder~ under~.
VII does not in
z-simplexes
of Li may
and therefore have the same image
If they do, then we can move one of them, A say,
sideways out of the way as follows. of Ki
Li
Choose a point VA E st(A, K')
modulo the z-skeleton.
such that ~vA
i
7U~~ which is possible bocause dim A
since x ~ m - 3.
=
z < x
=
dim V~
(This is one of the places where codimcnsion
~ 3 is essential.) the linear join
Let K' be a first derived
Since both VA and A lie in a simplex of Kt,
v.r!:"
the interior of
is well defined.
By property
(2) A lies in
and so there is a homeomor:phism
throwing A onto vAA, and keeping the boundary
fixed.
~r
of
·st(A~K')
Extend
1/r
to
a homeomorphism 1/r;V x I ~ V x I
fixed outside st(A, K'). ~vA
i
(z-skeleton
Since 1jr keeps Z-=-A f'i.xed,and since
of ~K')I we have
~~A n ~*(Z - A)
c ~1/rA.
z-simplex A ELi' such
Now choose a point v.1-1.. for each horizontal that the imagGs {~vA} are distinct. E.L'e
Since the stars {st(A, K')}
disjoint ~ we can define the homeor.iOrphism
all the A's simultaneously.
~ = 1/r keeps
~1/r-1:v x I ~ J, for the cylindGrlike
(V x I)" fixed. ~ =
so as to shift
Let
which is a valid alternative because
*
Let
g-1S(f)
=
1/rZ.
hypothesis,
- 53 Let
K =
L=
WKi1
V x I1~.
WL1.
Moreover
Then
R1
VII
t is a triangulation
any subdivision
if we replace
K in our original
will automatically
(4).
satisfy
satisfies
(L~)1
property
of t also satisfies
anc1 consequently ~1
L
by construction
of the pair
construction
(4).
by ~1
This completes
R
Therefore then L2
the proof
of
Subler.llna1. 9,p1.±§jrugj:,i?11_of 0n.~ -oipe. So far we have constructed
v
X
11 Z satisfying
barycentric of K3.
a triangulation
~()roperties (1 J 21 31 4).
first derived
Let K5 be a cylindrical
(3).
Let K3 be the
Let K4 be a cylindrical
of K2•
second derived
Let "'~1A~:: be two z-simplexes property
K21 L2 of
of
subdivision
K4.
of L2 identified
-by 'P1 by
I.• abeJ. thc~n so that one of the following
cases
occurs:
(i)
both vertical
(ii)
both horizontal
(iii)
A horizontal
One of these cases always
and A •.... vertical • occurs because
either ilOrizontal or vortical1 cylindrical
because
have 2UY (z + i)-dimensional get rid of. Lot
Ther0for~
J~be
be tho vertical
of K2 is
K2 is a subdivision
K1 (this was why we bothered
thore is no need to do any piping1
overy simplox
with Ki).
because
neither
stuff underneath
of the
In case (i) A nor A* will
them that we want to
we can assumo A is horizontal.
the barycontro interval
above
of .A1 and let P (p stands for path)
A
joining
A
to
~A x
O.
By
- 54 -
VII
,..
construction
A is a vertex of K3 and P is a subcomplex
of K4~ and
so the second derived neighbourhood DX+i == N(P~ K
5)
is an (x + i)-ball meeting V x 0 in a face~ DX say.
Let
DZ _ A ()DX+i, " in A5 (A which is a z-ball, being the closed star of A
5
subdivision
of A induced by K
).
5
7
I
<
(
p ..._....~ .....
x+i
D
'-....
(.
<
Bx+1 ....·..::;/\
\
< ~
1
•• __
.,.
.• _
being the
- 55 Let
*
DZ' =
which
.B.~l;
-1 cp DZ n cp
(3) abovcs
is a z-ball by property
a subcomplex
o~
KS'
Both P and
because
in K "
disjoint
5
embedcHng,
cp is not in general
~(Z-:~A) x I (4)"
disjoint by property
Their
second derived
=
..1 DX+1 cp cp
=
D
x+1
Let K6 be a subdivision
are
neighbourhoods
TherQ~ore
of KS' and
are
cplDx+1 is an
MS
a triangulation
Consequently
D~ becomes
of a
of K6,
Let K , M7 be barycentric second deriveds of 7 Then cp~K7 -? 1,17 is simplicial because cpis non-degenerate,
Bm :::N(cpDx+1, EX+1 = N(Dx+1, x+1
Eo Notice
o~ K
KS"
z u D~.
M such that cp:K6 -~116 is sim91icial.
K6s M6, Let
DZ•
on
4, and
are subcomplcxes
DX+1 n Z
Thcre~ore
simplicial
and
(5)
subcomplex
but is not in general
=
of balls,
7
K
)
7
z
N(D*, K7)·
that these aI'e three balls,
neighbourhoods
M )
because
they are second derived
Also BX+1 meets V x 0 in a face, BX say
'lJecause J,~x+1 dJ.'d): B~,':':+1 \ v ," lies in the interior of V x I (because D~ did); and Bm lies in the interior of M (because cpDx+1 did), From· ():)) we deduce (6 )
qJ -1 (Bm\)
:::
x+1
B
X 1 u B '" +
cp-1(:8m)= (:SX+1 -
BX)
·x+1
u B ..',',..
VII
- 56 -
By Theorem
6, V
I is homeomorphic
x
to closure
(V
I _ DX+1),
x
and so we can choose an embedding
l1:V x I tha tis
(v
X
T
tho id.cntity _
nX+1).
arc both proper ambient
01)
-7
V x I
tside :z?:+1 i m:.d maDS V x I onto closure
mh ,~e t wo ;[laps
and agree on the boundary.
isotopic
by Theorel~ 9, because
o:::dstsa homeomor'phism k:Bffi
-7
Brn,
Therefore
th0Y are
x ~ m - 3, and so there
keeping
the boundary
fixerl, such
that -nX cp I B x ::: kcph I Bx :.0
Extend k by the identity
-)
m B ..•
to a homoomorDhism
of M.
Defino
cp~ to be
the composition V x I
h
" V x I
__ .-52_._) M
_-E__
> M
-_._--,-"---_._----,-~ CPi
N o t·lce th U't
.~ OU t SlQO
1 c 1 "o.x+ D U B:::. -1-.
* ,
..p '"' an d th 0 on 1 y d l~~eronce X 1 x 1 . Bffi• b e twoen cp, 91 l·~ ~ to ~ltnr ~ tl1.e emb~dd'ng '"' l S OI~ B + , u .LJ;;: + In 9::: 91
Theref'ore 'Pi is homotopic
to cpo
Also 'P1 ::: 'P on (V x It
, because
tho o~ly place where they might not agree is BX, and hore they agree by choice
or. k.
Therefore
We have completed that the neighbourhood £:+1. .'..•
91 ~ cp keeping
the construction
of one pipe.
Notice
N I'eferred to in the digl"'essionabove is
irhe pipe VIas thrmvn up indirectly
rather than by drilling
(V x Ij fixed.
directlJT
by the homeomorphism
k,
along the path cpP (which might be
VII
- 57 -
= ~ -
. x pre tt y rugge d II Q
could then be locally
Recall
knotted
along p).
x
Since homcomorphically
=
onto ~B9
CPi
=
is com.muta tive.
Since l' = fj. 'NO
--1
'Pi
S ( f 1. )
>M
/'
/1~1 JI
/
outside
have f 1. (J - So)
i;B, and f( J - J 0)
o
c M, and
o
c Al as requil'cd.
Let
•
T,et.B -_ BX+1 U BX... ,+1. .... t';B.
into ~B, we can
-1
~
t;
B~ and sinco t'; maps B
'Pit; lli~ambiguously. Ther0fore
'0,
=
'P outside
anQ maps no other points
V x I ..------"
Z:l
.'1
the CO~;ll;lUtati vo diagpam
v
define f1
X 1 .oDx+1 ~ ~,,~n the:. _~ v~mb{~dc~l'J:~~ ~ ~ .~~ cpID + ~
7 ~,because
Therefore
Since h keeps Z - B fixed,
~.Lnen f ,1:1.~ agree
~ 'd e QUlJSl
- 58 -
VII
Now h~-1S(f1
GB) =
I~B), by
h~-1S(k~h~-1
1
= S(k~lhB)?
because
definitions
of ~1' f1,
h~-1 l~B:~B ~ hB is a homeomorphism,
=
S(~lhB), because k is a homeomorphism, z Oz z oz) - (D - D ) u (B* - L* "
= Let
f
°z U D~J °z Z - (D
each i we construct
3
. as rcqu1recl.
(Ai' l\;i)J be the set of pairs of z-simplcxes
of.'ty:pe (ii) or (iil), where
K ,
9
.. , K7 of K2"
i
runs over some indexing
a pipe as abovG, using We now verify
set.
of L2 ]'01'
the same subdivisions
that the constructions
are mutually
disjoint. Firstly
the paths
tP. 1J
are mutually
from UIA.. by pr'operty (L~) above" ·"1
disjoint,
Also these arc subcomplexes
and so their second dcr>ived neighbourhoods
[~D~+11 arc mutually
disjoint
their second derived neighbourhoods Therefore 1
subcomplexcs
(5) above, the
of'M6•
{B~J are mutually 1
we can define h, k so as to construct
B·~J. simultaneously.
As before def'1ne
~1
of K4
fD~+'l J are mutually
o.isjoin t? and_ o.is j oint fl"um UA;;~i" Therc fore, using images
and. disjoint
Therefore
disjoint.
a pipe inside each
= k~h? f:1. =
CP1
r;-1 •
VII
- 59 pro.2f..
Since h is an embedding
suffices by Lemma 38
it
to show that h(V x I) --"'';!h('i,r u T u Z1) --->h7J.
(4) each
By property
path P. is disjoint 1.
froD 0 u T1 and therefore
. d s nx.+1 1 BX the second derived nC1.ghbourhoo .+1 are a 1so d'" ISJoln t f rom 1.
W u T.
h keeps W u T fixed1 because
Therefore
the ~B~+1~.
1.
Therefore
it suffices
h only moves
inside
to show
..I.
h(V x I)~W
u T U hZ1~~.
Now h(V x I) is the sub complex of the cylindrical
triangulation
K5 obtaine d by rer:lovingthe open simpli cial ne ighbourhoods
fp.}.1.
h(V x r) is solid because
Therefore
meet UPi1 neither
sub complex of .ill =
K5
beneath
A1 and let
being the intersection Therefore
VI!
it.
A E L21 let A' denote the
z-simplex
UA", the union for all such
solids.
if a simplex docs not
does any simplex beneath
For each horizontal
f
.Ii.
It
=
ii.' n h(V x I).
Then A'
L.•
or the
Let
is solid, ard
so is L"
of solids, :cmd so is 13being the union of u T u E is solid.
We have
h(V x I) ~ VI! u T u E ~ V x 1 and so h(VxI)--,lWUTUE cylinderwi se by Corollary
We do this by examining according
to whether • 1.
2 to Lemma
each
Ail
Lt5.
Next we have to show
separately.
There are two cases,
A is the first or socond member
is the first member
of a pair.
Now A'
of a pair . is a
- 60 (z + i)-ball,
because
pro:9oI'ty' (2),
and so A' triangulatos
beneath
A is
VII to the prism
~
I by
x
(z + i)-cell
the convox linear
st(A., A') meets
The subball
.i..•
interior
A 'in
the commonface nZ,
and so tho complement AI1 is also
a (z + i)-ball
collapse
across
is
'.'
=
collapses
Thorefore
we can
,\ II J.1. .•.• ,
'.'
Z
froLl D.... or
In this
What is left
case after
all
is ~ u T U hZ1 by Suble~TIa 2. T
1:/ U
D
'~A
o
/
1'l..." •..•.. -
horizonta.l
hZ1--.. W u T by collapsing
U
°z
A -
nZ..•.." ' -.,..
• • ~ ~.~...• ..•-.
z-sim:plo::es A or A~;. in L2•
W u T-"j W cylinderwise.
a;:·
A').
the second member or a pair.
Next colla:psc
all
A')
with face F = lk(A,
h~ and so we can collapso
these
for
st(A,
-
from F.
All
A .. A~
:;: J.l.'
Finally
colla:pse
We have shown
rcquir·ed.
Defino 31 the thr00
To prove
:::
u i;(T-:-W).
8(f1)
We must verify
conditions:
(i)
dim J1
(ii)
dim (Jo
(iii)
J ---.30 u 31 ·~Jo.
(i)
observe
~
z
n 31)
that
~
z - 1
dim S(f1)
~
z and dim T < z.
that
31 satisfiJf'
VII
- 61 To prove; (ii);
n 8(f1»
Jo n J1 = (Jo
u (30
n F;(rr"-
= (30
n S(f»
u (1;W n r;(~f-
= (30
n S(f»
u r;(W n
dim
(30 n S(f»
dim
(W n
T~
w» w»
T- 11).
< z, by general
position
of f.
W) < dim T ~ z.
dim (30 n 31) < z.
Thercforo (iii);
To prove
JQ
U
J
=
i;(V
31
=
F;W
=
r;(W
=
r;'Fv.
30 Therefore
the collapse 37 because
by Lemmn
W ~
3"""" 3 0
S(r;).
I)
'x
u r;(ir- - w) u r;Zl
u
u
T
,J:1. ~
u Z:1.)
30
follows
The proof
from
Sublemma
of the piping.leIT~a
3 is
com:plete.
(c.f. Lemma 39) 1~:t~C_J?.~" .Q•. £lp~s*~g." ..l?-u..b.~.:g.Q...2..~. _'?J>~}:~l:L1.;lch t1~.:t.~iLtIILj.Q. ....D_. ~J_~~ I&..:t)c..J)§_C?..9~.P1LC~C_-:.U1~s ~n t.t~J-.~.,mLZ'"-..CLl!_c*S...~...._3_h..§.n. _th~_§iL.Qr·e, .. coona c t §~9.
G!dJ'd .•L 2~~..2..• s~L(~!tJlill.i X u C c y u C "-CS, Z u C ---=-"".--~~,.~~--,,-------...-.,.--...-----
Notice fol:ows
trivially
....
(X u C) n M
=
from
that
the interiorness
the other
(Y u C) n M
=
of the collapse
result~because
(Z u C) n M
=
C n M.
VII
- 62 The lemma is trivial if max (q, x) > m - 3, because
=
choose X
y
z = max (~9
=
Z.
Therefore
x) + x - m + 2.
assume q, x ~ m - 3.
then
Let
There are two cases according
as to
whether x < q or x ~ q.
Therefore
z = q + x - m + 2.
Vie can assume X = it then trivially the inclusion h:X x I
->
C,
because
if
it follows for X. X c M is homotopic
M keeping
not necessarily
'lye
loss of gener'ality
:prove the resul t for X .: c,
The C-inessentiality
Both f, h are continuous-maps,
linear, but before we make h piecewise
linear we first want to factor it through a mapping Without
loss of generality
of heX x I).
cylinder.
we can assume M to be com}?act.
For if not, re:Dlace M by a compact submanifold neighbourhood
means that
to a map f:X ~ C by a homotopy
;~ n C fixed.
piecewise
Without
(Construct
1\1:;:
M:.;;
containing
by covering
a
heX x I)
with a finite number of balls and taking a regular neighbourhood of their union).
Replace
C by C n
M:;.;'
If the result holds for Nt..•.
then using the same Y, Z it also holds for M, by excision. Therefore
assume that M is com}?act, and consequently
pair M, C is triangulable.
By the relative
simplicial
theorem, we can homotop f:X ~ C to a piecewise X n C fixed. J =
Triangulate
identifications
of the topological
cylinder.
approximation
linear map keeping
X, C so that f is simplicial,
~X u C be the relative mapping
the
and let
Since h realises
relative mapping
the
cylinder, we
VII
- 63 can ~actor h through J by Le~na 479
x
x I
h
"~ where glX
) lVI
/
/g
J/
u C is the identity.
Again using
the relative
make g:J ~ M piecewise
simplicial
linear keeping
approximation
theorem,
X u C fixed.
J
=
Let Y
g(~)9
which
is one of the spaces to be ~ound.
I,e t Y 0 = Av
Then Y ::) X an d d· 1m Y ~ x + 1 • Yo c
Y.
In particular
~he identity, position
glYo
Yo
fixed.
by the hypothesis homotopy
oJ>"
18 we can homotop
U
-Pv) .I..A
Y - Yo c M.
X n I\'I c C.
of g keeping
Therefore
The homotopy
X u C ~ixed, because
extends
it is
gl~X into general
0
There~ore
Then
•
because
At the same time we can ensure
o
g(~X - Yo) c M.
= g (v
is in general position,
and so by Theorem
keeping
U f-;r.J>..
