Lectures on polyhedral topology

Lectures on Polyhedral Topology; by John R. Stallings

Notes by G. Ananda Swarup

No part of this book may be reproduced in any form by print,, microfilm or any other means without written permission from the Tata Institute of Fundamental - Research, Colaba, Bombay 5

Tata Institute-of " Fundamental·..Research,^ Bombay 1967

TABLE OF CONTENTS Page

Chapter I

Chapter II

Introduction.

1

: Polyhedra

4

1.1. Definition of polyhedra

4

1.2. Convexity

5

1.3. Open convex sets

12

1.4. The calculus of boundaries

16

ΐ.5. Convex cells

22

1.6. Presentations of polyhedra

27

1.7· Refinement by bisection

30

: Triangulation.

35

2.1. Triangulation of polyhedra

35

2.2. Triangulation of maps

37

Chapter III : Topology and approximation.

Chapter IV

43

3.1. Neighbourhoods that retract

44

3.2. Approximation theorem

46

3.3. Mazur's criterion

51

: Link and star technique.

54

1. Abstract theory I

54

4.2. Abstract theory II

5δ

4.3. Geometric theory

61

4.4. Polyhedral cells, spheres and Manifolds

72

4. 5- Recalling homotopy- facts

79

Page Chapter V

: General position.

...

5. 1. Nondegeneracy

82

.

83

5.2. Ν D (n)-spaces. Definition and elementary properties 5.3. Characterisations of

· D (n)-spaces

...

5-4. Singularity dimension Chapter VI

86 89 98

: Regular neighbourhoods.

115

6.1., Isotopy

...

1l6

6.2. Centerings, isotopies,, neighbourhoods of subpolyhedra 6.3. Definition of 'Regular Neighbourhoods.

124

6.4. Collaring .. 6.-5· Absolute regular neighbourhoods

12?

and Newnanish theorem 6.6.

...

136

Collapsing

141

6.7. Homogeneous collapsing

...

...

...

145

6.8. Regular neighbourhood theorem

149

6.9. Some applications and remarks

I6I

6.10. Conclusion Chapter VII

118

. ''

...

165

: Regular collapsing.

I68

7.1. Regular collapsing

168

7.2. Applications to unknotting...

179

Appendix to Chapter VII ,

184

Chapter VIII : Handles and s-cobordism 8.1. Handles

...

191 191

Ill Page 3.2. Relative manifolds and their handle presentations 8.3.

193

Statement of the theorems, applications, comments

198

δ.4. Modification of handle presentations ... β. 5. Cancellation of handles

210 215

8.6. Insertion of cancelling: pairs of handles.

222

8.7. Elimination of 0 - and 1-handles

231

...

8.8. Dualisation

235

8.9. iilgebraic description

238

8.10. Proofs of the main theorems

246

8.11. Proof of the isotopy lemma

...

...

253

Introduction

The recent, efflorescense in the theory of polyhedral manifolds due to Smale's handle-theory, the differential obstruction theory of Munkres and Hirsch, the engulfing theorems, and the vrork of Zeeman, Bing and their students - all this has led to a vride gap between the modern theory and the old foundations typified by Reidemeister's Topologie der Polyeder and Whitehead's "Simplicial spaces, nuclei, and nir-groups". This gap has been filled somewhat by various sets of notes, notably Zeeman's at I^H.E.S.; another interesting exposition is Glaser's at Rice University, Well, here is my contribution to bridging the gap. These notes contains (1) The elementary theory of finite polyhedra in real vector spaces. The intention, not always executed, was to emphasize geometry, avoiding combinatorial theory where possible. Combinatorially, convex cells and bisections are preferred to simplexes and stellar or derived subdivisions.

Still, some simplicial technique

must be slogged through. (2) A theory of "general position" (i.e., approximation of maps by ones whose singularities have specifically bounded dimensions), based on "non-degeneracy".

The concept of n-manifold

is generalized in the most natural way for general-position theory by that of ND(n)-space - polyhedron Μ such that every map from an n-dimensional polyhedron into Μ

can be approximated by a

non-degenerate map (one whose point-inverse are a?.! finite). (3) A- theory of "regular neighbourhoods" in arbitrary polyhedra. Our regular neighbourhoods are all isotopic and equivalent to the star in a. second-derived subdivision (this is more or less the definition). Many applications are derived right after the elementary lemma that "locally collared implies collared". We then characterize regular neighbourhoods in terms of Whitehead's "collapsing", suitably modified for this presentation.

The advantage of talking about

regular neighbourhoods in arbitrary polyhedra becomes clear when we see exactly how they shoiild behave at the boundaries of manifolds. After a little about isotopy (especially the "cellular moves" of Zeeman), our description of the fundamental ^echniques in j'f^ polyhedral topology is over. Perhaps the most basic topic omitted is the theory of block-bundles, microbundles and transversality, (/).) Finally, we apply our methods to the theory of handle-presentations of PL-manifolds a la Smalls theory for differential manifolds.

This we describe sketchily; it is quite

analogous to the differential case. There is one innovation. In order to get two handles which homotopically cancel to geometrically cancel, the "classical" way is to interpret the hypothesis in terms of the intersection number of attaching and transverse spheres, to reinterpret this geometrically, and then to embed a two-cell over which a sort of Whitney move can be made to eliminate a pair of intersection.

Our method, although rather ad-hoc, is more direct,

avoiding the algebraic -complication of intersection numbers

(especially unpleasant in the non-simply-connected case) as well as any worry that the two-cell might causej of course^ it amounts to the same thing really. This method is inspired by the engulfing theorem. ./"There are, by the way, at least two ways to use the engulfing theorem itself to prove this point_7. We do not describe many applications of handle-theory; we do obtain Zeeman's codimension 3 unknotting theorem as a consequence. This way of proving it is, unfortmately, more mundane than "sunny collapsing". We omit entirely the engulfing theorems and their diverse applications. We have also left out all direct contact with differential topology. •«•

*

*

Let me add a public word of thanks to the Tata Institute of Fundamental Research for giving me the opportunity to work on these lectures for three months that were luxuriously free of the worried, anxious students and administrative annoyances that are so enervating elsewhere. And many thanks to Shri Ananda Swarup for the essential task of helping write these notes.

John R. Stallings

Bombay March, 1967

Chapter I Polyhedra 1.1. Definition of Polyhedra. Basic units out of which polyhedra can be constructed are convex hulls of finite sets. A polyhedron (enclidean polyhedron) is a subset of some finite dimensional real vector space which is the union of finitely many such units,

("Infinite polyhedra" which are

of interest in some topological situations will be discussed much later). A polyhedral map f ; Ρ — ^ whose graph is a polyhedron.

Q is a function f s Ρ

^Q

That is, suppose Ρ . and Q are subsets

of vector spaces V and W respectively; the graph of f, denoted by PCf),

is the set T C f ) = '^(x, y) I XC:P , y = f(x) t:Q ^

which is contained in V structure,

W, which has an evident vector space

"HCf) is a polyhedron, if and only if (by definition),

f is a polyhedral map. Constant functions, as well as identity • function Ρ - — ^ Ρ are polyhedral maps. The question whether the composition of polyhedral maps is polyhedral leads directly to the question whether the intersection of two polyhedra is a polyhedron. The answer is "Yes" in both cases. This could be proved directly, but we shall use a round about method which introduces useful techniques. /

It will be seen that polyhedra and polyhedral maps form a category. We are interested in 'equivalences' in this category,

that is maps f : Ρ

^ Q, which are polyhedral, one-to-one and onto.

When do such equivalences exist? How can they be classified? Etc... A finite dimensional real vector space V has a unique interesting topology, which can be described by any Euclidean metric on it. Polytiedra inherit a relative topology which make them compact metric spaces.

Since polyhedral maps have compact graphs they are

•epntinuous,. This· provides us with an interesting relaticHiship between polyhedra and topology.. We may discuss topological matters about polyhedra - homology, homotopy, hOTieomorphy,-: and .ask whether these influence the polyhedral categaiy and its equivalences*; • After this brief discussion of the scope of the subject^, we proceed to the, development of the technique. 1.2. Convexity. TIM IIFC ITJJII J

,.11,1 ,-; 1,1

(H, denotes the field of real numbers, and- V a finite dimenaional vector space over let a, b 6 V. "The line segment· betwem. a and b is. denoted by [[a, bj . It is defined thus: [a, b3 = ^t a + (l -.t) A set C (^V is called convex if

Ϊ whenever

a, b 6 C» ' Clearly V· itself -is convex, and.-the intersection of any family .of convex sets, is again convex. Therefore every set Σζ^ν is contained in a smallest -convex set --namely-the,intersexrtion of ali.,.convex. sets containing X;"" this .ia_called the convex hull of X, and is denoted by

t.2. 1. Definition.". A convex combination of a subset X of V is a point of V which can be represented by a finite linear combination k i^O where x^t 1, r^ e !R, r^^^ 0 for all i , and 1.2.2. Proposition.

The convex hull

k ^^ r. = 1. i =0 ^

K{1) of X is equal to the

set of convex combinations of X. Proof; Call the latter

Λ(Χ).

is convex and. contains X, hence If X

It will be shown first that· K(X) C λ(Χ).

X, then

1 • χ is a convex combination of X, k I · ' Let Ρ = J " r. x. , (j- = s. y. — 1 X ^ J J i = 0 j= 0

hence X

be two points of /\(X), A typical £oint of t f + (1-t) 5"= ^

(t r^) Xi +

= t + ( l-t) =1,

is of the form

((1-t) s^) y^, where 04t41.

k ^ ^i ^ ΣΙ ' ΣΙ i-O'

Λ(Χ)

= (k H ^i^ ^

j= o

i= 0

^i Σ. j= 0

and all the coefficients are ^ 0, t -P + (l-t) 1, then

That is ρ is on the line segmeht between x^ lind r Xo + .; - + JL X2 + +

•

'

iiiductionj the sedond point beldngs to

1 Gj hence ^ t G. Thus 7\{1)Q C. Therefore

arid

Λ(Χ)=ί((Χ).α 1. 2>5· Definition. A finite subset ^x^, ''' ^ \

^^

^^

to be independent (orfiffinelyIndependent), if, for real numbers TQ ,..., v^, the equations ^ ···· ^ ^k \ = ° r^ + 0 ^

+ r =0, k

imply that " · · · = ^k

0.