Y n M
=
that
YonM
trivially
~X n C = ~X.
c C9
to a
Since
VII
- 64 dim (~X) ~ x + 1, the general dim
position
S(glp,x)
~ 2(x :>;;
"beeause x < q. y
=
g(~X)
general
IIlay
implies + '1) - m
z - 1,
What we want is dim B(g) intersect
position
:>;;
z - 1, but as yet
C in too high e. dimension,
move is neces8ory. The ..! dim Co :>;; CJ. by hypo the eis.
Let Co = closure o
Since Y - Yo c M, by TheorG~ 15 we can ambient Yo f'ixed,
Qnd so another
l.Ultil Y - Yo is in generD.l posi tion
isotop
Y
keeping
with respect
to Co.
The:::>efore
= ~~ -.. ',.. Tr.:.ei sotop~r of' Y keeping Yo fixed x u C 1'ixed" that
does not alter
determines s(glp'x)
a homotopy of g keeping
hat
does reduce
S(g),
because g ( ~X)
n g (.J - p.X) ::: Y (, (C - f'X) ==
Therefore,
Ii
C
-- (y
f'X - X)
.-
1..-'-
Yo)
-
(y
Yo)
'~T
Ii
Ii
C,
since
J ==
Ii CO?
s~_nce Y
-
~X u (J - p'x), we see tlmt
- S(g Ip.x) u S(g IJ
p.X)
U
clusure
g-1 (g(p.X)
XnCcfX
C
8lnce g is non-degoner8te~
Writing S(g)
(Y - fX)
n g(.:J - p.X»?
0
Yo c M.
VII
- 65 of which
the
second
made of dimension
term
is
empty?
~ z - 1.
and ~he other
two terms
wa have
Therefore
dim B(g) ~ z - 1. Novytriangulate lJY the
J ~T
the
collapse
J'-;>C
to Lemma L~6)?
Corollary
u C by l,emma 45.
of the
and let
Therefore
T
mapping ::c
trail
cylinder B(g).
(given
Then
(T u 0) by Lemma 389 be cause
gJ ~g
T ::) S(g). Let
Z - gT n Y, which
is
the
other
space
that
we had to rind.
'l'hen dim ~ ~ dim r7
m .L
\ ~ 1 + dim Sf\.g},
~
=
Now Y u 0
gJ, [J.nd Z u 0
=
z.
(0"m <.;>-l.
n Y)
== (gT u C)
= =
by Lemma 44
u V,..,
n (y u 0)
g( '1' u 0) n gJ geT
u 0).
Therefore XUCcYUG~ZUC9
which
completes
the
Therefore th(; :piping simplicial, cylindel'
z
lemma. ~nd. let
proof'
=
of' case
2x -
ill
~LX
of f.
'Nitl1out
loss
dim 0 ~ :;;:; for
otherwise
Jet
This
ascume M9
As before ,..1 -
+ 2.
(3)
U
C be the of genernli C(x)
time
we shall
C compact, relative
make f:X
sim:plicial
ty Vie
c~')nnte the
need
C8.n
to use -+
mapping
aSS11Ine
x-skeleton
C
of C.
- 66 -
VII
Then fX c c(x)~ Qnd if we prove the lemma for c(x):
X
c(x)
u
c
Y u c(x)~ z u C(x)~
excision9 because C - C(x) c 1\.1 by the
then it follows for'C by o
hypothesis dim (C n M) ~
x.
Y. ~
Hence dim J ~ x + 1.
Therefore as surne dim C ~ x.
=
Xo be the (x - 1)-skeleton of X9 and 30 cylinder' of
f
IJet
~Xo u C the submapping
I Xo. As before construct from the homotopy h a
piecewise linear m3p g:3 ~ M such that g!X u C is the identity. In particu18r glX u C is in general position9
and 3 ~ X u 30
~
X u C,
and so by 'l'he(,rcif\ 18 Corollary 2 we can homotop g into general position for t~e pair J, X u Jo keeping X u C fixed. o
At the same 0
time we can ensure that g(J - X - C) c M.
Therefore g(J - C) c M,
o
because X - C c H by the hypothesis X n M c C. This tilne let Y = g3. y n M c C beco.use Y
-
C
:::
Then
- gC
g3
c g(tJ
x x X , 30 c 3)(+'1 is cylinderlH:::;, ~ g(3 Therefore
b;)'
the pj.ping 10m.Hie;
Y ~ X,
-
(LCl"fll.1t:t
-
dim Y
,,;;
x + 1, and
0
0) c M. 0
Now the triple x ~ m - 3.
30)
c M and.
Li.8)
we can hom.oto};) g l-::eeping
o
X u 30 fixed and g(3 - 30) c M~ and choose J1
~
S(g) such that
Trinll.gulatethe mapping cylinder collapse J 0 •••.•• '":1 C given by Lemm<: :+6 8or~(.)11nI~Y ~ and let T = trclil (30 n d.im
1":"1
J•
~
"
.
~ z..
dim
3:1.) •
(30 n cJ 1 )
Then
by Lemma
44
VII
- 67 Also
u C by Lemma l-\-5.
Jo~T
Jo n J1 c T.
because
Therefore
Z
dim J1
T9
~
=
z.
Jo
u J1.-;T
u C u 31
Therefore
Lemr:c.a38, g.J ~g
by
Now let dim
Therefore
(T u 31. U C),
g(T 'u J1); then We have
=
gJ
31 ::> S (g)
because
dim Z ~ z becQuse
=
Y
.
both
Y U C9 and geT u 31 u C)
=
Z u C,
and so
x u which
com};'letes
Definition:
the
y u
C c
froof
C"'-J Z u C,
of Lemma 48.
a 16gul:"'lr nei ghbourhood
~dmt9sibl-~
if
the
N~X
collapse
9L7~j·I'Jt~~_.l~e t :t:J2.e
If
adrnissible.
ar&
ElD.,,,;'l9l11issibler~£Lu.h1!'11.eJ1illbourhoQ.d
X·~ Y the
resul
~ is
Ther>e:foro ~ssume Y -2:,.x.
N -2;. X ~Y.
called
jJ1E: fl'_Q.n..t.Ler_,=9f~~\Lj.LL1!.~_~Jf ~.£.N - ~.2
z.:$Y orL$];.J_1Jl.E;:ll. N is .:;,Q....§.£..
~oj:.
is
N of X in IvI is
trivial
There
aro
because
then
two C'3.ses: the
o
absolute
case
when X c E and t:i:1e reL ..tive
case
when X meets
H.
o
In the
al)sclute
therefore
the
CDBC:
bath
re:;sul t follows
The rela ti ve case of the
proof, (i)
to
Lemma
-]4).
N, Y must
leaving
the
A derived
1 [3
also
li8
from Theorem similar
~etails
7
is
.•_
6 CorollaI'Y
q:nd we indica
to the
neighbourhood
in 1:1, nnd so F -
t3
~',
j'T
0
4. the
ste~9s
reader. admissible
(c.f.
Corollary
VII
- 68 (ii) ambient
two admissible
111.lY
isotopic
(c.~.
Theorem
If' N:l..is
(c.f'.
Theorem
adn.issible
that
8 Coroll~ry
Now~ given regular
the
I'egular
then
there
~(x~
0)
=
is
J.JGllliUa 50 is
regular
neighbourhood
a homeomorJ:1hism
.
0,
X E F and
tlle
lemma 9 let
~(F x
.
I) - MnCff-l{),
2).
s:~tuation
in
neighbourhood
c;}i":::"indel'vviseby
neighbourhood
are
8 (2».
of Y in N - F.
..p an a drnisslb'] ..e reg-u'1 8.1' neighbourhood. ~ oj.
Henco N~:N:t
neighbourhoods
8.1'lother admissible
or X in M~ and N:1.c N - F, N:l..such
regular
-v'
J).
In
-.,
IVl
(iii).
TherefOl~e
of Y because
N -S;N1 -~Y.
N1 be an Then
N:l..
oecm.lse
.1.'1
N is
",T'
is
also
---="C(, Y ~. a. v
an admissible
The proof
of
complet.e.
We now prove In :Qarticul[~l~
thlS
Theorem eovers
2-j
in
tbJ:; case
the
8pecial o
1."-:11 or:. X c M.
case
that
X n M c C.
- 69 f~!l~~
Let
VII
C 12.e~a __£1:"colJ.~siQJe
~_:-~~~c;>£~e~9f--,~-,~~n
~~ __1;>8 . ~.Q.L1ill.~ . ~.J..~C_-:igesSel1J;ja~~:~LD.,Ji_c C.
Le,1
o
c = dime C n ~,~).
Lot tl1~ .I._·t
o
(d,
('Iv•••••.••. •
0
r""
.r1-1c> -.~
We consider
dJ'• m1 (0.V_",;, n nI 1.7) -" q • .ll. ""
separately
the
three
The Q-collapsi"bili c > q in
Of course
(1)
cases:
ty
general.
(2)
c ~ q
means
c > q > x
(3) c > q ~ x. (c.f.
x == -
Thl~ ~9roof' is
')Y induction
Assume th0
::>e8u1t true
1.
y~ Z such
choose
the
proof
of Theorem
on x, for
starting
trivially
dimensions
< x.
with
By Lemma L.l-9
o
tlLi.t
X c Y U C-----:lZ U C, Z n 1·,Ic C9 and
z == dim Z ~ maz (q, rrhere:fore
19).
x +
z :r; k by th.e hy:,;othesis
TheI'efoJ'(:3 Z is C-inessential.
x) + x -
~il3.X
(q,
x)
b;y th2
Also z
ill
+ 2.
~ rn + It - 2. hypot:1csis o
X
~
with to
m -
3.
ThereforE
b;! inductjon
d:.m (}~ - C) ~ z + 1 ~ the
A,
Z c E ~C;.
we can enguli'
and
L n
iVl
C n H.
:::
Apply
Lemrria Lj.2
situQtion o
y. u C'~Z
ancl a' nbient .
(y u C) u
H, to
isotop
E
u C c E,
l\:eeping
Z
U
C fixed
so that
0
Since
E';"---'IE;;;.
D.mblent isotopy
preserves
interior'
<)
we h:cive 3:---..,;C.
coll::psibility o
we hnve
D = Y u C u E,.,.~
Thcre:~-'ore putting
-.'
0
D --:I E,.. --.:-''::;
j
2nd
80
)~c
o
ri--~C.
Also dim (D - C) ~ x + 1 because D - c -
(y -\
E.) "j
u (E.
- 70 dim Y ~ x + 19
VII
=
and dim (E.;; - C)
dim E - C ~ x.
Qas_q_l?J.; .. _._CL~L..9..2~"~ o
Let C-~Q be given by q-collaJ?sibili Q so that
Xn C
=
X n Q; for
some triangulation
by Lemma44, and C~~,
fact
it
o
C--~ Q,.
u T by IJemmaL~5.
as Q in the definition
we cell 8u:p:poseX n C = X n Q.
C ~ Q ensures
Q-inessential
also.
Q
u T is as
of q-collapsibility.
Therefore
retraction
(X n C) under
Therefore
becnusa X n C = X n (Q u T).
o
can choose
Then dim T ~ x + 1 ~ q
is better
X u C ---=-X u q by excisioit9
'Ne
T = trail
not let
of' the collapse o
good a candidate
if
ty.
In
Therefore
and X n M c "~' The deformation
that
if
Therefore
X is C-inessential
by case
then X is o
(1) we can engulf
X c E ~Q.
Apply Lemma42 to the situ&tion o
Xu
and e.mbient isotop
:ill to Eo"
'j'
o
(x u C) u E,., --....." E.,.• •'
o
E '~~.
Let F
.J.~
Therefore
C"-,.."X
F :J C
QcE
IJ
keeping
=
X u (~fixed,
X u C u E~. 0
:J
so that o
0
Then F-;.K,.; ~Q9
because
0
':J9 F ~.~
ane. C -'''; Q,. o
If we could deduce th:J.t F --.-) C we should be finished'9 this
cannot be deduced,
Chapter 3.
Therefore
as is
shown by the example at the end of
,"V8 have to get I'm.met
a regulRr
neighboul"hood of F;
but it
attention
to a compact subset
in order
C19 F19
1'.1;1
be a regular
that
neighbourhood
~h denote th0 intersections
the diff'icul
is necessary
shou.ld exist. Let
but
first
tbB r8gular
-_.-~----
ty by talcing to restrict neighbourhood
of F - q in M.
Let
of M1 with Cjl F9 Q respectively.
VII
- 71 o
Then Fi --~
l~hand Cl
in Mi.
C:h in VIi because
of Hi.
meet the frontier of Fl
0
~
Then Cl
Let:n be an admissible
Adding C - Cl C
of N because
also have N an admissible
regular neighbourhood
c F
regular neighbourhood
does not meet the frontier
-r.! ~l'
x
none of the collapses
Fi
of Cl
by Lemma 50.
to both sides, N u C ~C C c N u C.
U
in Ml,
the collapse N u C--"'/C.
Let D
=
by excision.
trail X under
44.
..
N-->Ci•
Now
some triangulation
of
by Lemma 45, and
Then X c D~C
dim (D - C) ~ x + 1 by Lemma
Therefore
By Le~lla
X u C fixed so that D n 1'.1= (X u C) n M
=
C
43
isotop D keeping
n M.
This completes
the proof of case (2). C~ s~j)~~~->
..9.. ~ "_x
o
Let C ~~~
be given by the q-collapsibility.
Triangulate
X u C· .:.~Q, so that X is a subcomplex,
and subdivide
that C ~Q
Let TX be the trail of the
collapses
(x - 1)-skeleton collapses
of C~
of dimension
X u C because
although
of some elementary and therefore
where Xi
Q.
~ x + 1.
T u Q by elementary
It is valid to perform
an x-simplex
it is nevertheless
principal
a free face when X is added.
(C - (T u Q»)
u T.
simplicial these on
of X may occur as the free face in X,
Therefore
We now want to engulf Xl from
Q. Observe
so
o
Then C ~
collapse,
remains
= (X -
simplicially.
if necessary
that dim Xl ~ x because
dim T ~ x.
Also
VII
- 72 -
x~ n
M c (X u C) u M
=
=
C n M
so is X u C, and therefore r'etr'action C ~ Q, ensures
Q n M c Q.
Since X is C-inessential,
so also is X~.
The deformation
tha t X~ is Q-inessential.
case (1) we can engulf X~ c E ~
o
Therefore
by
Apply Lemma 42 to the s1 tuation
Q.
o
X u C -;:I and proceed
as in case (2).
The problem
Q c E1
The proof
of proving
.that the deformation type collapses,
X:1. U
of Lemma
the general
case of Theorem
of X into C may involve
and so the engulf
interior-to-boundary
involve
type expansions.
at the end of the proof
introduce
a device
of Moe Hirsch
some boundary-to-interior
But when we try to expand
stuff out of the way.
situation
21 is
the inverse process
interior-to-bolil1.clarywe hi t an obstruction, room to push other
51 is complete.
because
there is no
We drew attention
of Lemma 42.
Therefore
to this we
to cope with the difficulty.
InwaL.C!&. 9gl:l;.apsing.
~~JEi_tjoA: simplicial
Let K ~ L, J be complexes. collapse
every subcomplex Example
K~L
x
O.
J
if J n trail W
=
J
n W for
W c J. Let K be a cylindrical
Then any cylinderwise
X
is ~vau.fy~
We say that an (ordered)
colla:;?seK ~base
triangulation
of X x I.
X x 1 is away from the top
VII
- 73 Let s denote induced
collapse
simplicial induced
K'~L
collapses
a subset;
K--~L,
o~ t are a subset
set of simplexes, therefore
while
V.
Definition. ..~ --
Let X~Y
to V
by adding
Therefore
t V c J n trails V Therefore t is also.
s is away from J.
by adding
trailt V is obtained
V c trails
trailt
The elementary
of those of s, with the
J n V c J n trail
because
and t the
u trail W given by Lemma 45.
If V c K, trails V is obtained
ordering.
an ordered
the collapse
=
in the manifold
J n V,
M.
We call the
'
collap se l.llwa.r.d~9 and wri te x....:f..;y if, given any tria.ngula ti on of
X~~,
there exists
.......,~.-=-..•..- ~;~-'~ X - Y
'A
definition
-
':{
a subdivision
n Y away from
;~':II.~ .A -
that y is invariant
x '~X
n
and a simplicial Y n
TIt'(
.'.u
under Y
It follows
collapse at once from the
excision:
<=> X u Y ~
Y
;t0c...§mp 1e • Let M be compact, of' the collar. there exists
X a collar
Then X --1;; Y, because
a cylindrical
subdivision
on M, and Y the inside boundary given any triangulation and a cylinderwise
of X
collapse
away from M. L-iLI}ll!l~L~J __(JLi 1's.£hl L~..t,Q..J?..9_~0....9..::.9~9l:tan sibJ-a k..::Sl9t::'L ot:.JvL~.9JJJ2.Il.o_sa y~~ Z.z.. aJ:~~
.c 21f1E.a,c b_.:J.
:J_,j~_.Q. ,9_::::S.;.z...-1L~q..&-,g;....J.l=L_Q.::.i.ll§ s~en tJ1?-l 9 anQ..~-Ll. ..rt_s.....9• L~ t
- 74 -
VII
/
Z\J
M
I
y
The proof is by induction with z
= -
19 for then choose D = Y u C.
true for dimensions Since z ~ Y9
the hypothesis
Therefore
assume the lemma
of Lemma 51
is satisfied for Z9
dim (~ - C) ~ z + 1.
and so we can engulf Z c E~C9 £9
starting trivially
Z9
less than z.
o
ambient isotoD
on
By Theorem 15
keeping C u Z fixed9 lli~tilE - (C u Z) is in
general position with respect to Y. Let W
=
closure (Y n [E - (C u Z)]).
Then
dim W ~ y + (z + 1) - m. Now N c '(Yu C) - -(Z u
cr.
so that W is a subcomplex. subdivision T = trail W9
and a simplicial
Therc:fore triangulate By the definition collapse Y u C~Z
.
then T n M = W n M.
(Y u Cr-:
(Z u C)
of y there exists a u C such that if
VII
- 75 -
~e claim that Y~ T~ E satisfy the hypotheses for Y~ Z~ 0 in the le~na.
Once this claim has been established~ we can appeal
to induction~ because t == dim T ~ 1 + dim W~ by Lemma ~ y +
44
z + 2 - m
< z~ by the hypothesis y ~ m - 3.
Therefore by induction engulf Y c D~E. o
E ~O.
Thererore Y c D~O
Therefore Y c D~
because D~E~O.
0 because
Finally
dim (D - 0) ~ max (y~ z + 1), because dim (D - E) ~ max (y, t + 1), by induction~ and dim (E - 0) ~ z + 1~ by choice of E. There remains to establish the claim about Y, T~ E satisfying the hypotheses. o
First E is q-collapsible because E --~O. k-core becauso 7C.(E~ 0) == O~ all i. J.
is compact.
Next E is a
Next T is compact~ because W
Next Y ~ T because Y ~ W and so~ taking trails under
the collapse Y u 0 -"1 Z
U
0,
Y == trail Y ~ trail W == T. Next Y u E '~T
u E by excision~ because Y u 0 -y"'T u Z u 0 by
Lemma 52, and (Y u 0) u (T u E)
=
Y u E
(Y u 0) n (T u E) == (Y n (T u E)] u 0 ==
T u (Y n E) u 0
=
T u [tV u Z u(YnO)]uO
==
T u Z u O.
VII
- 76 Next T is E-inessential
=
t
because
E is a k-core~
y + z + 2 - m
~ k, by the hypothesis Finally
the dimensional
hypotheses
y + z ~ m + k - 2.
are satisfied because
t for z~ and t < z.
change is to substitute
and
Therefore
the only
the proof
of Lemma 53 is complete.
3e=1a.t:i..y_~__~olJar:s~~ Let M be compact~ V mod C in M as follows.
C c M, V c M.
We construct
a collar on
Let c:M x I ~ M, c(y~ 0) = y~ Y E M be
a collar on M.
Choose a cylindrical
a triangulation
of M such that V, C arc full subcomplexes
simplicial.
triangulation
(This can be d.one as follows:
triple M, V, C~ noxt take a first derived, c simplicial~
then subdivide
extend the subdivision
V - C to
vertices
the
next subd.ivide to malte and finally
Choose 8 > 0 such that M x (0, 8]
Let f:V -~ [0, .s]be the simplicial
by mapping 8.
and c
first triangulate
to make M x I cylindrical,
to M).
contains no vertices. determined
of M x I, and
of V n C to 0 and vertices
map of
Define
Vi
=
V2·-
tc(v, t); V E V~ 0 ~ t ~ fvJ tc(v, fv); v
E
vj.
We call Vi a ~oll.§..r o.n":'[J!l...2.¢l. .Q.J.lL1!, and we call V2 the ip.§J"g.e
bOBn~£Y
of the collar.
VII
- 77 -
c
V
Suppose, ~urther,
.
dim (X n ,Iii) < .,I.'U.
t.')
X~
that we are given XX c M such that
and that we chone the tri~ngulation
o~ M so as
have X a sub complex.
1en~.
o
ii
i.L-V 1. ::;::->:::!_y _ilL
1Yl__ ili!r.~.J. X
('1"..
~11·
0 V 2 )~jC-:..
P~q-.2f.. i)
o
If v E V - C~ then v E A, some si~plex A
.
i
C.
The
fibre v x [0, fv] c int [st(A x 0, M x I)] because the triangulation choice of
8.
is cylindrical,
Thererore
and because
of our
the image c(v x [0, fv])does
VII
- 78 not meet C. V2
n
C = V
=
V2 n M ii)
V:1.
--r....,.
Therefore
n C because
=
1'-10
V
n
V
=
C
V1. ()
n C c
\[2
l{ext Finally
C V1..
V 2 because take a cylindrical
that V1. is a subcomplexs
C.
V n C is full in V.
because
C9
V ()
subdivision
such
nnd collapse cylinderwise
away from V. iii)
Let p:M x I ~ I denote the projection. meets
V1.
-
V
then pA is 1-dimensiona15
is a convex linear cell of dimension
If A a E M and A n
V2
a - 1 separating
one of which is A n V1..
A into two componentss
CQ11crpse across A n V1. from A n V2 for all such simplexes A not in Xs in order of decreasing o
and we have the collapse V1. --, iv)
If
v
.r!. E
.lI.,
then either
.M) V
V u
eX
dimensions
n V1.).
~!!s vvhence
A Ii
2
Ii
< Xs by hypothesis~
C
dim (i~t
Ii
else
n V2 contains an interior points whence
"
.L.~
V' ~ dim eX 2 )
< dim
dim (A n V2)
A
~
x.
Therefore
or
dim eX n V2)
We m.'8 given X c M: to engulf, where ;~ is C-inessential, C is a q-collapsiblc
k-cores and
q~m-3 .i~
:.f;
m -
4
q + x ~ n + k - 2 2x ~ m + k -
3.
< x.
- 79 -
VII
Let Y be a col12r on X mod C, with inside bOill1dary Z.
We now
want to apply Le~~a 53, and so let us check the l~potheses. Y u C
~z
u C by excision,
Lemma 54 (i) and (iv) • furnishes because
a homotopy
because
from Z to X keeping
hypotheses by
.
Iv~?
Finally
43.
Finally
the
y :::Z + 1, z :::x. Therefore
Next
z + 1) ::: x
we can choose D so that D n
ThG proof of Theorem
H~~
because
Lemma 53 we can engulf Y c D~C.
dim (D - C) ~ max (y,
Lemma
X n C :::Z n C fixed, and
by LemlTl1J. 54 ( i ).
are satisfied
XcD~CbecauseXcY.
the collar
by hypothesis .
Next Z n C :::Z n M c
Therefore
Z and Y n C :::Z n C by
Next Z is C-·j.nessenti2.lbecause
X is C-inessential
dimensional
Y ~
First
..Q.. . c ltl..L X £.91~ ct. t
"+1
A
!£1_eIl~r~r~~i si. l!iT
21 Case
CL Q -:-i ne
cU}.•
+
n M by
is complet3 .
9..§_e.~]..i,t~J~.2~ ..a.n.CI._,.§.j1!L.1J;..
x
~~
W .•1L
9 ::;Y__V-_.9%-:S;'~~-.Jd,..,.Q
-k~ti~l_.Q=-~
/S--£"
(2)
.M :::(Y n1 . C)
.. §.§!F~!-...LaJ--1-. §l!-cll. that
.
n_ID_<: ~
VII
- 80 •
---
---
The important part of the le~~a is to get Z n M c C and the v-collapse. Without
P£.99f. compact5 in M5
for otherwise
and perform
loss of gen2rality
we may aS2ume M
replace M by a regular neighbourhood
all the constructions Xo ==
--~~-
eX
therein.
of X
Let
n l~) - C
Co == Xo n C. We assume Xo Triangulate
J ~~
otherwise
the lemma is trivial: X
M such that Xo and M n Care
first derived.
=
subcomplexes
Let Vo be the closed simplicial
=
W == Y
Z.
and take a
neighbourhood
of
Xo - Co; in other words Vo is the union of all closed simplexes of M meeting Xo - Co' points
(Notice that Va may not be a manifold
of CO~ and is thorefore not a regular neighbourhood
general) .
'Ne deduce
in
at
VII
- 81 (1)
Vo·--'" Xo
(2)
Va
n
by Lemma 1 L.j. Corollary.
9
C = C09 because
of the first derived.
Let V~ be a collar on Va mod C in M9 X a subcomplex appli cnble) •
l,et X~ be the subcollar on Xo of the collars.
n X -.: Vj.
V~
(4)
dim (X n V2
(5)
V~ u X u C~X
(6)
V~ ~
=
Fa
on Fo'
U
Xj.
~
X
X2
denote
We claim:
V2
-
1
u C.
F:t n X Vj.
V2 -....,.Xj.
frontier of Va in M9 and let F~ be the
n X
= =
Fo
=
Fo
Therefore by construction
=
Co
n Xo
009 because Va is a neighbourhood
F is a collar mod X as well as mod C.
by IJemms 54 (i).
n X=- V:t
u (X n Vj.)
of
n C.
= = = =
Therefore
X n (frontier of Vj. in M) X
n (F:1. u V2)
Co u (X n V2)
X n V2•
follmvs from Lemma 54 (iv).
V:1.-~Vo
and let V 29
Then Fa
Therefore
)
n
X
9
0
(3)9 let
subcollar
(4)
=
(3)
that we have
(so that LemIJ1s54 is
during the construction
the inside boundaries
To prove
with the proviso
To prove
by Lemma 54 (iii).
(5)
observe that
Next Vo u (X n V~)~X
n V:1.
VII
- 82 by excision
from
Therefore
(1),
Next
Vo,
V2
(6) ~X2
n (X
U
n x\I.J
U
=
~r) n ),.
U Co
that
by
because
(1),
(Vi
V:t--Y.JXi
observe
There f ore
o V ::3.-.....-I X2
Xi
X2•
n V2 =
Let trio.ngulation
and so Vi U X U C ········~x U C
C) = (Vi
the
Also V2 n M = Vo n d
Xo.
X n Vo = Xo.
because Vi
To prove
=
Vo n (X n Vi)
n Vi by composi tion,
Vi---.,lX
by excision,
because
=
Co
=
X2
V2
pair'
V2,
X2
Lemna 54 (i)
by
(2)
again
by
(2)
cylinderwise
by
n M,
by
is
Le~~a
( \1
away from
homeomorphic
Vo•
to
54 (i).
o •
= trail of' the
W
Y Z
Xi
Therefore
The proofs Ti
U
.)
n C) by Lemy)1a54
(Vo
(3,
of
(X n
Vi)'
4,
5, 6)
T2
=:
X:t
u (.ri
U
(X
- Vi)
Xl
U T2
U
(X
- Vi)
X2
U T2
U
(X
- Vi)·
by
are
trail
collapse
composite
= = =
U V2---~X1
(6).
excision
because
complete.
(X n V2) Let
under
some
.-
83 -
VII
•
M M
/' x:t.
.......... ~ Y
·x
c: ". I
I
~
X:a
I
T:a
'~
z
w
We must now show that W, Y, Z satisfy the properties
"
in
the Lenlina. First dim ~N ~ x + 1 and dim Y, Z ~ x because dhl X:t ~ 1 + dim Xo, ~
because
x, by the hypothesis
dim Xs ~ x - 1 , because
X2
Xi is a collar
dim (X
. () M)
on Xo,
< x.
to Xo· is ho:t:i.eomorphic
dLn T1 ~ 1 + dim (X n Vi) , by Lemma 44 ~ 1
dim T2
~
+ x
1 + dim (X n V2)
~ x,
by
(4).
Next Vi u X U C is (X u C)-inessential C-inessential
because
also C-inessential
x
by
X is C-inessential.
because
(5),
and therefore
Therefore
they are subspaces
of
V:t
is
W, Y, Z are U X U C.
Next
c Vi! because
o
Next we have to show W u C-~Y
u C.
First observe
that
VII
- 84 U T2 by Lemm~ 45 Corollary; moreover
X~ U T~~X1
is interior for the following reasons. under the first part of
.
If
this collapse
rri = trail (X ()Vi)
(6), then
Ti n M = (X n Vi) n M, bJi7the propertJ.
T "(,
= X n Va
=
Ti
Also T~ (6),
Xi U V2,
C
because it comes frow the second part of
.
flnd so (T:l.- Ti) n lV1 C (x~ U v2)
Thorefore (X~ u T~) n M Xi
-,r
'/:"'0·
o
U
T~--~ X~
U
T2•
=
Xo
=
=
by Lemma 54 (i) .
.Xo,
(Xi U T2) n M.
We can add C u
=
Co,
C
T2•
lr i n X"~ V;
n M
by
X""":v;..-
Therefore
to both sides because
Leoma 54 (i)
n
X- -
v7
C
V~
=
X n V2 by (3)
o
Therefore iN u C "---; Y u C by exci sion. X2 by Lemma 54 (ii).
Next X~ ~ C u T2
U
X - -v7 to both sides because Xi n C = Co
x~
Ther'ei'ore
We can add
n
X· - V"";
C
X2,
C
Xi
by Lemma 54 (i)
n
V2
=
X2,
by
( 3)
•
Y u C --Y.,. Z u C by oxci si on.
Next X, W, Y all ~eet M in the same set because X n Vi' Xi U T1' Xi U T2 nll meet M in the same set, namely Xo.
Finally
- 85 Z
n
VII
M c 0 because (X2
u T2) n M c V2 n
lVI
by Lemma 54 (i)
~ Vo n C c
C,
and
(X - V1) n M c (X - Xo) n
i
c C, by definition The proof
of Lemma 55 is
of Xo.
complet6. ~,.
9£_ TEE3.2Fem li~Q~~,§~~~
Proof We are X is
given
C-inossenti~l
.
where dim (X n M) < x.
X c M to engulf,
where C is
a q-collnpsible
k-core,
and
3
q, x ~ rn -
q + x, 2x ~ m + k - 2. B;y
Lemma 55 choose
Since
Z is
C-inessential
by Lenuna 53. and ambient
yX, ZX such
vyX+1,
and Z n M c C we can engulf
A}?ply Lerllfl1a42 to the isotope
('IVU C) u E,;,--.E,,:. X c D-~ C because
X c \IV U C -~.}y u C ~:£.":" Z U C.
that
Let D == W u D -~,E·~C. D
-
W U C ~y
(Y u C) u M fixed, (W u C) u
==
E;;;.
u C c E, so that
'l'hen
Also dim (D - C) ~ x + 1, because Ti'., C
~".
dim (D E~> '6"
E;;
o
situation
E to E~;:, keeping
Y c D ~C
-
"1,1
v
-
Y,
E.,. ) ~ dim "
C '" E
-
V{
==
x + 1,
C,
dim (R., - C) == dim (E - C) '" ~
lilax
==
x
(y,
+ 1•
z + 1)
- 86 Finally
VIr
we can choose D such that D n M :: (X u c) n M by Lemma 43. This completes
Theorem
the proof
of our main engulfing
theorem,
21.
We conclude a theorem of En.
this chapter with an application
of John Stallings
More precisely,
which
given
implies
the uniqueness
two piecewise
linear
(::pOlystructures)
on En (there arc obviously
they are piecewise
linearly
pieceWise
linear
trivial,
manifold
homeomorphic, structures,
n :: 2 is classical,
of Moise.
We shall prove
and n
Conjecture
proof
structures
infinitely
provided
and n
I 4.
of structure
many) then
that they are The case n :: 1 is
3 is the Hauptvermutung
Theorem
the case n ~ 5 •
.Q.~_stj.9.p.~~ ~~2.Y_s.E.J The proof below
=
of engulfing,
t
_t£:H~_!2J' n_=
41-
fails for the same reason
that the Poincare
fails.
This is the Hauptverwutung
for manifolds.
The obvious
case
to look at is:
~~2..1:L.i:
~~.jJ1e_. d_'2..\!_ble s.us.]ensi91L.2f_ a PotI}ca~_~...Jillllere ~2.lL()}_9,g.~gall::Y_.Acmlq2llorJ2l1i.q .to_S 5?
By a Poincare is a homology
3-sphere,
. sus~9Emslon 0 f'
1\T3.
m
sphere we mean a closed
3-manifold
but not simply-connected.