Ex., 1.2.4. The subset ^x^ ^...., x^ ^ of V is independent if and only if the subset

0

of V

IR. is linearly

independent. Q Ex. 1.2.5. The subset ^x^ only if the subset ^x^

x^

x^

of V is independent if and

»*, x^ - x^

of V is linearly

independent. Q Hence if ^x^ ,..·., x^^ C ^^ x »· * > is independent if and only if ^ + x^ ,..., χ + x^^ is independents These two exercises show that the maxunum number of independent points in V is (dim V + 1)»

8

The convex hull of an independent set ij^x^ is called a closed k-simplex with vertices denoted by jc^

x^^

x^^ x^^ and is

The number k is called the ''imension

of the simplex. The empty set 0 is independent, its convex hull, also empty, is the unique (-1)-dimensional simplex. A set of only one point is independent; li_x3 =

is a 0-dimensional simplex. A

set of two distinct points is independent; the closed simplex with vertices

y^ coincides with the line segment [x, y^) between

X and y. 1.2.6. Proposition.

If ^^^

, ^Jr^Cj, then ^x^ ,..·.,

independent if and only if every point of K ^ Q

is ^

unique convex combination of \x χ 7 . L0 η5 Proof; Let ^ ^ x ^ b e independent. If

f^A · · · ^ ^ = s X +....+ s X , with 5" r. = 1 = Τ s., ' 0 0 n n O O n n /L χ ^ χ' then (r - s ) χ + ... + (r - s ) χ =0, and . '' υ U 0 η η η (tq - Sq) + .. + (r^^ - s^) = 0. Hence and the expression for

(r^^ - s^) = 0 for all i ,

is unique.

If ^XQ ,..., x^^ is not independent, then there are real numbers r^ , not all zero· such that r„ x^ + ,-..+ r X = 0 and 0 0 η η r_ + . + r = 0. O n Choose the ordering ^ ^ x^^^ so that there is a it for which

Γ,^Ο 0

if i < 1 if i ^ I,.

Since not all r.^ are zero, r^ +

^ = (-^i)

···

^

Let this number be r. Then 12 X, •y> 0

... V.

-rs X t + .... + -r. ηX

X, t-1

But these, are two distinct convex combinations of ^x^ which represent the same point, a contradiction. 1.2.7. Proposition. The convex hull

^

Κ(X) of X is equal to the

union of all simplexes with vertices belonging to X. Proof} By 1.2.2.,, it is enough to show that a convex combination of • V X belongs to a simplex with vertices in X. Let 'P = r^ x^ + of

+ r^ x^ ; x^ C X,. '51 ^^ =

j

^ Qj

point

(X). It will be shown by induction on η that^belongs to

a simplex with vertices in the set ^x^

, If η = i.,

then ρ = x^ £ jx^ . So let η > 1. If ^x^ >«·.·> Xj^^ is independent, there is nothing to prove. If not, there are s^ s^, not all zero, such that s. X. + . . , + s X = 0 and s + 11 η η 1 i>. define

+ s =0. η

When s. = 0, i '

_ qq,. ^j^^gj^ ^^ ^an be supposed that x,

arranged such that li s..,

^ ...

Then s^ y 0. Hence x^ = ~

^n η

^s^ x^ + ... + s 'n-1

x„

is

10

Therefore

+

+ (r - s ^n) χ •t · η-1 η-1 —η' nSince for all i < n,

- ... Sn

this expresses Ρ as a convex combination of L1 n-1 i Ρ is contained in a simplex vdth vertices

By inductive hypothesis, in l^x^

α The following propositions about independent sets vd.ll be

useful later (See L. S. Pontryagin "Foundations of combinatorial Topology", Graylock Press, Rochester, N.Y., pages 1 - 9 for complete proofs). Let dim V = m, and

be a euclidean metric on V.

First, proposition 1.2.4 can be reformulated as follows: Ex. 1.2,8. Let

be a basis for V, and jx^ )*")

a subset of V. Let x. = =1 a e + ,.. + a. e ; 0 4 i 4 n. Then 1 1 1 1 m' the subset jx^ >·'•> ^^ independent if and only if the matrix ^

a

1

a

0 1

•

1 has rank (n + l)·

·

2

0

0 2

. a.m

«

a η

1

a η

2

m η

D

1.2.9. Proposition, Let ^x^

X V be a subset of V, η 4, m. n3

11

Given any (η + 1) real numbers

(c^ > 0, 0 ^ i

y^eV, such that ό (x^ , y^)

3 points

and the set ^yQ

y^'^ is

independent. Sketch of the proof; Choose a set ^UQ

u^^ of (n + l)

independent points and consider the sets Z(t) = t u^ + (l-t) x^^ , 0 ^ t 41,

u^ + (l-t) x^

Let N(Z (t)) denote the matrix

corresponding to the set Z(t) as given in 1.2.8. (the points being taken in the particular order).

Z( 1) =

>··· ^^^ ' hence some

matrix of (n + l)-colurams of N(Z(l)) has nonzero determinant. Let D(t) denote the determinant of the corresponding matrix in N(Z(t)), D(t) is a poljmomial in t, and does not vanish identically. Hence there are numbers as near 0 as we like such that D(s) does not vanish.

This means that N(Z(s)) is independent, and if s in

near 0, Z(s)^ will be near x^. Q Hence in anyarbitrary neighbourhood of a point of V, there are (m + 1) independent points. The above proof is reproduced from Pontryagm's book. The next propositions are also proved by eonsidering suitable determinants (see the book of Pontryagin mentioned .above). Ex.

1.2.to. If the subset

then there exists a number of V with

x^ ^ of

V is independent,

^ 0, such that any subset ^y^ ,..., y^^

h (x^, y^) < Τλ for all i, is again independent,

1.2.11. Definition.

A subset X = ^XQ )··">

of 'V is said

to be in general position, if every subset of X containing points is independent (where m = dim V).

m + 1

12

Ex. 1,2.12. Given any subset X = ^x^^ ,..., x^ ^ of V and (n + 1 )-numbers with

0 4,1 ^ n, there exists points y^, O ^ i ^ n

,7 ^ ) a n d

such that the subset Y = ^Q, ..., ^n

V is in general position. Hint; Use 1.2.9, 1.2.10 and induction,

Q

1.3· Openconvex sets. 1.3.1. Definition. A subset A of V is said to be an open convex set if 1) A is convex 2) for every x, y ζ A, there exists £·> 0, such that -ζχ + (1 +€)y6A.

(e= 6(x,y) depending

on X, y).

In otherwords the line segment joining χ and y can be prolonged a little in A. Clearly the empty set and any set consisting of one point are open convex sets. So open convex sets in V need not necessarily be open in the topology of V. Clearly the intersection of finitely many open convex sets is again an open convex set. 1.3.2>

Definition. Let ^x^ >·.·>

13

An open convex combination of ^^ > · ·. > x^^^ is a convex combination r^ X| + ..i + Tj^ Xj^ such that every coefficient

^ 0. The set

of all points represented by such open convex combinations is denoted by 0(x^

x^). It is easily seen that 0(x

χ ) is an open convex η

1 set. 1.3.3. Definition. If ^x^ j.-..^ 0(x^

is independent, then

x^) is called an open k-simplex with vertices Jj^x^

The number k is called'the dimension of the simplex 0(xq

x^^, Xy^).

If ^ig >'·'·> ^^ C ^ J · · · J η ^ , then the open simplex ) is called a s-face (or a face) of 0(xq

x^).

If s < k, then, it is called a proper face. Clearly, the closed simplex x^ > · · · > xj^ i® '^'be disjoint union of 0(XQ

x^^) and all its proper faces.

We give another class of examples of open cpnvex sets below which will be used to construct other types of open convex sets. 1.3.4. Definition.

A linear manifold in V is a subset Μ of V

such that whenever x, y ζ;· Μ

and r

then r χ + (l-r) y 6 M.

Linear manifolds in V are precisely the translates of subspaces of V; that is, if V' is a subspace of V, and Ζ 6 V, then the set ζ + V' =(z + z' z'

is a linear manifold in V,'

and every linear manifold in V is of this form. Moreover, given a linear manifold Μ the subspace Vjyj· of V of which Μ is a translate in imique, namely f

ΊΑ

- y ζ ζ.Μ,

- ζ' ζ eM, ζ' a fixed element of M | .

Thus the dimension of a linear manifold can be easily defined, and is equal to one less than the cardinality of any maximal independent subset of Μ (see 1.2.5·). A linear manifold of dimension 1, we vdll call a line. If L is a line, a, b

L, a

b, then every other

point on L is of the form t a + (l-t)b,

If Μ is a linear

manifold in V and dim Μ - (dim 7-1), then we call Μ a hyperplane in V. 1,3.5. Definition. Let V and W be real vector spaces. A function (D : V — ^ W is said to be a linear map, if for every t €{K,and every x, y ^ V, (p(t X + (l-t)y) = t (p(x) + (1-t) φ(y). Alternatively, one can characterize a linear map as being the sum of a vector space homomorphism and a constant. Ex. 1.3.6. In definition 1.3.5, it is enough to assume the 9(t X + (-l-t)y) = t φ(χ) + (1-t) Cp(y) for 0 ^ t ^ 1.

•

If A is a convex set in V and (p ί A — ^ W, (¥ a real vector space) is a map such that, for x, y ζ: A, 0

t W a map. Show is convex, (graph

consisting of " (x, y), χ t A, ys cp(x)).D

15

Ex. 1,3·8· The images and preimages of convex sets under a linear map (resp. open convex sets) are convex sets (resp. open convex sets). The images and preimages of linear manifolds under a linear map are again linear manifolds. D A hyperplane Ρ in V for instance is the preimage of 0 under a linear map from V to j]^ . Thus with respect to some basis of V, Ρ is given by an equation of the form ^

x^ - d,

where x^ are co-ordinates with respect to a basis of V and deiR not all theJ^. s being zero. Hence V -· Ρ consists of two connected components

(

^

"

^

i

^i

which

we will call the half-spaces of V determined by P. A half space of V is another example of an open convex set. 1.3.9. Definition. A bisection of a vector space V consists of a triple (Pj-H , Η ) consisting of a h3φerpl·ane Ρ in V and the two half spaces H"*" and Η

determined by P.

These will be used in the next few section. remarks: Let the dimension of V = m and V

A few more

be a (m - ,k)-dimensional

subspace of V. Then extending a basis of v' to a basis of V we can express v' as the intersection of (k-l) subspacejof V of dimension (m-l). Thus any linear manifold can be expressed as the intersection of finite set (non unique) of hyperplanes. Also we can talk of hyperplanes, linear submanifolds etc. of a linear manifold Μ in V. These could for example be taken as the translates of such from the corresponding subspace of V or we can consider them as intersections of hyperplanes and linear manifolds

16

in V with Μ. Both are equivalent.

Next, the topology on V is

taken to be topology induced by any Euclidean metric on V. The » topology on subspaces of V inherited from V is the same as the unique topology defined by Euclidean metrics on them. And for a linear manifold Μ we can either take the topology on Μ induced from V or from subspace of V of vrtiich it is a translate. Again both are the same. We will use these hereafter without more ado. t.4»

The calculus of boundaries.