.. 8'1.,.-,J[3 1S t he same as t1,. 110 J01n "'lla
•
M3, which
The double This car..not be a
VII
- 87 polyhedral
sphere, because
suspension
ring 81 is M3, not S3. M 9.~n
Call a manifold boundary,
the link of a i-simplex
if it is non-compact
and its structure has a cOlmtable base.
equivnlent complex,
to saying there is a triangulation
in which the links of vertices The key idea of Stallings
Let M be 2-connected.
is the following
We call M 1-connect~d
i8 2-connected. structure
s8quence~
The property
This is
of M by an infinite
definition.
a~ i~fin~t[
if given
Q c M (Q is not necessarily
such that M - Q is i-connected.
by the exact homotopy
without
are (m - i)-spheres.
compact P c M there is a larger compact a subpolyhedron)
on the
This is equivalent,
to saying that the pair (M, M - Q)
is topological~
independent
of any
on M. If m ~ 3, Em is 1-col~~ected at infinity. Whitehead's
gamplCL1...~ after Theorem
19 Corollary
example M3 (given in Example
3) is a contractible
1
open 3-manifold
not
A
i-connected
at infinity.
In fact, if S' is the curve not contained in a ball~ and Q is compact ~ 81, then the fWldamental group of
M - Q is not finitely
generated. The interior
after ~~itGhead's
example)
1-coru~ected at infinity.
example M4 (given
is a contractible open 4-manifold not In fact, if D2 is the spine, and Q is
compact ~ D2~ then the fundamental
~i(i~)as a subgroup.
of Hazur's
group of M - Q must contain
The dimension
4 is not significant
in
VII
- 88 Mazur's
examples,
because
Curtis has given similar
examples
for
~ 5.
dimensions
It is no coincidence illustrate
that we use the same examples
non-engulfability
fact the idea behind connectedness
and non-connectedness
the proof
at infinity
of Stallings
implies
a certain
theorem
to
at infinity;
in
is that
engulfability.
T.hg...ol'-.?m .1~_._~~1lJ11~)
L~j:,MID be~_~9.9ll.i£ftCLti ble _o~n manif.'old, i:_9.Q.nIlecYed __at :m
iD%J11i.tx. If ..E12~ Proof. --.-".' -=-=--
tp.~n M
= ~m
Let P be a compact
of the proof is to show that Pis engulf P directly Therefore
because
of M.
The main step
con tal.ned in a ball.
it is not in general
we have to start indirectly
M l1away from P".
subspace
We cannot
of codimension
by engulfing
~ 3.
a 2-skeleton
of
So choose a tl""iangulation of M by an infinite
complex. By hypothesis,
choose compact
i-connected.
It is important
subpolyhedron
in general
i-connectedness for the moment.
Q ~ P such that M - Q is
to observe
that Q is not a
(in order tl1at the definition
at infinity
be a topological
of
invariant).
Forget P
Let
x
= union of all 2-simplexes
meeting
Y
=
not meeting
union
of all 2-simplexes
Q Q.
VII
- 89 -
In the diagro.n the 2-skeleton
is represb~ted by
2-conn0cted h0li1oto:picin M; keeping assume f is piecewise approximation
X
1'1
Y fix0d~
linear,
ambient
tv a map t':X
-7
thE;crem in M - Q~ keeping
X n Y
linGor manifold).
in ~ 5 dim0nsions~
vYe can
(Since M - Q
By Theorem
X n Y fixed;
that fX - (X n Y) is in general posi tion wi th respect Since both are 2-dimensional
X c M is
simglicial
fixed.
isotop the inagc fX in ii/I •• ;~ keeping
i-skeleton.
M - Q.
oy using the relative
is open in M it is also a piecewise 15
th0 inclusion
P.
so
to X n (:/I - I.;J).
they are disjoint.
VII
- 90 Therefore
fX n X
C is connected
=
X n Y.
because
Let C
Then C c M - Q.
fX u Y.
it is the image ~~der f u 1:X u Y ~ fX u Y
of X u Y, whicll is connected,
being
~1(M, C)
~o(C) = 0 and
Therefore
=
the 2-skeleton
= O.
Therefore
of cOllilectedM. C is a 2-collapsible
i-core. Now X is C-inessential the inclusion ?~tting
X c M is homotopic
k = 1,
q
are satisfied.
=
x
=
to f:X ~ C keeping
Then U :J 11 -
Q
:J C.
h:M ~ M isotopic
such that hU :J X.
Therefore
hP does not meet the 2-skeleton
notice
of Z u Y.
hP is contained
neighbourhood compactness,
X u Y.
a regular
:J X U Y.
Choose
hP in a ball.
19 Corollary
1
it does not
a second derived neighbo~rhood
of M such of X u Y.
second derived Again using
in the second derived
of Z, because
Therefore
our first
th8t since hP is compact
subspac-: Z of the dual
By Theorem
(1)
This mLl1ceshP "effectively"
(m - 3)-skeleton.
hP is contained
neighbourhood
21 part
by Lemma 41
We have achiev0d
in the complementary
of the dual
of some compact
of Theorem
hU :J X U C
that hP does not meet the second derived Therefore
X n Y fixed.
Therefore
of codim 3, and so we c~n now start engulfing
meet a neighbourhood
X n Y, and
to the id~'l"l. ti ty keeping
P off the 2-skelcton.
More precisely,
=
we can engulf X fro~ C.
there is a homeomorphism
objecti ve of pushing
X n C
2, m ~ 5 the hypotheses
Ther8fore
Let U = l':~ - P.
C fixed,
in M because
neighbourhood
(m - 3)-skeleton. Z is compact,
Z is contained
the N
N is now
and so N--;Z.
in a ball.
- 91 Therefore
N is contained
P c h-1B9 because
VII
in a ball,
hP c NeB.
B saY9 by Lemma
We have completed
the proof, which was to show that any compact
37.
Therefore
the main
subspace
step of
is contained
in a ball. We now use this result of balls
{B.] as follows.
Choose
1.
is connected
to cover M by an ascending
the triangulation
a triangulation
is countable,
A. u B. 1. 1
contained
For i > 1, define
to be a regulQr
Then
1.-
neighbourhood
{B.] is an ascending
of its successory
The proof
of Theorem
of a ball containing
sequence
1.
in the interior
UB.1. = UA.1. = M.
Since M
and so order the
.... B.1. inductively
of M.
sequence
of ballsy
each
such that
22 is completed
by:
~e2llin~2h If. Mm .J, s th~_ un:Lon in thEt,~nterior Proof.
Let Em
each in the interior f1:B1
~ ~1'
combinatorial
-_
then --_ Mm ~ Em. ~--
of its --. successor, ......•.••....•
= U~., the 1.
m....a 52
sequence
of m-simplexes9
Choose a homeomorphism
extend fi_1 to fi:Bi ~ ~i by the
theorem
(Theorem
---~ ~ sm-1 I that Bi - Bi-1 ~ ~i - i-1 x. The two corollaries to Theorem Q.Q£ollar;y:1.
..~.~
ascending
of its successor.
and inductively annulus
....
8 Corollary
3) which
says
22 are also due to Stallings.
.. Th _.ep1.ecowJ. so l' lnear _~t ruc t ure. of ~IIm !\
is uni..@e up to h0.11!£2.rn.2£l?hi sm.
- 92 -
VII
m q Le-.i.1V1,Q: be contrScctible _2lli'3n ma:g,ifolds. m q If m + q if: 5 the11.lL~_("l E + • Copolla"rX 2.
=
This result examples
is interesting
above. It suffices
at infinity,
to show that M x Q is 1-connected
and this is done using
if m ~ 2 then
compact
in view of the non-trivial
Mill
algebraic
H~(M)
has one end because
topology.
= Hn_1(M) =
First,
o.
Therefore
if both m, q ~ 2 we can find arbitrarily
large
subspaces
A c M, B c Q such that M - A, Q - B are connected.
Therefore
(M is 1-connected
x Q) -
(A
B)
x
= M
by Van Kampen's
with amalgamation
both
x (Q -
Theorem,
sides are killed
On the other hand if m > q so choose
B to be an arbitrarily
is homotopy
equivalent
again is 1-connected While Theorem
=
u
(M - A)
becQus0
by Van Kampen's
x Q
in the free product
by the amalgamation
1, then Q
=
large interval.
to two copies
discussing
B)
.•
the real line9
and
(M x Q) - (Ax B)
Then
of M sewn along M - B, which Theorem.
Em, we mention
the analogous
result
to
10 for spheres. ~mma
5..L..
!:mL.S?£.ierl1f.rt.:L2.n __pres~£'yjtlg l.l
i§2.t2l?i£.to ._ the ideIl~1;l t,y~ Proof. f to g, where
Given a homeomorphism g keeps a bell Bm fixed,
f, first
ambient
as in the proof
isotop
of Theorem
10.
VII
- 93 Nmv
embed Em - 13 in n simplex
barycentre~.
/J.monto the complement ID
glE
The restriction
h:/J.~ /J. by mapping
pi_ecewise linear except h is isotopic
extends
to a continuous-
"
,...
homeomorphism
B
-
/J.~ /J.,that keeps
" at /J.. By l•. lexanderts
to 1 by an isotopy
of the
/J.fixed and is (Lemma 16)
Theorem
H:/J.x I ~ /J.x I that keeps
!J.and
,...
~ fixed,
and is piecewis0
linear
except
on !J.x I.
Therefore
the
,...
restriction . t opy lSO
of H to !J. -
0f ~m ~
. movlng
Therefore
which deformation
the subgroup
retracts
of Em.
However
because
the Lie group topology
induced
from P.
The higher
linear
of linear
onto the orthogonal
both P, L have two components,
the two orientations
linear ambient
the group of piecewise
of Em, and let L denote
honcomorphisms, group.
a piecewise
1
Let P denote
g,emarlt..:.. homeomorphisms
go.t
/J.deter~ines
corresponding
this is a deceptive
to
remark,
on L is not the same as the topology
homotopy
groups
of P arc not known,
but they ure known to differ from those of L.
1
INSTITUT DES HAUTES ETUDES SCIENTIFIQUES
1963
Seminar
(Revised 1966)
on
Combinatorial
by
E.C. ZEEMAN.
Cha.Qter8
Topology
EMBEDDING AND ~~~OTTING
The University o~ Warwick, Coventry.
INSTITUT DES P~UTES ETUDES
1963
Seminar
Chapter
(Revised
SCIENTIFIQUES
19661
on
Combinatorial
by
E.C. ZEEMAN
Topology
8
In this chapter we wish to classi~y of one manifold
in another.
into equivalence The natural
IIClassi~y" means
relation
In Chapter
o~ algebra9
notion
is a horrible
a nice embedding
the virtue
o~ homotopy
something
is done by
computable, interesting.
Geometrically idea, because
gets all mangled
the during
up.
But
theory is that the homotopy
o~ maps are o~ten ~inite
--- ..
theory.
a homotopy
~requently
Listing
and the way to pass into algebraic
is via homotopy
o~ homotopy
quality
5 we saw that ambient
isotopy was the same as isotopy.
topology
to use is ambient
this has the same geometric
as the embeddings.
means
sort
classes and then list the classes.
equivalence
isotopY9 bec~use
embeddings
or ~initely
generated,
classes
and
and so out o? the mess we get There~ore
our classification
- 2 -
technique will be to map the ambient of embeddings
(geometry)
maps (algebra). the algebra
isotopy classes
into the homotopy
classes of
If this map is an isomorphism
classifies
the geometry;
then
if not then we
have a knot theory to play with. Let M be a closed manifold manifold
without boundary
M ~P2t£
Otherwise
A knotted
isotopic
,\'=---4
We say
M c Q are
homotopi c.
we say M knots in Q.
E~a~~l~_iil
circle.
(open or closed).
in Q if any two embeddings ambient
and Q a
The classical curve is homotopic,
Similarly
kind of knotting
by Corollary
example is S1 knots in but not isotopic,
s3.
to a
Sm knots in Sm+2, m ~ 1, and this
is characteristic
2 of Theorem
9
of codimension
in Chapter
2.
4.
It is the latter example that we want to generalise to arbitrary
manifolds,
and in Corollary
1 below we
giVE: ou.L'f'icicl"J.t condi tions for M to unknot proving
an unknotting
an embedding between
theorem it is natural
in
Q.
While
to prove
theorem in the same context, the relation
the two being explained
as follows.
Let
- 3 -
lso(M
C
Q)
=
[My Q] = [M c Q] =
ambient
isotopy classes of embeddings
homotopy
classes of maps (algebra)
homotopy
classes of embeddings
(geometry)
(hybrid).
There are natural maps A
l-L "._.~---~ [M c Q] injective ----~[M ", Q]. surjective To say that M unknots in Q is the same as saying that l-L
lso(M c Q)
u
The main results of this Chapter are
is bijective. Theorems for A and
23 and 24 below, which give sufficient l-L
to be bijective.
In other words conditions
for there to be a classification lsoOd Remark. ~..
<--="
"='~'
.•. ."
C
Q)
conditions
isomorphism
Q].
~~ [l\!I9
We think of a "knot" maJ2.wtst?,9 as an isotopy
class of embeddings. to think of a "knotll class of subsets.
Other authors,
notably Fox, prefer
s~t\IViseas an isotopy
Clearly
the mapwise
(or homeomorphism)
definition
is
finer than the setwise9 because potentially it gives more knots. Therefore our mapwi se unt:notting theorems are stronger.
However
our preference
rather than a setwise approach
for a mapwise
is dictated by our aim
to classify knots in terms of homotopy. St§ttt?l1L~toJ~-=m_a.in th.e()rELms. (with or without boundary). 11 is compact.
Let
Ir
9
Qq be manifolds
We shall always
We shall state the theorems
suppose that
in relative
- 4 -
~orm;
the absolute
M =~.
Throughout
~orm can be deduced this chapter
=
d
The letter
let
2m - q.
d stands for double-point
this would be the dimension an a~bitrary
dimension
because
o~ the double points were
map ill~ Q put in general
Ern» E? d..c1 iDE. The ore1!l23.
by putting
( Irw in)
position.
Le t .~ :M :-+ Q.. t?JL a" II§.J2~
.s.Ecll~JJ1a ~ 1~1 ts a1l.~e.Q.cli..nJL_oL 1~ in~~~TheJ}J~.J,? h,9I11otopic. to iL pr:9;Qel?_.e.rnl2.e_ciCiing lCT~~tPK. ~lLtx~d7
J2X'ovi-
{m ~ .•£1.-.3
{J M 'T~~_4.::.5?..?~1J:11~e.s,~~ d
1..5? is .1-d+ll:-cC2p.ne ~t.e
.~---~
Remark,
piecewise
As usual we always linear,
to the contrary. Because
to assume
f'IM is a piecewise the existence
Theorem
simplicial
linear
In other words
In the following
23 is an exception.
approximation,
embedding; linear
theorem
we can still deduce embedding
==)piecewise
everything
it is only
map such that
this is the strongest
hypothesis
to be
draw attention
that f is a continuous
of a piecewise
continuous
everything
we explicitly
However
of' relative
necessary
to~.
unless
assume
homotopic
way round:
linear
is piecewise
thesis. linear.
- 5 -
lL~:qE~t.tinKTheorem
k~_tf'?_Jl:~.1 ..:::... Qp_~"~t}'V_q~
24.
W_<m.er SlTJ?!~.£..?JJ1~ __~u~h_ t~?.!,LIiL~~~
~f ~.E.._~1'~
homot()]i c lfeeJ2ing r:1J'ixed--.1;~e1L.the.x. are ambJenJ....J; ~Q.!2.Q..t.£
Q
keeping
f'ixed, provided { !1L__~Jl==-.2
{&1-),S . (g.+1l-_<;Lonn~£~~q.
i 'i is
(dt2.l-co®ecJ~~d.
\,
In the absolute
case the two theorems
can be combined
to give. Cor_ollarY 1.
LE?t_1L"t>~_c;lo~~9--!llLg .~t!,ithouDo~E~qa~l..:..
'rl.ten.M.unli.n.Sll§ .. 2n Q.L.Qnd Jso(M~~...Q.)~
.b1,
QL
provid_~d
r ill...;·~•..JL ..::•.2 " M. is,l d+1J ... _~,OYl.l2-~~cteq
LQ j)L.Lq._T_~.=qOIln ~~c t 82, . The proof's of' the theorems ingredients balls,
of' the last f'our chapters,
covering
and we give
isotopy,
general
some remarks
some problems, dimensional illustrate
namely
position
of' the unknotting
and engulf'ing,
the proof's at the end of' this chapter.
bef'ore we give them, we deduce make
are a mixture
about
some more
restrictions in Theorem
corollaries,
f'urther developments,
and give counterexamples
suggest
to show that the
are the best possible.
25 how the theorems
But
We also
can be used to
- 6 -
classify
certain
of spheres follow
links
of spheres
in solid tori.