1.4.1. Definition. Let A be an open convex set in V. A point X ζ: V - A is called a boundary point of A, if there exists a point a ξ; A such that 0(x, a) ^A.

The set of all boiindary points of

A is called the boundary of A and is denoted by δ A. A number of propositions will now be presented as exercises, and sometimes hints are given in the form of diagrams. In each given context a real vector space is involved even when it is not, explicitly mentioned, and the sets we are considering are understood to be subsets of that vector space. Ex. 1.4.2. A linear manifold has empty boundary. Conversely, if an open convex set A has empty boundary, then A is a linear manifold.

D

Remark; This uses the completeness of real nimbers. Ex. 1.4.3. If (P 5 H^, H~) is a bisection of V, then d H^ = "6H~ = Ρ and δ Ρ = 0. 1.4.4. Proposition.

Π

If A is an open convex set and χζτ'^Α, then

for all b e A, 0 (x, b)(^A.

17

Proof; Based on this picture: There is

'a' such that 0(x, a)(^A.

Extend a, b to a point

c ζ: A. For

any q ^ 0(x, b), there exists a ρ 6 0(xj a) such that qtCKc, p).. Since c, ρ ^ A, q 6 A, Hence o(x, h)QA. Ex.

α

5. Let Cp : V

^W

open convex set in W. Then then equality holds. t.4.6. Definition.

be a linear map, and let Β be an φ

ςΒ). If Cf is onto

• The closure of an open convex set A is

defined to be k u h k ; it is denoted by A. Ex. 1.4.7. If k Q B , then "Α^Β-

•

1.4.8. Proposition. If a, b ^ A and a ^ b, where A is an open convex set, then there is at most one χζ^^λ

such that b t:

x)·

Proof: If b e 0(a, x) and beO(a, y) ; x, y OA, x ^ j, then 0(a, x) and 0(a, y) lie on the same line, the line through a and b and both, are on the same side of a as b. Either χ or y must be closer to a i.e. either χ

y) or

y€:0(a, x). If x^.O(a, y), then x6 A, but Anbk - φ. Similarly y 1.4.16

^x^

^ finite set in vector space V. By ~ ^

x^)· It is enough to show

that 0(x^ }.·.) x^) is an open convex cell. Let Μ be the linear manifold generated by ^x^

· Let dim Μ = k . Write

A = 0(x^ J... J x^) -J A = K. ^ ^ ,..., x^^ . A is open in M. To prove tl:^e proposition it is enough to show that A is the intersection of half spaces in M. Step 1. A and

A are both union of open (hence closed) simplexes

with vertices in ^ ^ 1.2.7.

. The assertion for A follows from

26

Step :2. If Β is a (k-1)-simpiex in ^ Λ and Ν is the hyperplane in Μ .defined by B, then A cannot have points in both the half spaces defined by Ν in Mi Step 3.. It is enough to show that each point of closed

(k-1 )-simplex with vertices in

^A belongs to a

-^x^

x^^"· ·

Step 4. Each point χfcbA is contained in a closed vd-th verticer in

^x^

» To prove this let C^ )»··>

be the closed simplexes contained in x^ jii.j x^

which contain x^

(k-1simpleχ

c) A with vertices in

and Dj 3..., D^ ( C"^^) which do

not contain x. by Step I) U C. Ο D. - ^ A^ Consider any point i j , a ^ A and a point bfe0(a, x). Let C^ (resp. D^ ) denote the closed simplex whose vertices are those of C. (resp. D.) and 'a'.. ' J By the remark following 1.4.18. U C.' W D.' ^ A, Show that r i ^ j ^ then

C^

is a neighbourhood of b. If dim G^^k-1

for all i,

dim C^'^ k-1 for all i. Use 1.5.2 to show that in this

case [J C^

cannot be a neighbourhood of b.

Q

Since the linear image of convex hull of a finite set is also the convex hull of finite set, 1.5.?, immediately gives that the linear image of a closed convex cell is a closed convex cell. If A is an open convex cell in V and ψ

a linear map

from V to W, then φ(Α) = that φ matter what (p U6iU

no

is.

Definition. A polyhedral presentation is a finite set (p

whose elements are open convex cells, such that A ζ (p Implies bAVp. 1.6.2. Definition. sentation ^

A regular presentation is a polyhedral pre-

such that any two distinct elements are disjoint, that

is,

implies A Π B = 0.

Ex, The (p of proposition 1,5.4 is a regular presentation. t.6.3· Definition.

A simplicial presentation is a regular pre-

sentation whose elements are simplicies and such that if A then every face of A also belongs to (p . If

(ps·^® polyhedral presentationsj we call Q j

subpresentation of ^

, If ^

a

is regular (resp. simplicial) then(^

is necessarily regular (resp. simplicial). The points of the 0-cells of a simplicial presentation will be called the vertices of the simplicial presentation. The dimension of a polyhedral presenΡ tation^is defined to be the maximum of the dimensions of the open cells of (p . 1.6.4. Definition. li" (p is a polyhedral presentation

|(pl will

28

be used to denote the union of all elements of

» We say that(J)

is a presentation of ^(J·^ or that ^(p^ has a presentation

,

Recall that in 1.1, we have defined a polyhedron as a subset of a real vector space, which is a finite union of convex hulls of finite sets. It is clear consequence of 1.5.4, 1.5.5 and 1.5.6 that t,6.5. Proposition. Every polyhedron has a polyhedral presentation. If (p is a polyhedral presentation, then

is a polyhedron. D

Thus, if we define a polyhedron as a subset of a real vector space which has a polyhedral presentation, then this definition coincides with the earlier definition. 1.6.6. Proposition. The union or intersection.of a finite number of polyhedra is again a polyhedron. Proof; It is enough to prove for two polyhedra say Ρ and Q. Let φ and Q) be polyhedral presentations of Ρ and Q respectively. Then

is a polyhedral presentation of Ρ \JQj hence Ρ U Q

is a polyhedron. To prove that Ρ Π Q is a polyhedron, consider the set

consisting of all nonempty sets of the form A Π B,

for

and Β t©^·

It follows from 1.4.13 that

polyhedral presentation. Clearly \ Ρ Π Q is a polyhedron. If X C Y

is a

| = Ρ Π Q. Hence by 1,6.5

Q polyhedra, we will call X a subpolyhedron'

of Y. Thus in 1.6.6, Ρ Γ\ Q is a subpolyhedron of both Ρ and Q. 1.6.7. li* (p

0"» sre two polyhedral presentations consider

the sets of the form A K B ,

A ^(p ,

Clearly A X Β is

29

an open convex cell, and by 1.4.20 b (AXB) is the disjoint union of

bA X

the fom if both ρ

B, A / S 5 b

and ό A > 0 B. Thus the set of cells of

A X B , A^(p., Β ^ (3i is a polyhedral presentation, regular and

are. This we will denote by (p y.

.. As above,

we have, as a consequence that Ρ K Q "is a polyhedron, with · presentation ^ X

.

Ex. 1.6.8, The linear image of a polyhedron is a polyhedron (follows from the definition of polyhedron and the definition of linear map).Q Recall that we have defined a polyhedral map between two polyhedra as a map whose graph is a polyhedron. 1.6.9· Proposition.

The composition of two polyhedral maps is a

polyhedral map. Prooft

Let X, Y and

Ζ be three polyhedra in the vector spaces

U, V and W respectively, and let f ί X polyhedral maps. Then ~P(f)(^ U polyhedra.

By 1.6.7,

^ Y, g : Y

V and P ( g ) C _ V

and X A p ( g )

are

are also polyhedra

ζ)Γ\( 'ί )tsV> Ζ be

' is a polyhedron.

This intersection is the set S : ^(x, y, z) ( X ^ X, y = f(x), ζ = g(y)j in U A V X W .

By 1.6.8 the projection of U y ^ V X W

to

UKW

takes S into a polyhedron, which is none other than the graph of the map g ο f : X .

^ Z. Hence g ο f is polyhedral. Q

If a polyhedral map

f :Ρ

^ Q, is one-to-one and onto

we term it a polyhedral equivalence. Ex. 1.6.10. If, f 5 Ρ

^ Q is a polyhedral map, then the map

30

f': Ρ

^P(f) defined by f'(x) = (x, f(x)) is a polyhedral

equivalence. 1.6.11. Dimension of a polyhedron. The dimension of a polyhedron Ρ is defined to be Max. dim C

j C ^(p » where (p is any polyhedral presentation of P. Of course we have to check that this is independent of the

presentation chosen. This follows from 1.5.2. Let Ρ and Q be two polyhedra and f : Ρ a polyhedral map. Let

^ Q be

: Ρ X. Q — ^ , Ρ and jJu : Ρ ^ Q

be the first and second projections. If "PCf), then the open cells of the form

> Q

is any presentation of (C), CT

presentation of P, regular if and only if ^

is. a

is regular. If f

is a polyhedral equivalence, then the cells of the form |Jl(C), C ^

is a presentation of Q. This shows that

1.6.12. Proposition. invariant.

The dimension of a polyhedron is a polyhedral

α

1.7. Refinement by bisection. 1.7. 1. Definition. If say that (p refines

and

are polyhedral presentations, we

, or (p is a refinement of

provided

(a) (b) If

Aei(p, and

In otherwords, ^

and

B ^ C ^ , then AAB = 0 or

ACb.

are presentations of the same

polyhedron and each element (an open convex cell) of (p is contained in each element of

which it intersects. Hereafter, when there

31

is no confusion, we vd.ll refer to open convex cells and closed convex cells as open cells and closed cells. A polyhedral presentation is regular if and only if it refines itself. Let (3 = (P ; H^, H~) be a bisection of the ambient vector space V (1.3.9); G^ a polyhedral presentation offi.polyhedron in V, and let A ξ (2.. We say 'that

admits a bisection by

(B at A, provided; Whenever an open cell A, ζ. Q. intersects B a (i.e. A, n d A i Φ), and dim A^ < dim A, then either A^

Ρ or

or

(in particular this should be true

for any cell in the boundary of A). If (j^ admits a bisection by (J^ at A, then we define a presentation Q

as follows, and call it the result of bisecting

by (3 at A; consists of

with the element

the nonempty sets of the form, Α Π Ρ or

A removed, and with

E* or A n h"^

that is d

=U(I-iA})

A O H ^ A π η"}

^

I

By 1.4.13j and the definition of admitting a bisection, Q , a polyhedral presentation.

Clearly

refines

is

, if ^ ^ is

regular. We remark that it may well be the case that A is contained in Ρ or h"^ or H~. In this event, bisecting at A changes nothing at all, that is

= CS- . If this is the case

we call the bisection trivial. It is also possible, in the case of

m

32

irregular presentations, that some or all of the sets ΑΛΡ, A Λη·*·, A Π Η

may already be contained in Q . - ^A] in

this event, bisection will not change :§,s much as we might expect. Ex. 1.7.2». Let A and Β be two open cells,, with dim A.^dim Β and A / B. Let

(B

: fp J h!

hTI

be bisections of

space such that A is the intersection of precisely one element from some of the

(β .'a.. J

that Β Π Ρ

BOH

If A Π Β / 0, then 3 .and Β Λ H~

an S.