First
in spheres,
the corollaries:
irnnediately from the statement
and are obtained corollary
we put Q equal
C9rol,l2-~rx~2.
to Euclidean
they
of the theorems,
M or Q.
by specialising
and knots
In the first
space. m
Any__closedk-co:q'pect~~§.JJLnifold
M_"-.2...
k ~.m -. 3-2_c.?~~. be ._e~m.E=edded~nE2m-~? .§Ed unknots. in one p_igher .diJ!lf?ns ion .~ In particular
any homeomorphism
by an ambient
isotopy
S1 can be embedded
can be realised
of E2m-k+1,
Sphere
in the same way that
in E2, and any homeomorphism
by an ambient
The next corollary to a sphere,
M ~ M can be realised
isotopy
is obtained
and is a higher
of s1
of E3 (but not of E2).
by putting
dimensional
M equal
analogue
of the
Theorem.
9_0.ro,~la~. m ::;;' ...9...,-=.-
If Qq is l2]1:9.±jl.::.cOn!L~2~Y~_.\~_~Fe
:2. _tJ:1\3n~1fx..."el=e~lJlent-9..L7i:m(~can b_El._r_EU?I'es~nt_~~q o
Pl.: ..all.n~::sJ2.l1ere e~b~(~l.c~te_d iA. Q .nI(j:~ is_one connecte~Uhen
?em C~.L£;J..as §lif.ie. s tlle.ar£l."9iELl)...1.i sotC)]?y ..J?lass~.s
m gf, S S:~l2.rovill.ed 0
By Theorem
.IT.!.
.?:..~J..:..
24 the ambient
same as the homotopy no base point
.htgher
trouble.
classes,
isotopy
classes
are tl~
and since m > 1 there
If m ~ 1 then the isotopy
is
classes
-7are classified words
by the free homotopy
the conjugacy
classes
of ~1(Q).
Co£olJ-ary 4.
A special
case
If Qq is m;::",connegte
,:ttJ-ell..EPYJ!!..oel!lbeddin~Sm The next
corollary
c,9llreambi~nt
is obtained
to a disk9 and is in some ways
~._~_.
in other
3 is~
of Corollary
analogue
classes9
is_o~t.2.,]J~
by putting
a higher
M equal
dimensional
of Dehn's
Lemma and the Loop Theorem. . m-1 . m 1 L.eJ1., S_ - S Q and suppo se S LS
---.-
Corolla:t:X 5. .•
iIl(;sse,!,ltJ& i~L,,_~"=Jf. ,g 1-l?~U~1!l~.g •.t1J-~gnnected
9
_wh~r_~
m ~_9." ~ 3, t11&JtSm-1 caI}_Re.s.:illlnnY.9-~.lL.IL~oper1Y. m ~1J1bed.9:ed~9-~~k D c ..;(h..-1.f Q.. is op_~_tJ-igh~~_c0I:J.ll~c~~2:. ttLen_~m(9}
cl.§.s§..ifi.e_s~the.~§:nib,l...e-ILLJ_~c>t9J;lL. cl..§sses of
.§..~£l1~§J~f?Jc_fi2_k.e elL~~
The correspondence and elements
rl fUC~' between
of ~ (Q) m
isotopy
classes
is not natural
but is obtained
by choosing
not exceptional
because
of disks
as in Corollary
29
a base disk9 D~ saY9 and associating with any other disk Dm the difference element of ~m(Q) given by Dm u D~. This time the case m = 1 is
on
we can choose
a fixed base point
Sa. We now make
some remarks
about
the two main theorems.
- 8 -
Remark_l:
Hud~on's
improvements.
John Hudson has improved both theorems by weakening
the hypotheses:
manifolds
to be connected he re~uires
to be connected. k-connected
instead of re~uiring
only the maps
We say that the map f:M ~ Q is
if the pair (F, M) is k-connected
F is the mapping
the
cylinder
of f.
This is e~uivalent
to saying that f induces isomorphisms for i < k, and an epimorphism
where
~(M)
?c. (M) ~'](. l
l
(Q)
~ ?Ck(Q). As before
let
Hudson's
=
d
=
2m - ~
t
=
3m - 2~ = triple-point
improvement
double-point
dimension dimension.
in the Embedding
Theorem
23 is to
replace M d-connected
:
} by
Q (d+1)-connected and in the Unknotting
Theorem
{
f (d+2) -connected ~ by
Q (d+2)-connectedJ In both cases the connectivity Q
because
of homotopy
(t+1) -connected.
24 to replace
M (d+1)-connected1
for
L'lL
(d+1)-connected
{
M (t+3)-connected. of M implies the same
t ~ d - 3 and so f induces isomorphisms
groups in the range concerned.
Hudson's proofs are too long to give here, and so
- 9 -
we content stated.
ourselves
with proving
His main idea is combine
given here with those developed smooth case.
Another basic
Two embeddings is a proper
by Haefliger
cQncoJ:dant if there
F:M x I ~ Q x I that agrees with f there is no requirement
that F should be level-preserving In codimension
than isotopy:
for the
idea is to use concordance.
at the top and g at the bottom;
weaker
as
the techniques
f~ g:M ~ Q are called
embedding
is in isotopy.
the theorems
in between~
2 concordance
for example
as there is strictly
the reef knot
-:::J --y J
is concordant because
to a circle by a locally
it bounds
a locally
flat disk in the 4-ball~ but
the reef blot is not isotopic flat isotopy.
However
to a circle by a locally
in codimension
shown that two embeddings concordant
flat concordance~
are (~~9isotopic~
~
3
Hudson has
-
and so the unknotting
10 -
theorem becomes
of the embedding theorem.
a corollary
However we shall prove the
two separately. Remark_
:So
Co¢iiIIlens ioIl 1.
Our results are essentially ~ 3.
in codimension is fundamentally
The situation
different~
group.
2
occurs and
In codimension
is again different, because
Sn knots in En+1 because
results
in codimension
because knotting
is detected by the fundamental the situation
unknotting
1
of orientation.
two embeddings with opposite
orientation are homotopic but not isotopic, and therefore Iso(Sn c En+1) contains at least two elements. We do not know whether the piecewise unsolved
there are more than two elements because
linear Schonflios
for n ~ 3.
equivalent Co&e_cture:
In fact the Schonflies
is still Conjecture
is
to: Isol~~.!i:'~ta§~_t1lJ2.-eJeme.~tf:>..
Again the situation different,
Conjecture
in codimension
zero is quite
and there are many more unsolved problems.
If M is a closed manifold namely the quotient
then Iso(M
C
M) is a group,
of the group of all homeomorphisms
of VI by the component
of the identity.
It is called the
- 11 -
~J1le.9~topx group of 111"
Only three example s of homeotopy
groups are known. n
Ex_argple_ iJ)
.
of Chapter
4.
determined
by the degree ~ 1.
The isotopy
n
Iso(8 J3lxampJ" e 1) i~. of 81 X
2.
0 Q
n c 8 )
x)
=
n c 8 ] ~ Z2"
Pe is rotation of 82 through
Recently
Browder 1
the same result
for 8
from a theorem
group of a 2-manifold group of ~1(M) modulo
example
3.
of Baer that
is isomorphic
to the
inner automorphisms. unknots
In
in itself9 but
shows that this is not true in general.
~:~m~l~llvl.Browder
3
has shown that 8
the homeotopy
although
known.
He gives a homeomorphismh
in
of 83 x 85 onto itself
that is homotopic
but not isotopic
Choose an element
U E
~3(80(6],
x 85 knots
group of 83 x 85 is not yet
itself9
representative
has proved
n x 8 9 n ~
each of these three cases the manifold the following
group
Th e f" lrs t two f ac t ors correspon d
(69 PeX)9 where
It follows
automorphism
words
2 of 81 and 8 9 and the third factor 1 by the homeomorphism h:8 x 82 ~ 81 x 82
6 about the poles.
the homeotopy
n
[8
0 ther
is
reversals
Z2 is generated
(unpublished)
In
=
Z 2 x Z 2 x Z 2'
to orientation
angle
class of a homeomorphism
Gluck has shown that the homeotopy
lS
given by h(69
is Z2 by The orem 10
The homeot opy group of 8
to 1.
We sketch the proof.
~ Z9 choose a smooth
f E u9 and use f to twist the fibres of the
- 11a -
product
bundle
83
x
~ibre-homeomorphism
85 ~ 83•
i~
F5
degree 1~ then
~3(F5)~ ~8(85)~
h is not topologically is not piecewise non-concordance
to h.
to 1.
(x~ 0)
I~ h were concordant
P1(Th)~
that a E Z classifies
and so if a
o~ Th is non-zero. S
3
X
J
hi
To prove the
=
torus, obtained (hx~ 1) for all
s5
x
s1.
the Pontrjagin
0 then the rational
But the rational
But lot
class
Pontrjagin
Pontrjagin
~ 1 SJ X 8 is zero, and is a topological
so we have a contradiction.
then shows that
to 1 then Th would be
. 11y h omeomorp h"lC t 0 T 1 -- 83 x t opo 1oglca transpires
linear
to 1~ and therefore
the mapping
from S3 x 85 x I by identifying x E S3 x S5.
Browder
isotopic
let Th denote
To place outsclves
choose a piecewise
concordant
linearly
o~
Z24~ and so a is killed
category
hi concordant
to 1~
o~ 85 to itself
~3(80(6)) ~ ~3(F5).
in the piecewise-linear
We claim
o~ 24 then h is ~ibre-homotopic
denot8s the space of maps
by the homomorphism
homeomorphism
is a smooth
h o~ 83 x 85 onto itsel~.
that i~ a is a multiple because
The result
class
class of
invariant~
and
- 12 Examj2le lv). In smooth theory can knot in itsel~. homeotopy
group o~
6
group of 8 preserving by using
For example
S6
In piecewise
by the Poincare
group D
o~ S
linear
spheres
Conjecture9
linear
The orientation
to exotic 7-spheres9
6 to glue two 7-balls
theorY9
on the other hand 9
(at any rate in dimension which we shall prove
~ 5)
in the
chapter.
Remark
k.
Higher
h0l!lQ..ts2J1Y. grollPlL_'J\i OJ c
Our so called classi~ication Q has only touched generally
the sur~ace
Ql.
of embeddings
o~ the problem.
of M in More
we can study the space (M c Q) o~ all embeddings
o~ M in Q9 regarded (RS
•
28
228 corresponds
the homeomorphisms
there are no exotic
the piecewise
is 229 but the smooth homeotopy
is the dihedral
subgroup
together.
next
it is well known that a mani~old
in Chapter
particular 'J\.(Mc Q). l
something
either as a piecewise
2) or as a semi-simplicial
linear space complex.
we can study the higher homotopy
groups
SO far in Theorem 24 we have only said about the zero homotopy
~O(M c Q)
=
group
lso(M c Q).
For example we might generalise
Theorem
9 the unknotting
o~ spheres by: Conjecture.
In
'J\i (£~ c
S~.2 =
09
l2.rovidedi + m :::;; ~.
- 13 -
Remark,..,.j.. An 9PsJ;ruction th..§..ory. In the critical dimension, not sufficiently developed
connected
when the map is just
for unknotting,
Hudson has
an obstruction
for homotopic maps to be isotopic, a quotient of in/the first non-vanishing homology
with the obstruction
group of the map (with certain
coefficients).
would like to develop a more general and fit it into an exact sequence, the terms 1t.(M l
25
to Theorem
C
Q) ~ 1t.(M,
Q],
l
One
obstruction
perhaps
i ~ O.
theory,
including
In Corollary
below we give a non-trivial
example
2
that
looks as though it ought to fit into an exact sequence. We now discuss dimensional
counterexamples
restrictions
to show that the
in the two main theorems are
the best possible.
In each of the six cases we relax
a single hypothesis
by one dimension,
theorem
then becomes
f
EmbCddi~-g ~odimenSion
theorem
false.
2
i
:4 \--------~5T-,-
Unknottin~
Codimension
theorem
----------------
and show that the
2
2~ only
\~
only
~--"-"-l
I d-connected - .----~--!-.---------------
I (d+1)-connected
1----
I M only
I I 1
\-~-
. Q only
d-connected
(d+1)-connected.
..---, .._--_._.._-----------!...- .._--_._-~----_..... _._-~,---!
- 14 -
This is the only one o? the six cases where the counterexample than proved. 4-nani?old
is conjectured
Let D2 be a disk9 and ~+ a contractible
with non-simply ?:D2 ~
connected boundary.
embeds D
onto an essential
in Q4.
Such a map exists because
4 in Q.
By the Conjecture
79
Let
Q4 ·2
be a map such that?
in Chapter
rather
curve
the curve is inessential
6 a?ter Theorem 21
in Example
the curve does not bound a non-singular
disk in Q4 and so ? cannot be homotopic kee~ing
the boundary
the connectivity
?ixed.
to an embedding that n2 and Q4 satis?y
Notice
conditions because
they are both
contractible. p01.11rtereXam..12~e~ 2.. Let M
=
Let m be a power of' 2 and m
pm9 real projective
space9 and let Q
Then d = 1 and P just f'ails to be i-connected. Q is 2-coruLected and the codimension caru10t be embedded homotopic
is
;<:
=
;<:
4.
E2m-1. Meanwhile
3. Since pm
. 2m-1 m 2m-1 ln E , no map P ~ E can be
to an embedding.
9o~p~~~exa~~e"~.
(Irwin)
Let m
;<:
3, and let ?:Sm ~ S2m
be a map with exactly one double point9 where the two sheets o~ fSm cross transversally. We shall show that such a map exists in a moment.
I?
ill
were allowed to
- 15 -
equal 1 then the figure 8 would give a correct picture. Let Q2m be a regular neighbourhood
of f'Sm in S2m.
claim that f':Sm ~ Q2rn cannot be homotopic
=
Notice that d
a and Sm is a-connected,
to be i-connected.
In f'act ~1(Q)
=
We
to an embedding. but Q just f'ails
Z, generated by a
loop starting from the double point along one sheet and back along the other.
Notice also that the codimension
is ~ 3.
m m Wrl·t e Sm = D a u Sm-1 x I U D l'
J?ro9.:L.Jhat f exists:
m
m
Embed the two disks transversally as DO x 0 and 0 x D1 in a m m 2m 2m little ball DO x D1 c S . Let B be the complementary ball, and extend the embedding
± ~
disks Sm-1 x
of the boundaries
i32m to a map Sm-1
x
I ~ B2m
of the
Now use
Theorem 23 to homotop this map into a proper embedding, keeping
the boundary
f'ixed. The result gives what we want.
FRoof _tha~ fi~mbedding. f'was homotopic
Suppose on the contrary that
to an embedding
g:Sffi~ Q2m.
Let p2m
denote the universal
cover of Q, which consists of a m b e d together coun tab Ie n~~ber 0f· coples OL~ Sm x D , p 1urn
in sequence.
We can lif't f',g to a countable nmnber of ,.,ffi
maps fi, gi:o embedding, once.
By construction each f'i is an and, for each i,f.Sm cuts f'. ism transversally ~
Meanwhile
P, i E Z.
1
1+
g.Sffiis disjoint from g. ism because 1
1+
g
- 16 -
was an embedding.
Here we have a contradtction,
because
the intersection of the f's is homological, and must algebraically be/the same as the g's because f. !:>! g., each i. 1
More precisely,
1
of H (8m)
if ~ is a generator
III
and '
is Poincare Duality (where Hc stands for compact cohomology) then in H2m(P, p) we have the contradiction c
o =
u
Dg.~ 1
Counfer.§xample
Dg.1+1~
.l±..
=
Df.~ 1 u Df.1+ 1~
f o.
8m knots_ i1l.sm~2..:.-lIL ;:::. 1. 1
m-1
.
8 x S knots 1n 8 Q_o_ujltel:exlLmJ2.Le-2,. (Hudson) Notice that d = 0 and 81 x Sm-1 just fails to be ...."
~~~~
2m
m ;:::
.•.•••.:.o.__ ~'~
3.
(d+1)-connected. Given an embedding we shall define a knotting that it is an invariant
knotting
numbers
Let T
=
~.n()t
isotopy
two embeddings
class of
fO' f1 with
0, 1 respectively.
f(S1 x Sm-1), the embedded
a E 81, let Sa ;[..emmp 5 8..
number kef) E Z2' and prove
of the ambient
f. We shall then describe
f:81xSm-1
=
torus.
Given
f(a x Sm-1), an etlbedded (m-1)-sphere. 11 A·ln 82m• spanning 8a ~here_,js.an m":;:pa
1fl~:tin€.£J:~at1l.
-+
82m
- 17 -
proqf.
Since m ~
3,
Sa is unknotted
can be spanned by an m-ball, ambient
isotop D, keeping
D say.
D =
S
in s2m, and so
By Theorem
15
o
fixed, until D is in
a
o
general position
with respect
T in a finite number
of points,
one by one as follows. Choose YES,
a
to T.
D meets
which we can remove
Let x be one of these points.
choose an arc acT
arc ~ in D joining xy.