1 ^

m, such

are all non vacuous. Q

What we are aiming at is to show that every polyhedral presentation ^

has a regular refinement,, which moreover is obtained

from (p by a particular process (bisections).

The proof is by an

obvious double induction; we sketch the proof below leaving some of the details to the reader. 1.7.3. Proposition

REFI ( (p ,. (P' j {s^)

There is a piOcedure) whidh) applied to a polyhedral presentation (p , gives a finite sequences cells by bisections of space), which start on and

of bisections (at , give end result (p ,

is a regular presentation which refines ^ ..

Proof: Step 1. First, we find a finite set

(B . ;·("ρ 5 H'^j H ] , j = 1,.,.n

J

j

3 jJ

of bisections of the ambient space, such that every element of (p is an intersection one element each from some of the (3 j's». This is possible because every-element of ^

is a finite intersection of

hyperplanes and half spaces,, and there are only a finite number of elements in (p

33

Step 2. Write (p = (p^, index the cells of (p^ in such a way that the dimension is a non decreasing function. That is define p^ to be the cardinality of fp , arrange the elements of Μ 0 such that dim D^^ dim for all - 0 < k ^ κ >«5t

fp as D^,..., D*^ , vJ 0 0 , Pq ρ - 1. Ο

Step 3. SQ ^ denotes the process of bisecting ^ ^ Inductively, we define S^ tPojk: ^^ i+1

to be the process of bisecting

G^i 5

(Po k+1

defined since the elements of nondecreasing dimension.

Step 4. Write

at D^ by

result. This is well

^ are arranged in the order of

This can be done until we get S„

i = (PQ ρ

and

^ repeat step (2) and then the step (3)

with bisection

(j^j instead of 2 And so on until we get

is clearly a refinement of ^

=

is regular. Each element by

φ

. 1 (O , when the process stops, (p ^ η y η ; it remains to show that belongs to some

fp

^

and

each element of (p -is a finite intersection of exactly one element each from a subfamily of the (p) s. It is easily shown by double J induction that if A ^ Ο , then for any j, i ^ 0 $ either α Γ P. ^ η J or A(2h. or H^. That is 0 admits a bisection at A by · J j V η (Β. for any j, but the bisection is trivial. Let C , D

e (P

,C

C ο D ^ 0, dim C ^ dim D, Then since C is an intersection of one element each from a subfamily of the (j^ .'s, by 1.7.3, there exists an

such that D Π Pj|_ , D Π H^ and D O, Η £

But this is a contradiction.

Hence (p

are all nonempty.

is regular. Write

D

34

+ Ρ^

(Pn-

j· This gives the

Ό

"REFI ((P . p', {s.p".

Q

We can now draw a number of corollaries; 1.10.4. Corollary.

Any polyhedron has s. regiolar presentation. Q

U10.5.

Any two polyhedral' presentations (p »

Corollary.

of the same polyhedron X have a common refinement obtained from

^

, which is

and from G^ by a finite sequence of bisections.

To see thisJ note that ^ of X. The application REFI

is a polyhedral presentation ^

provides (p. .

Let ^T^^ and ^U^ | denote the subsequences applying to respectively; observe that they both result in (j^ .

and

D

1.10.6. Corollary. Given any finite number

(P

polyhedral presentations, there is a regular presentation ^ with

^l(prl ' (p j_| =

^

subpresentations

of of

>·.·>

>

for all i and φ) . is obtained from

_

by a finite sequence of bisections. This is an application of R Ε F I ( φ and an analysis of the situation.

Q

y .. ^ ^

,

35

Chapter II Triangulation As we have seen, every polyhedral presentation φ) regular refinement.

has a

This implies that any two polyhedral presentations

of X have a common regular refinement, that if Χζ] Y are polyhedra there are regular presentations of Y

containing subpresentations

covering X, etc.. In this chapter we will see that in fact every polyhedral presentation has a simplicial refinement, and that given a polyhedral map Ρ

f : Ρ ——^ Q, there exist simplicial presentations of

and Q with respect to which f is "simplicial",

2.1. Triangulation of polyhedra.. A simplicial presentation

of a polyhedron X is also

known as a linear triangulation of X, We shall construct simplicial presentations from regular ones by "barycentric subdivision". 2.1. 1. Definition.

Let ^

be a regular presentation.

of (p is a function Τ| : (p

> such that

A centering G, for

every C ζ; p . In other words, a centering is a way bo choose a point each from each element (an open convex cell) of (p . 2.1.2. Proposition.

"^q

^]

^k ^^^ elements of (p ,

ordered with respect to boundary relationship, then

(C^),,..,

is an independent set for any centering T^ of ^ » Proof; Immediate from 2.1.3. Proposition.

1.4.9?

by induction.

D

Suppose that φ) is a regular presentation and

36

C^^p. C is the disjoint union of all open simplexes of the form

where A^ ^ ^ Proof;

, ^o < ^^
|

·

We have proved 3.1.4.