Therefore
joining xy, and an
We can choose the arcs so as o
to avoid the other points S
a
only in y.
of T n D and so as to meet
Let ~2 be a 2-disk
in S2m spanning
a u ~, and not meeting T U D again (this is possible
- 18 -
by general position
3).
since m ~
Triangulate 2m
and take a second derived neighbourhood S2m, which
is a ball
B
of
since ~2 is collapsible.
everything
A2 In . Consider
the set
2m
x= Now X consists 2m
common face B intersection
n (T
B
of three m-balls
n
a
at the interior
consists
together
we have removed moved back.
n (T
2m
in B
point x.
in B
•
with one self
U D)
()m-2
2m
X
-sphere
- 2m
B
faces glued
n
Sa'
the torus meanwhile,
point x;
but We have
and so we must now move
U
71)'
are the m-balls
Y
=
2m
B
n T, B2m n D.
and the (m-i)-ball
-2m
B
n D.
Since~,
with the same boundary,
Corollary fixed.
Similarly
YT U YD
where Y , Y are the cones on the (m-i)-sphere T D
B2m
it
We can write
where 7~, ~
2m
Let
If we replace X by Y, behold
the intersection
X = ~
in B
along the
Let
of the three complementary
along the common
Y be a cone on
glued together
S , and embedded
X = B2m which
D).
U
1 we can ambient
B2m
n T
YT are two m-balls
and since m ~ 3, by Theorem
isotop YT onto XT keeping
This moves the torus back
into position.
9
- 19 -
Meanwhile
the isotony
replacing
D by
carries Y
D to Y' D say.
Then
n
(D - XD)
U Y
has the e~fect o~ reducing
the intersections
of
o
T n D by one.
After a finite number
A as required.
This completes
and we return
to the construction
o~ steps we obtain
the proof
of Lemma 589
of the knotting
nunmer
k(f). Choose
1 three points a9 b~ c E S .
choose three m-balls respectivelY9 choose
number
and not meeting
the balls
another~
A9 B9 C spanning
in general
By the lemma
Sa~ Sb~ Sc
the torus again.
position
relative
and so each pair cuts transverBally
of points.
Let AB denote
of A and B9 modulo
2.
=
k(f)
the number
We can to one
in a ~inite of intersections
Define AB + BC + CA.
We have to sho~ that k is independent
of the choices
made.
First we show k is independent
of A.
denote
the interval
denote
the immersed m-sphere SBC
=
of S1 not containing
B U f([bC]
Let [bc]
a9 and let SBC
x Sm-1) U C.
' I l'lnk'lng number mod 2 o~ T hen the h omo I oglea ~ Sm-1 and a
m , S2m, , b SBC 1n 1S glven y L(Sa~ SBC)
=
AB + AC9
- 20 -
which
is independent
because
of A.
isotopy of S
a
L(Sa' SBC).
B, C, b, c. an ambient
carries with it the whole
different
k is independent
because
numbers.
of an embedded
A, B, C disjoint
Construct follows.
m s1 x D •
fO' f
1
with
one given by the
Then we can draw
as in the picture below.
the embedding
isotopy
the embedding
fo:S1 x Sm-1 ~ S2m to be the obvious boundary
Clearly k is
of A, B, C.
embeddings
Define
of
any ambient
construction
we have to produce
knotting
[be]), then the
k is well-defined.
isotopy invariant,
Finally
of a1
does not alter the linking
Similarly
Therefore
Therefore
Also k is independent
if we move a (without meeting
resulting number
x Sm-1) •
A does not meet f ([] bc
because
1
f ~S 1
Therefore
x Sm-1 ~ S2m as
- 21 -
- 22 -
First
link two (m-i)-spheres
with linking number
1.
into each hemisphere.
in the equator
Then collar each o~ these Finally
the two collars by a cylinder
connect
only moment
the tops o~
I x Sm-1 in the tropic
o~ Cancerp and connect the bottoms by similar cylinder
S2m-1
in the tropic
o~ the two collars o~ Capricorn.
o~ doubt OCGurs as to whether
The
two linking
spheres can be coru~ected by a cylinder, but this doubt is resolved by glancing under
the identi~ication
at the image o~ the diagonal map
u Sm-i x Sm-1 x I ~ Sm-1*Qm-1
To compute k(~1)p in the equator, bottom
x I
_ _ S2m-1
•
choose Sa to be one o~ the (m-1)-spheres
and choose Sb' Sc to be the top and
of the other collar.
Form B by joining Sb to
the north pole,
and C by joining Sc to the south pole.
Then L(Sa' SBC)
=
S
1, because
we can compute it by spanning
a with an m-ball in the equator, that meets the other
(m-1)-sphere Meanwhile
Be
and hence also SBC' in exactly
=
0 because
k(~1) This completes gemapk
=
B, C are disjoint.
There~ore
L(Sa' SBC) + BC = 1.
the proof of Counterexample
o~ knott~d
5.
tor~.
Hudson has shown that i~ the codimension then his knotting
one point.
number
described
is even,
above is in fact
- 23 sufficient
to classify the knots, but if the codimension
is odd then an anologous Z is required.
knotting number
in the integers
More generally he has proved that
Iso(SP x Sm-p c S2m-p+1)
= {_Z2'
codimension
Z, codimension provided m ~ 3 and 1 ~ P < m - p. dimension
for knotting
higher dimensions, The case p
=
odd
This is the critical
tori, because
by Corollary
even
they unknot in all
1.
0 turns out to be exceptional.
Here
the torus Sa x 8m consists of two spheres, and the ambient isotopy classes of links of spheres in the critical dimension
are classified by their linking number, as
we shall show below:
=
Iso(80 x Sm c 82m+1) Z, m ~ 2. m . 1 2m Counter_~xample 6. ~lS.not~J1l __8 _x 8 .. JLm ~ 2. Notice that d = - 1 and 81 x 82m just fails to be (d+2)-connected. froo£' qf the kn2tting. We shall give two embeddings m 8 C 81 x 82m such that one bounds a disk and the other doesn't.
Therefore
they cannot be ambient isotopic,
but they must be homotopic because homotopic.
any two maps are
It is trivial to choose the first one.
For the other choose the embedding after Theorem
19 in Chapter 7.
described
in Example 3
It consists of two little
- 24 -
linked m-spheres
1
connected by a pipe round the S •
We showed in Chapter 7 that this embedding
cannot
bound a disk. RemaI"k. -~-~
We shall shortly furnish a large class of
alternative
counterexamples
by giving conditions
, the SOIl'd torus Sr x E q-r . Sm to knot ln are given in terms of homotopy Corollary
for
The conditions
groups of spheres, in
2 to Theorem 25 below.
The simplest example
is that S4 knots in S3 x E4. We shall show there are an infinite number of knots, although
~4(83 ) =
Z2.
Notice that here d = 1 and 83 x E4 just fails to be (d+2)-connected.
- 25 This completes
the six counterexamples
designed to show the dimensional
restrictions
that were in
23 and 24 were the best possible. next We want/to classify the links of two disjoint
Theorems
spheres Smy sP in a larger sphere sqy up to ambient isotoDY.
The classical
situation of two curves linking
in 83 is somewhat deceptive because
knotting
with linkingy but it does illustrate
is confused
the three types
of linking that can occur.
L1J .
Homolog~callJJ}kin~.
Each curve is non-homologous of the other.
to zero in the complement
By duality this is a symmetrical
situation.
~
/
Here Ay Bare
homoligically
in the complement because,
of B.
unlil~edy
but A is essential
This situation
can be unsy~netricy
as we have drawn them, B is inessential
in the
- 26 -
complement
.01..
of A .
(}eC?Eletris~ .ljpking.
Two curves are geometrically ambient
isotoped
geometrically unlinked.
unlinked
with opposite
linked
if they can be
hemispheres.
We illustrate
curves that are homotopically
- 27 -
Summarising we have: (homologically\ \
)<
linked
In higher dimensions
(hOmotoPicallY) -
=)
-
linked
)1~eometricallY)
/ ~.\
linked. ~ 3~ so
we shall stick to codimension
as to separate knotting and linking and be able to concentrate on the latter.
Therefore
we shall assume
so that each of Sm~ sP is unknotted
in sq.
There are three cases. Then Sm~ SPare unlinked
by Corollary
1 to Theorem
geometrically
24.
In this case homological
linking
can occur~ and this is the only case in which it can occur. We shall show in Corollary
1 to Theorem
classified by the linking number9 more precise
25 that the link is
which is an integer.
To be
there are two linking numbers which differ only
by the sign (_)mp+1.
However we shall not bother
the 'homology linking nllinbers~because of the more general homotopy
linking numbers9
case.
2m + p ~ 2q -
linking can occur in
We shall define the homotopy
and show in Theorem
the link~ provided
they are special cases
linking numbers. Homotopy
both this and the previous
to define
4.
25 that one of them classifies
- 28 -
~
hom9topy
0t
linking numbers
Since each sphere is unknotted
_c.~~
S~Sp
we have
sP ~ Sq-p-1 x EP+1 Sm ~ sq-m-1 x Em+1• We assume that all three spheres Sm, sP, sq are oriented, are induced on Sq-p-1, sq-m-1 (we shall
and so orientations
examine these induced orientations Therefore
the linl: determines 0, E
f3 Notice
E
more carefully
in a moment).
homot~o£:¥_JiHkin~".nUIIlbers
1C (sq-p-i) m 1C (sq-m-1). p
that both these are in the (m+p-q+1)-stem,
and we
shall show in Lemma 62 that they have a common stable suspension (to within sign). homotopy
groups.
We call
f3 st~qle if they lie in stable
0"
Recall that 1C.(Sj) is stable if i ~ 2j - 2. 1
Therefore 0,
is stable if m + 2p ~ 2q
f3 is stable
• .t:>
1-1.
2m + p ~ 2q
4 4.
Since m ~ p, we can have a. unstable while f3 is stable, and this will be a particularly
S3, S4 c
interesting
situation;
for example
Let L; denote the suspension homomorphism,
and
L;p-m the composite suspension 1C (sQ-P-1) -+1C 1 (sq-p) -+ .•. -+7\.(sq-m-i) . m m+ p When there is no confusion we shall abbreviate L;p-m to L;.
- 29 -
:t:f13_ iS2table~~en
a,
classitt~JL~ille=link.__ l_1f_~~t]1~_:r~?rd~
:!..:.:~
m
J so (8 • 8! c 8 ~ ~ ~ 2:1Cm.l9...~ A:L~()~.ii_:::-L:-lm+p+q+prn~~ Before we prove of examples
the theorem we deduce a corollar.y and a couple
and prove
If
Q£r9].la~J...:.
two lemmas.
=_
m .+ p
9 -:. 1..2.. the..n
ISOC§..m..J_8P~~sLqJ .. ~ 7Cm(8~1_~ Hm{8m)~~_~, and the )~0!10logi$a.)]..inldnK.number .9lClS~ ifi~ Two 50-spheres dimensions9
can be linked
but in 97~ 96 they become
linked again in 95~ 941
'0'
???
'00'
the linl~" in 101~ 100~ 999 98
unlinked9 52.
and then can be
The explanation
is
that the words link/unlink
~ nonzero/zero
of certain
stable homotopy
groups9
correspond
to the vanishing
and the unlinking
is 97~ 96
of the stable 49 5 stems, There are exactly two links of 89, 810
One .is geometrically
unlinked,
linked as in the second diagram
J 0,
i3
=
a, E
7C
816,
and the other is half-homotopically above 9 because
(85) ~ Z20 9 6 09 13 E 7C10(8 ) = 0,
a,
C
- 30 -
J!.2mm~_59~ •.
1~Sm
>
i..?_~unkn-?tt.ed in ~Cl,-!l1.~nan oriEglt_alio=n
S.:
l?£~s2FBnRJ"l.£,mE;0I!L-srJ2..h,J.§El_p! k~e12:i-F$~~~lJJSeriis tS_9.to12 ..tS. !-o_the iq,e,nJi ty keeping. Sr: f_i_xe_cl. proo(
by induction
the induction
on Cl, keeping
beginning
trivially
the codimension
with m = - 1.
h:SCl ~ SClbe the given homeomorphism. K1 L of SCl1 Sm and a vertex
x E L.
Choose Choose
fixed,
Let
triangulation
subdivisions
such
that h:K1 ~ K2 is simplicial. Let BCl9 Bm be the closed stars of x in K19 L1• Then h maps BCJ.9Bm linearly into st(X9 K) 9 st(x. T.) and so by pseudo we can ambient kBCl
radial projection
isotop h to k? keeping
BCJ.. Now Bm is unknotted
=
induction
Sm fixed,
preserving,
we can isotop klBCJ.to the identity
By Alexander's the isotopy
trick
89 Chapter
3)
such that
in BCl, since Sm is locally
and klBCl is orientation
unknotted9
(see Lemma
(c.f. the proof
to each of BCJ.1SCl -
and so by keeping
Bm fixed.
of Lemma 16) we can extend
BCl keeping
Sm fixed9
and so
isotop k to the identity. For the next lemma we want to compare with knots S
Cl
=
rl-p-1
S~ ~
right-hand denote
in solid tori. 1"\
*S~.
Let g:S
P
links in spheres
Write rl
~ S~ denote
the embedding
onto the
end of the join, and let e:SCl-p-1 x El?+1 ~ SCl
the embedding
onto the complement.
f:Sffi~ SCl-p-1 x El?+1 determines
a link
Then any embedding
- 31 -
~P-in
o~~~~:~~":*~
Let ~ denote ambient
Proof.=~~.~
...
~:'c_s*c:J~
E~~~)_ : __Isol~9
x
If f ~ f', then we
isotopic.
-=-
can choose the ambient Theorem
12 in Chapter
Therefore
the ambient
keeping
sP sP
support 9 by
59 and it can be extended
to
sq.
rp(f) ~ rp(f'). Therefore
rp induces a map9 rp saY90fisotopy classes for, given rp(f) ~ cp(f'), then the end of
rp is injective~
keeping
isotopy to have compact
isotopy gives an orientation
preserving
fixed9 which by Lemma 59 is isotopic fixed.
The restriction
of sP gives f ~ f'. lilLk ~9 ambient
homeomorphism
to the identity
of this to the complement
Finally ~ is surjective~
for given a
of sP onto g (using
isotop the embedding
p :::;; q - 3), and hence k ~ (ef 9 g) for some f.
f£20f' ():f 'l'he_orem?5 ... Sm unknots in S q-p-1 x E p+1 because d + 2
=
(2m - q) + 2
~ q - p - 2 by the stability Therefore
Iso (Sm 9 Sp
C
sq)
~
Iso (m S
c
So-u-1 D+1) by Lemma 6a ~ ~ x E~
~ ~m (sq-p-1) by There remains
to show that ~
~ust be more explicit
=
of ~.
Corollary
2
to Theorem ~
(_)m+p+q+mPZa9 but first we
about our orientation
Suppose we are given orientations
conventions.
of Sm, SP,
sq.
We
- 32 -
define the induced orientation
on Sq-p-1 in
n-"O-1 "0+1 as follows. Sq - Sp '" S~ ~ x E~
=
choose a local coordinate linear subspace. orientation
At a point of sP
system in which sP appears as a
Choose axes 11
of sq so that 1,
."1
q to give the
P gives the orientation
".1
"0
of S~.
Then p +11 ..•, q determine an orientation in a transverse disk nq-p, and hence induce the require orientation
on
nq-p =
Sq-p-1 c sq-p-1 x EP+1•
topological
invariant definition,
To see that this is a observe that it can be
expressed homologically:
if x E H (SQ)1 Y ~ HP(SP) are the q 1 P then oy E H + (Sq1 SP) and the cap product
given orientations,
x n 6y E Hq-p-1(sq._ gives the induced orientation.
SP)
We use the given orientation
on Sm and the induced orientation linking nillfiber a E ~ (sq-p-1). m
on sq-p-1 to define the
Similarly
define ~.
Sus~e~~eqltnk?
Suppose that we are given a link Sm, sP c Sq1 with linking numbers a1~. Let 2Sm1 sP c 2Sq denote the link formed by suspending
Sm and SQ1 while keeping sP the same. Orient the suspension 2Sq by choosing axes 0, 11 •••, q at a
point of sq so that 0 points
towards the north pole and
1, •.. , q gives the orientation
of sq.
Let a*, ~* denote the
linking numbers of the suspended link. LeIl)ma~2.h P~.oo!~.
First look at
~*.
At a point of Sm we can choose
- 33 -
axe s so that 0, 1, •.• , or~ents ~sq.
••. , q
Therefore m+1, •.. , q orients the same transverse
disk as in the unsuspended Therefore
link.
Meanwhile
sP is unchanged.
~ is unchanged.
Now look at a~. At a point of sP we can choose axes , q orients zsq.
so that 1, ... , P orients sP and 0,1,
...
By the prescribed
the axes so that
rule we must reorder
1, ••• , p come first.
(-)p
Therefore this introduces
into the orientation
0, P + 1, •.• , q. the suspension link.
determined
induced on the transverse
For the transverse
of the transverse
In the unsuspended by homotoping
a factor of disk by
disk we can choose ~Dq-P,
disk Dq-P in the unsuspended
link the class a E ~m ( S Q-p-1)
is
the embedding. Sm c sq - sP into a map
f·.Sm ~ D"q-P say.
In th e suspen d e d 1"Ink we can h omo t op ZSm c zsq - sP into the suspension ~f:ZSm -+ znq-p, which -r
determines
the class ~a E ~m+1(sq-P).
Adding in the factor
(-)p we have
The linlc consists
Proof. and g:S
m
-+
of disjoint
embeddings
SCl, where a is the class of f:Sm m 13 is the class of g:S
-+
sq
gSm
-+
sq
fSm.
f:Sm
-+
sq
- 34 "1 .t e SCl = Srn*SCl-rn-1 i'rl , and 1·et J:S m the left hand end of the join. and assume
denotes
f'rom now on that g = j. sq _ Sm Em+1 x sq-m-1. then ef:Sm
projection,
Sm x SCl-m-1 denote of the join. ef.
Ambient
=
complement
S Clbe the inclusion
-?
-?
-?
isotop g onto j,
Then f'Sm lies in the If e:Em+1 x sq-m-1 -? sq-m-1
SCl-m-1 represents
the torus half-way
Let r(ef):Sm
between
keeping
Therefore
gSm fixed.
we can ambient
ConseCluently assume
We have reached
a situation
Let
the graph
of
a, and since ~ is stable
a classif'ies the link, by the part of Theorem already.
a.
the two ends
Sm x SCl-m-1 denote
Then both f and reef) represent
proved
of'
25 that we have
isotop f onto reef)
f = reef) from now on.
where both
spheres are Gn~
embedded
in the complement
of the right hand!
More precisely f:Sm -? Sm x Eq-m is given by fx g:Sm
-?
Sm x ECl-m is given by gx
= =
Scl_ SCl-m-1 = Sm x ECl-m.
(x, efx) (x, 0).
Let T be the antipodal map of SCl-m-1. There is a homeomorphism h of' Sm x Eq-m, isotopic to the identity such that hf
=
g j
hg = (1 x T)f.
If we were
content
_ -fI
" fx
to have a topological -
____ -: •
homeomorphism, describe: X
x
a-ill
then h would be easy to
for each x
E
E ....by the vector
Sm merely - fi.
translate
However
such
I/'\ -::::+=-
..•..
\
.
(C'.
I
---L
\<-L~
i
_
- 35 -
an h is not in general piecewise to construct
h is as follows. m
(1 x T)f keeping are homotopic of a1
~
fS
First ambient
fixed, which
m
(1 x T)fS
ambient
isotopy
Theorem
12, it can be extended
~ =
because
isotop f to g
=
compact
Since the support, by
to sq, and so the link is
In the new position [ehg]
they
and the stability
1
fixed1 for similar reasons.
can be chosen to have
But the antipodal
~=
of fS
Then ambient
keeping
isotop g to
is possible m
in the complement
ensures unknotting.
unchanged.
linear and so the best way
[e(1 x T)f]
hfSm,
we see that, removing
=
[Tef]
=
T[ef]
map T of S q-m-1 has degree
=
Ta.
(- )q-m .
Therefore
(_)q-ma•
Com1Ll~tion of the_proof
?5.
of ~e?rem
We are given Sm1 sp c sq with linking numbers the smaller sphere p - m times, with linking numbers
~*
= =
~.,'.'
=
a... .•.
a~:;9' ~ 'I'... say.
Then
(_)p(p-m)zp-ma ~
(_) (qp+-m)-pQ,~i: ~ is stable.
Therefore
~ = (_)q-m+p(p-m)zp-ma
= This completes
Suspend
to give a link Zp-mSm1
by Lemma 61 , and
by Lemma 62, because
as~'
(_)p+m+q+pmZp-ma.
the proof of the theorem.
sP c zp-msq
- 36 -
I~ neither
o~ a, ~ are stable then we can show
they have a common stable suspension
to within
the sign
(_)m+p+q+pm by suspending both sides of the link sufficiently
and applying Lemmas 61 and 62.
Consider
Sm c Sr x Eq-r•
embeddings
us plot against
coordinates
different
types o~ knotting
attention
to q ~ m +
Keeping m fixed, let
q and r the three regions can occur.
in which
As usual we restrict
3.
r
! rn+3
unstable range
;>
<:
metastable range
> ~------stable
range
- 37 -
C'f
1. UllJ.a1,ottJ-J1..E.. Region which
is the condition
Theorem
24 Corollary
are classified
by g +r ~ 2m + 3, in Sr x Eq-r by
is bounded
by Sm to unknot 1. r
Therefore
the classes
by ~ (8 ). m
C0
Region 2. Homotop¥-~ ta.Q).enlcl1,?ttill.E.. q + r < because ~m(Sr)
2m + 3 and r ~ m/2 + 1. the condition to be stable.
homotopy-stable
r ~ m/2 + 1 is exactly In the next corollary
Region
the condition
for
we classifY
range.
in the following
Sm c s2m x E1 described
there are no analogous
that region
Therefore
corollary
situation
is the complement.
6 above.
example
in the unstable
~
like the knotted
It is interesting
in spheresy
by
knots.
Here we can tie knots
smooth
is bounded
I'Vecall these homotopy-stable
3. [~~2t02y-un~tabl§kr~t;in£.
in Counter
of embeddings
(g) lies entirely
all the knots
lie in the unstable
results
in the smooth
is complicated
range. category;
by the presence
and it is difficult
to disentangle
that we classify So far the
of sphere knots them from the
situation. Ham 0t
Iso(Sm
c Sr x Eq-r)
e 7C
d~§J.N.lill!.
1'"
(Sm-r-1 ) ..-
~""
~
(Sr).
..-7 m
q-r-1
There:t~£.. s~ u~n~o_tf3 gL§r __x-.m.~_:~tt_.1?-nd ..Q!lIY iL the susJ?ensJ_C?ll
L~.§..a_T!l0E2m0.rl?.h~ sIT!.~
- 38 -
Pro.£.f' •
Notice
that conditions
for region
QL) imply
<1- r - 1 ~ (2m - r + 2) - r - 1 9 be caus e 2m - r ~ <1- 2 < m9
because
m ~ 2r -2.
Also it is easy to show r ~ 2.
Therefore
Iso(Sm c Sr x E<1-r) ~ Iso(Sm9
S<1-r-1 c S<1) by Lemma
60
~ Iso(S<1-r-19 sm c S<1) putting the smaller sphere first. ~ ~ Define
by Theorem Finally
e
the isomorphism 259
~e
= ~
we can write
<1-r-
1 (s<1-m-1) 9 by Theorem s t abl e.
= (_)(<1-r-1)+m+<1+(<1-r-<1)ma• Then~
and so the diagram ~
=
A~
[Sm c Sr x E<1-r] A injective
in Sr x E<1-r <
Sm unknots
is commutative.
where
Iso(Sm c Sr x E<1-r) ~ 7 surjective Therefore
25 since ~
»>
'T-
(Sr).
m
~ ~ injective
~;.c~
~
injective
<----.~ ~ monomorphism.
This completes
the proof
of the Corollary.
isotopy
classes
homotopy
classes
sphere c torus of embeddings epi
Z ---7
x E10 I
of maps
Z2
z ---- --) 0 2 --+ ---------------- -------.---------------.-I i
i
---- ...-.------
- 39 In the last example there are three knots all homotopically trivial, and no realisation homotopy
as an embedding
of the other
class.
It would be interesting
(i)
th e
to extend the results
. (---3 " reglon ",,--j.
(ii) knots of Sm in arbitrary neighbourhoods
q-dimensional
regular
of Sr, rather than just the product neighbourhood.
(iii) knots of Sm in an (r-1)-connected is not big enough for unknotting. of an obstruction
manifold,
where r
This would be the beginning
theory.
We now return to the task of'proving which occupies
to:
Theorems
23 and 24,
the rest of this chapter.
We are given a continuous map f:M ~ Q which we have to homotop into a piecewise
linear embedding
in the interior,
and we are
given that m :;-;; q - 3 M is d-connected Q is (d+1)-connected, where d
=
2m - q.
The first step is to make l'piecewise approximation. follows.
Next homotopy
linear by simplicial
l' into the interior
Let Q1 be a regular neighbourhood
of Q as
of fM in Q.
Since
- 40 -
M is compact shrinking
so is Q1' and therefore
this collar
Q1 has a collar.
to half its length
(the inner
o
homotop
By half)
0
Q1 into Q1'
This homotopy
carries
fM into Q19
0
c Q.
o
Now homotop Corollary
f into general
1 of Chapter
position
6. Therefore
in Q by Theorem
singular
18
set s(r) or r
will have dimension dim s(r) ~ d, the double is contained
point
dimension.
in the following
The next main
step of the proof
lemma. o
I;~rruna_63. TE~..rEL_~~..Lst co11aJ?_sibl$=subt3J2.§tc~s _C2 D of.M,4 Q re~.l?estJVE?)~1l.c:q.J;haL~i.fJ,.c
c. =:..- r-J2..
D f
c
>
- 41 -
The main idea of the proof' is to use the engulfing
Proof. Theorem
7
20 or Chapter
several
Since M is d-connected, a collapsible
times in an inductive
we can start by engulfing
process. S(f) in
C1d+1
subspace
S(f) c C1 c M. Oi' course when C1 is mapped by f into Q it no longer remains collapsible 9 because bits of S(f) get glued together to i'orm non-bounding
cycles.
we can engulf fCi
Nevertheless,
since Q is (d+1)-connected,
in a collapsible c Q.
1 yet9 because
although
it may contain
other stuff as well.
as to minimise
the dimension
it.
More precisely
the ith induction
D~+29
o
fC1 c D We are not finished
subspace
f
-1
D1 contains
The idea is to move D1 so of this other stuff and then engulf
we shall define an induction statement
C1'
on i9 where
is as follows:
There exist three collapsible
subspaces
C. in M and l
o
D. :) E. in Q9 such that l
l
S(1') c C.
( 1)
l
-1
E. c C.
(2)
f
(3)
dim (D. - E.)
l
l
l
The induction
l
begins
above, and choosing ends at i S (l') c C. l
= =
~ d - i +3•
at i
l
19 by constructing
E1 to be a point of £'e1•
d +49 because f-1 D.
=
then D.l
as required.
=
C19
D1 as
The induction
E.l and so we have
- 42 -
There remains to prove the inductive aSSllffie the ith inductive
statement
Then fC. U E. c D. by (2).
=
is true, where i ~ 1.
Let
111
F
step, and so
D. - (fC. U 111
E.).
Then dim F ~ d - i + 3 by (2).
Using Theorem
15 of Chapter 6 ambient isotop D. in 1
Q keeping fC. U E. fixed until F is in general position 1
1
respect to fw.
with
Then
dim (F n fM) ~ (d - i + 3) + m -'1 ~ d - i, because m ~ '1 - 3. dim f-ip ~ d - i,
Therefore because
f is non-degenerate,
being in general position.
Let Ei+1 denote the new position Then E. 1 is collapsible 1+
Since M is d-connected, 0f
the closure
",-1 F )
I
because
of Di after the isotopy. it is homeomorphic to D .. 1
we can engulf f -1F ( or more precisely
by pushing
out a feeler from C .• 1
That
is to say there exists a subspace C. 1 of M such that f -1FcC.
1+ 1 ......,., C.
1+
1
dim (Ci+1 - Ci) ~ d - i + 1 . Then C. 1 is collapsible because C. 1~ C. ~ 0; l+
1+
S(f) c C.1+l' because
S(f) c C., by induction 1 C
1 f- E.
1
C.1+1.
1 c C.1+l' because f-1E.1+1 = C.1 U f-1E.1 U f-1F
1+
=
C. U f-1F, by induction 1
c Ci+1' by engulfing.
- 43 -
Since Q is (d+1)-connected
we can now engulf f(C. 1 - C.) 2+ 2 That is to say there by pushing out a feeler from E.2+ 1" o
exists a subspace D.2+ 1 of Q such that f( C. 1 - C.) cD. 1 --'--;l E. 1 1+ 1 1+ 1+ dim (D. 1) ~ d - i + 2. 2+ 1 - E·2-t Then Di+1 is collapsible
because Di+1 ~
Ei+1~
O.
We have
constructed the three spaces, and verified all the conditions of the (i+1)th inductive
fCi c Ei+1, since the isotopy kept fC i
This follows because fixed9
statement excent C.2+ 1 c f-1D i+1· .t;'
and so fC.1+ 1
= C
=
fC.1 U f(C.2+ 1 - C.) 2 E. 1 u D. 19 by engulfing 1+ 2+ Di+1"
This completes the proof of the inductive
step, and hence the
proof of Lemn~ 63. We return to the proof of Theorem 23" submanifold
Q~ of Q containing 'l'
Triangulate
M9
subcomplexes.
Q~;;
Choose a compact
fM u D in its interior.
such that f is simplicial
If we pass to the barycentric
and C9 Dare second derived
complexes
then f remains simplicial because f is non-degenerate (being in general position). Let Bm, Bq denote the second
derived neighbourhoods
of C, D in M, Q* respectively;
are balls by Theorem 59 because Lemma 63 implies S(f)
C
Bm
=
these
C, D are collapsible. Then f-1Bq• In fact the lemma implies
- 44 -
more: it implies that f Om I maps B
f'
,JI
embeds
-+
Bm
0q B -+
-
\ embeds M - B
Bq m
-+ Q -
q B .
Now we see our way clear: we have localised singularities them out. that g .mB
of f' inside balls, where
More precisely
let g~Bm
all the
it is easy to straighten
Bq be an embedding
-+
such
= f 10m B
, obtained by joining the boundary to an ' point in some linear representation of' Bq. Extend interior g to an embedding g:M -+ Q by making g equal to f outside Bm. Then g ~ f.
Notice
that the homotopy is global, but takes place inside the ball Bq• This completes the proof of Theorem
23
in the case M closed. prs>of.. ()t~_TA~or.emS2_ wht:;n Lt§..._boulld~d. We are given a continuous piecewise
linear embedding
map f:M of M in
Q such that flM is a
-+
Q.
First make f'piecewise
o
linear keeping M f'ixed by relative
simplicial
The next thing to do is to straighten boundary. .
If g
-1
0
Call a map g:M •
Q = M.
-+ Q
proJ2~
approximation.
up the map near the (as in the case of embeddings)
- 45 -
•
0
E fixed, such that S(g) c Q. -.,.=-~··"_m.~~·~~"""""_==",,,,, ",r',-" .. ~",
•••....•. ", ...,__
After
Once the lemma is proved we can apply the same arguments in the unbounded
case to eliminate
working
in the interior of Q.
entirely
the singularities
as
of g,
The proof of the
theorem will therefore be complete. Pro~f 91'L.§mma
64.
Since M is compact we can choose a to Lemma 24, that is to sayan
collar by the Corollary c
M
embedding
:111 x I ~ M,
a) = x, for all x E r~1. Let Ma = closure (M-im ~~)• ,).be a homotopy Let ht:M ~ M/that starts with the identity and finishes with a such tha t
C~ff(X,
map hi that shrinks the collar onto the boundary homeomorphically by stretching particular
onto M.
Such a homotopy
and maps Ma
can be easily defined
the inner half of a collar twice as long.
In
- 46 -
h1CM(X9 for all x E M9 U E I.
=
x9
Then ht keeps
.
f ~ fh
u)
M
fixed9 and therefore
1 keeping M fixed.
Now let Q1 be a regular neighbourhood
of fM in Q.
Since
M is compact so is Q1' and we can choose a collar cQ
such that c(y, 0)
=
:Q.1 x I ~ Q1
y for all y
Let kt:Q1 ~ Q1 be the homotopy inner boundary
a ~
E
.
Q1'
Let QO
shrinking
and keeping QO fixed.
=
closure(Q1-im
the collar onto its
More precisely
for
t ~ 1 define t + u),
a ~
u ~ 1- t } Y E Q,1
1),
We now use k to construct into QO and sketches cQ'
a homotopy
gt:M ~ Q that moves MO
the collar cM out again compatibly with for a ~ t ~ 1 define
More precisely gtc~:I (x , u )
1-t~u~1
= {
cQ (fx , u), a ~ u ~ t
•
X E
cQ(fx, t), t ~ u ~ 1
1i
gtlMO = ktfh1lMo' Notice
that gt keeps M fixed.
Define
g
=
g1' and we have
f ~ fh1 = go ~ g1 = g, and the collars Meanwhile gMo c Q09 all keeping ~l fixed. cM' cQ are compatible with g in the sense that the diagram
cQ ).
- 47 -
000
•
is commutative.
Therefore
So g is proper.
Also the restriction
an embedding,
gM
C
Q1 c Q and gM
=
fliI c
Q,
and
of g to the collar is
and so
S(g) c closure Qo = QO
0
C
0
Q1 c Q.
This completes the proof of Lemma 64 and Theorem
23.
Proof of Theorem=-- 24 when M is__.,-~~ closed. ,~~_.~-.... ... ~ :=lI;-."
""
. ..-..-...-.
+
-
o
We are given homotopic
embeddings
f, g:M ~ Q, we have to show
they are ambient isotopic keeping Q fixed, prOVided m ::;; q - 3 M is (d+1)-connected Q is (d+2)-connected. Without
loss of generality
because
if we prove the result for the unbounded
f, g are ambient isotopic possible
o
in Q.
case then
Then, by Theorem 12, it is
to choose an ambient isotopic with compact support,
which is therefore
Q fixed.
we can assume that M is unbounded,
Therefore
extendable
to an ambient isotopy of Q keeping
assume Q unbounded.
R~marko If \lITe had Hudson t s concordance
. :>isotopy result available,
then the theorem could be deduced immediately as follows.
The homotopy
from Theorem
gives a continuous map
23,
- 48 -
F:M x I ~ Q x I
I)" is an embedding in (Q
I)""
such that FI(M
x
connectivities
o~ M~ Q have to be increased by one each~
because the double-point 2 (m + 1) -
dimension
(q + 1)
=
x
The
o~ F is
(2m - q) + 1 = d + 1 "
By Theorem 23 homotop F into an embedding
G:M x I ~ Q x I keeping
UI
x
I)" ~ixed"
between ~ and g"
In other words G is a concordance
There~ore
they are ambient isotopic.
as we have not proved the concordance
>isotopy
However
in these notes~
we give a separate proo~ o~ Theorem 24~ similar to that o~ Theorem
23 above.
Begin the proo~ by ambient isotoping with respect to ~M~ by Theorem continuous
15.
g into general position
The given homotopy
is a
map h:l1 x I ~ Q
which we can make piecewise by relative
linear keeping
(M x I)" ~ixed~
simplicial approximation.
fJ_~_~.d. '_we__c_aJl~in_d.col~.§l..1?.~:!:J:L~_~=f2.1.l.9_~_ge s.G..9 D __ ol'Jk_.~~h S(ll.l~_
C~"
=
~p_at
1
h- D.
We prove the lemma in two stages in order to make the proo~ more translucent. by assuming
In the ~irst stage we prove a weaker result
the stronger hypotheses
- 49 -
m~q-5
M is (d+2)-connected
Q is (d+4)-connected. The proof follows the pattern of the proof of Lemma 63. In the second stage we show how to sharpen the proof as various points in order get by with the correct hypotheses of
Theorem 24, namely
m ~ q - 3 M is (d+1)-connected Q is (d+2)-connected. We achieve the sharpening by using the piping techniques
of
the last chapter. Proo~ of First S~age.
Let ~~M x I ~ M denote projection.
Notice that hl(M x I)· is in general position because we have already isotoped g into general position with respect to fM. By Theorem 18 homotop h into general position fixed.
keeping
(M x I)·
Therefore dim S (h) ~ 2(m + 1 ) - q
= There is an induction
d + 2.
on i9 as follows.
There exist collapsible
subspaces C. in M and D. ~ E. in Q such that 111
( 1)
S(h)
(2 )
1 h -1 E. c C. x I C h- D.
(3)
dim (D. - E.) ~ d - i + 6.
C
Ci x I
111 1
1
- 50 -
=
The induction begins at i hypotheses
1.
Since under the stronger
we are assuming 1\1 to be (d+2)-connected,
engulf
~S(h) in a collapsible
subspace c~+3 of M.
to be (d+4)-connected9
engulf h(C1 x 1) in a collapsible
d+5 subspace D1 of Q.
and the conditions
Since Q is assumed
Choose E1 to be a point of h(C1 x 1),
=
for i
For the inductive
1 are satisfied.
steP9 assume true for i.
Obtain E.1+ 1
by ambient isotoping D. in Q keeping h(C. x 1) U E. fixed9 III
until the cODplement F is in general position with respect to
1).
heM x
Then dim ~h-1F ~ dim h-1F dim [F n heM x 1)]
=
~ (d - i + 6) + (m + 1) - q ~
d
-
i + 2,
because we are assuming m ~ q - 5. Engulf in M AA-1F c C. 1 ~ C. 1+ 1 C. ) ~ d - i + 3. dim (C. 1 1+ 1
-
Therefore dim (C. 1 x 1 - C. x 1) ~ d - i + 4. 1+
Engulf in
1
Q
h( C 1+ . 1 x 1 - C.1 x 1) cD. 1+ 1 ~
E.1+ 1
dim (D.1+ 1 - E.1+1 ) ~ d - i + 5. Verify the (i+1)th induction The induction S(h)
C
ends at i
C. x 1 ::; h 1
-1
D.,
1
=
statement as in the proof of Lemma 63.
d + 7 with D.
as required.
1
=
E'9 and consequently 1
- 51 -
When homotoping positiony
the full strength of Theorem
now use the additional
information
for the pair M x I, (M x I)·. that the (d+2)-dimensional interior
18 was not used.
We
that h is in general position
In particular
this implies
stuff of S(h) all lies in the
of M x I, at places where exactly two sheets of M x I
cross one another. top dimensional
7.
The trick now is to punch holes in this
stuff, by piping
one of the sheets over the
More precisely
we use the piping Lemma 48 of
o.
free-end M x Chapter
h into general
The triple M x 0, M x 1 c M x I is "cylinderlike"
in the sense of Chapter 7, and Q has no boundary Therefore
by assumption.
by the piping lemma, we can homotop h keeping
(M x I)·
fixed, and then find a subspace T of M x I such that (1 )
S(h) cT
(2)
dim T ~ d + 2
(3)
dim
(4)
M x I ~(M
[(M
x 1 ) n T] ~ d + 1 x 1)
u T '-:ol.M x 1 .
By being a little more precise
in the proof of Lemma 48 at
one J.)oint,we can factor the first of these collaJ.)ses
(5)
M x I ~
(M xi)
u (1tT x I)
~
(M xi)
u T.
engulf
it is J.)ossible,using (3), to d+2 (M x 1) n T in a collapsible subspace R of M x 1.
Define
C~+2
Since M is (d+1)-connected
I
=
~(R
U
T).
Notice
that comJ.)aredwith the dimension
of C1 in the J.)roofof the first stage, we have scored an
- 52 -
improvement
of 1.
Then
=
C1 x r
(~R x D)
U
(~T x r)
'~R u (-JeTx I), cylinderwi se ~ RuT Therefore
by
(5).
heR u T) by Lemma 38 and (1) above.
h(C1 x I)~
But dim h(C1 x I) ~ d + 3 dim heR u T) ~ d + 2. h(C1 x I) can be "furledil
Therefore
in the sense of Chapter
7.
Since Q is (d+2)-connected engulf h(C1 x r) in a collapsible d+2 subspace D1 of Q, of the same dimension, by the furling Corollary to Theorem 20. As before define E1 to be a point of h(C1 x I).
Notice
the dimension
that we have scored an improvement
of 2 in
of D1•
We can write this improvement statement by replacing
into the ith inductive (3) by
condition
dim (D. - E.) ~ d - i + l
2
4.
We now have to do some more piping and furling for the inductive step.
As before
obtain E. 1 by ambient 2+
isotoping D. keeping
h(C. x I) U E. fixed, until the complement l
position
2
with respect
to h~M x I.)
Let W
2
F is in general
=
dim ~W ~ dim W
=
dim [F n heM x I)]
~ (d - i + 4) + (m + 1)-q
~ d - i + 2
-1 closure (h F).
Then
- 53 -
because m ~ q - 3.
We now want to engulf
with a feeler of the same dimension, is to furl ~W. dimensional piping
heM
x
We furl
D)
of
heM
x
punching
top dimensional
C
holes in the top triangulation),
piece of F off the end
After "
l\iIxI
pipe
J~
-
c.~
Notice
by
I).
Before
M
but 1 and the way to do this
(of aome suitable
simplexes
the relevant
~w by
~w from
,
l'
hole
that S(h) c C. x I, and F docs not meet h(C. x I), 1
and so that the self-intersections
1
of heM x I) do not get in
the way of the pipe. More precisely,
we can adapt Lemma 48 to give the following
- 54 -
result.
We can find an ambient isotopy of D. keeping 1
h(C. x I) U E. fixed, such that in the new position 1
~Wd-i+2
1
can be furled to a subspace Xd-i+1 relative to C., and 1
(Ci x I) U (~W x I)~(Ci By the Corollary
x
I) U (X x I) U W.
to Theorem 20, engulf in M
~w c
C. 1"--; 1+
C.
1
d - i + 2. dim (C. 1+ 1 - C.) 1 ~
Let yd-i+3
=
closure
(C. 1 x I-C. 1+
1
Then Y can be furled to Z relative
(C. x I) U Y 1
=
X
I), Zd-i+2
=
(X xl)
UW U (C. 1+1
x
to C. x I because 1
C.1+1 x I
... ~ (C. x I) U (~W x I) U (C. 1 x 1) cy 1ind erwi S E 1 1+ ~(C. x I) U Y by above. 1
Therefore hY can be furled to hZ relative Lemma 38, because to hZ relative
S(h) C Ci x I.
Moreover
to h(C.1 x I) by hY can be furled
to E.1+ l' because
hY n E.1+ 1
C
heM x I) n E.1+ 1
=
h(C.1 x I) U hW
C
hZ n E.1+ l'
and therefore hY n Ei+1 = hZ n Ei+1• hY
C
D.1+1 ~
Therefore
engulf in Q
E.1+ 1
dim (D. 1 - E.1+1) ~ d - i + 3. 1+ Verify the (i+1)th inductive
statement as in the proof of the
first stage, and the proof of Lenma 65 is complete.
- 55 Lemrna 66.
1:;h9. t
Tl?-e1Z.e_~~ts.:LEa~].J32m c ~L__ l?q c~suc]l
f t
13m x
WDS
h
~_Bq
I
0
embeds
Bm
~:beds
1M- Bm.).,,, I
BCl
x I ~
.:'._Q _
B'l,
The obvious way is to take derived neighbourhoods o~ the collapsible
subspaces
run into the technical triangulations
66.
C, D o~ Lemma
di~~iculty
However we
o~ not being able to ~ind
such that both maps h
M x I ----?Q
1~ M
are simplicial o~ Chapter
(as is illustrated
1).
Therefore
in the Example
~irst choose triagulation
M x I, M such that 1C:K ~ L is simplicial, subcomplex
o~ L.
such that A-10 speci~ied
later.
C.
Choose
De~ine
Bm
=
it is a regular neighbourhood
~ Q1 is simplicial.
deriveds
K
1
map
< 1 o~ a smallness
to be
8] which is ball, because 14.
o~ K, and a triangulation
I~ K2, Q2 are barycentric
2 ~ Q2 remains
then h:K
non-degenerate.
A-1[O,
simplicial
Q1
o~ heM x I) in Q such that
o~ a regular neighbourhood 1
8
o~ C by Lemma
Now choose a subdivision
h:K
0 <
8,
K, L o~
and C is a ~ull
Let A:L ~ I be the unique
=
at the end
Now choose
B
simplicial
such that
8
because
~irst h is
< A1CV ~or all vertices
Call a simplex o~ K2 e~ceptionaJ. i~ it 2 not in C x I. meets C x I, but is not contained in C x I. Then (A1C)-18 meets V E K
- 56 -
only exceptional
simplexes~
and meets each exceptional
simplex
in a hyperplane. Let K3 be a rirst derived or K2 obtained by starring all exceptional
on (A~)-18~
simplexes
and the rest barycentrically.
Since S(h) c e x I~ no exceptional any other simplex by h.
simplex is identiried with
Thererore we can derine a rirst derived
Q3 of Q2' such that h:K3 ~ Q3 remain simplicial, by starring images of exceptional simplexes at the image or the star-point, and the rest barycentrically. Derine Bq == N(D, Q3)~ which is a ball~ being a second derived neighbourhood subspace D.
of the collapsible
Then
h -1B q
==
I, K3 ) == q m x I h-1B == B
N(e
x
S(h)
C
Bm
x I.
The proor of Lemma 66 is complete. eon~inJ1J1J.Jt the l?.1~o~()fco:r~T.l1.e.9r_em _24. So far we have the picture:
III
B
x I.
- 57 -
~ ~
Me
BrnXI
~
Bq
B~
Since hiM -
Bm
is a proper embedding
this means that by Theorem
~IM - Bm
is isotopic
Om
of M - B-
0q
in Q - B
Om
to g I M - B.
9
Therefore
1 of Chapter 5 they are ambient Extend the ambient isotopy of Q - q arbitrarily 12 Corollary
isotopic. over Bq to give an ambient
B
isotopy of Q.
The latter moves
Om
m
~ to f' 9 saY9 where f' agrees with g except on B 9 and r' IB 9 glBm are proper embeddings of Bm in Bq that agree on the Since m ~ q - 39 by Theorem 9 Corollary 1 of Chapter 4 we can ambient isotop Bq keeping Bq fixed so as to
boundary.
move f' IBIDonto glBm• to an ambient are ambient
This ambient isotopy extends trivially
isotopy of Q9 moving f' onto g.
isotopic.
when M is closed.
This completes
Hence f and g
the proof of Theorem
24
- 58 -
We are given embeddings
. M . Q
keeping keeping
f, g:M ~ Q that are homotopic
fixed, and we have to show they are ambient fixed.
isotopic
The first thing to do is to make them agree
on a collar. Let Q1 be a compact
submanifold
image heM x I) of the gi7en homotopy
of Q containing h.
the
Choose a collar
By Theorem 10 of Chapter 5 choose two collars cfQ'
of M.
f
Q1 such that c ' cQ are compatible M compatible
with g.
k of Q1 keeping fixed, k extends ~ fixed.
13 there is an ambient isotopy . f g . Q1 fixed, moving cQ onto cQ' Since Q1 is kept
O
M
C
to an ambient
if we replace
on the collar
wi th both f and g. Then fMO' gM
By Theorem
trivially
Therefore
then f, g agrees
g with f, and cM' cQ are
QO'
isotopy
f by ~f,
of Q keeping
and write
cQ
=
c~,
im c ' and cM' cQ are compatible M
Let MO = closure The picture
(M - im cM), Qo = closure (Q1- im CQ)
now looks like:
- 59 -
J:,~I£1Ela_ 67.
ThC:;; ... 2T1fl:PSY-1Mo?~LMO ~MO
::"Q.o
are hQmo_toJlJc __:ij1~Qo
?ee~ing MO fixed. Let et~MO ~ M be a homotopy inclusion
and ending with a homeomorphism
collaX' of Mo over that of M. all x E M? tEl.
is obtained by
of the three homotopies
jfet, jhte1? jge1_t· the proof of the lemma. of what we have done so far is to push the
singulaX'ities of the homotopy that the boundaries
homotopy
that shrinks
jCQ(Y? t) = cQ(y? 1)
Then the required homotopy
composition
The purpose
However
etcNr(x? 1) = cM(x? 1 -t)?
Let j:Q1 ~ QO be the retraction
all Y E Q1? tEl.
This completes
that stretches a
In particular
the collar cQ onto its inner boundary?
timewise
starting with the
into the interiors
of M? Q so
do not interfere with the engulfing.
there is the trivial technical
difficulty
that a constant
of the collar is of cour se a singular map of (collar) x I.
Nor can we ambient
isotop glim cM away from flim cM for two reasons~ firstly we have got to keep M fixed? and secondly there is an obstruction in H2q-2m(Q). Therefore we get round this
difficulty
by defining
the homotopy
to be a map of a reduced
product M # I? obtained from M x I by shrinking point for each x
im cM. More precisely? identify
x x I to a
E
(but not the complements
the collars
of M x O? M x 1
of the collaX's) and define M # I to be
- 60 -
the relative
mapping
Mx0--7Mx1.
cylinder
Let 7\.# :M:I+ 1
of the homeomorphism M
--7
denote the projection.
collar
M
=
If X c M, define X:I+ 1
=
M:l+I
7\.;1 x.
Then u (collar).
0,1.[0#1)
Define h# :M# 1--7 Q by.rapPing
110 # I by Lemma
lembedding
67
the collar by f7\.#.
0
Then 7C#S(~) c Mo c M. 0
0
h#S(~) Therefore
c Qo c Q1 c Q.
we can apply engulfing
of M, Q, as in the unbounded Bq c Q, such that 1
)
embeds
0
c M,
On
(M -
Ell
--7
Bm) # 1
Therefore,
as before,
By Theorem
12 they are ambient
proof of Theorem
m
B';1.
--7
embeds 13m #: 1
l
in the interiors
case, and obtain balls B
Om
map s B :1+ I
arguments
--7
Eq .
Q -
f and g are isotopic isotopic
24 is complete.
.
keeping M fixed.
keeping
Q
fixed.
The