If X is a subpolyhedron of Y, there is a polyhedron Ν

which is a neighbourhood of X in Y, and there is a polyhedral retraction r : Ν — ^

X. Ο

3.2. Approximation Theorem. We imagine our polyhedra to be embedded in real vector spaces(we have been dealing only with euclidean polyhedra) with euclidean metrics. • Let X, Y be two polyhedra, ρ ,

ρ' be

metrics on X and Y respectively coming from the-vector-spaces in which they are situated. If cjC , p) ·

^ ^

functions,

we define p(oC,i5 ) = Sup f (oC (x), ^(x)) xeY If A is a subset of X, we define diam A=-sup ρ (χ, y), X,y e A and if Β is a subset of Y, we define diam Β = Sup

f'(x,y) Β

47

We can consider X to be contained in a convex polyhedron Q. If X is situated in the vector space V, we can take Q to be large cube or the convex hull of X. Let Ν be a second derived neighbourhood of X in Q and r i Ν — ^ X be the retraction.' Now Q being convex and Ν a neighbourhood of X and Q/ for any sufficiently small subset S of X,

Ν

(recall that < (S)

denotes the convex hull of S),. This can be made precise in terms of the metric; and is a uniform property since X is compact. Next observe that we can obtain polyhedral presentations (p of X, such that diameter of each element of Φ

is less than a prescribed

positive number. This follows for example from refinement process. Now the theorem is 3.2,1. Theorem. Given a polyhedron X, for every ^ > 0, there exists a ί ^ 0 such that for any pair of polyhedra any pair of functions f : Y

^ X, g : Ζ

Υ, and

^ X with f continuous

and g polyhedral, if p(f| Z, g) < (S j then there exists a g polyhedral, i j ζ = g, and Proof;

fCf, g K ^ , .

We embed X in a convex polyhedron Q, in which there is

a polyhedral neighbourhood Ν and a polyhedral retraction r ;Ν

^ X as above. It is clear from the earlier discussion,

that given 6 >

there is a "v^ ;> 0,. such that if a set A

diameter < Υ| , then Κ Define

=

X has

Ν and diameter r ( / and any

two open intervals 0(x, y) , 0(x', y') are disjoint, where Xj

0- J

y' ζ: Τ » ^ ^

the union of open

1-simplexes 0(x, y), xt C

In this case> yt

·

is Q

This is easy. Actually it is enough to assume

0(x, y) r\0(x', y') = 0 for x, x' e 6" , y, y ' 1

x / x' or y ^ y'

55

That it is true for points of (j" and

and

^

r\

= 0 follow

from this. Ex. 4. U3.

i'ftien ζ- cf

same as

is defined, the faces of

' , where

respectively.

ff ' and

If either

' ^

is a proper face of Ex. k. 1.4» In V ^

are the

faces of

and·^

Γγ'

, then

qf '

. 13

Let ^

and c^ be in ambient vector spaces V

W % iK , let

^

or

S" Τ

= Γ X 0 )< 0 and ^

=0

and W.

«f X ' · Then

is defined. Q

4. t.5. Definition. Let an element of

be a simplicial presentation, and «5~

, Then the link of ζ" in

denoted by ' Lk (

,

is defined as =

is defined, i

Lk (G- , ^ ) - ^

if r* =

Obviously Lk (

,

) is a subpresentation of

In case ζΤ is O-dimensional, we write Lk (x, Lk(

ί

. ) for

) where χ is the unique element in ^ .

Ex. 4.1.6. If Τ ζ Lk(

/Ϊ ), then

LK(CR , 4.1.7. Notation. If

)) = LK( R T , 4). D

is an open simplex, then ·|ςΓ~ j- and ^^

denote the simplicial present'ations of n- and

made up of faces

of (Γ . Ex. 4.1.8. If

= p ^ Lk (

, and dim p ^ 0, then

p , [d^} ) f^ L r ] ) =

will

r]

α

)

56

A..t.9. Definition. Let QL and (j^ be simplicial presentations such that for all if

QL , q;

6"'

or

£) / Γ Τ

defined, andQ-TOC

ff '. Then we say that the join of

is defined, and define the join of QLand (J})

0 and;^

, denoted by QL·^*· (R)

to be the set ί 5" ^ I Γ 6 O t J ft t (β ^ „·

By 4.1.3

(Tor cr may be empty"! but not both. j

^ simplicial presentation.

If 0

is empty, we define Q l * 0 = 0 * 0 ^ = (JL. In case 0 L and (j^ are presentations of polyhedra in V -V/ and W, then we construct, by 4.1.4, QX

and (p which are isomorphic

to 0Land(/6 5 and for which we can define 0 L ·ϊ Q^ are simplicial

f ^^ g '· P,

is simplicial with respect to φ ^ ·ί 0, Q

—1,

p5 : C (Q) — ^ [ 0 , j], the join of ν — > 0, Q

) 1.

Simply speaking, p((l~t)x+ty) = t oC( (1 - t) V + t y) = t , for χ

y e

L^ takes

to

· • The restriction of h^ to

let

Lg* x ) refining

ζ; ^

55-

; the three points

is quasi-linear. For,

a^, a^, χ determine a plane

and in that plane an angi^ar region ^

, which is the union of all

rays from χ through the points of • by definition, takes Pa , a Γ ^ • 1 2-J geometrically obvious, is just

· to L.

standard mistake, X

, which, it is

h^(a^), h^Cag)^.

This, "together with 4.3. 14, enables us to define I=

tr )

C" t A '·

presentation refining ^

, and to see that this is' a simplicial

.

Similarly, via the standard mistake h^ ί | Λ

—^

L,

we have

Λ

2

Since, clearly, h 5 L^

L^ is h^ · h^

and the

composition and inverse of quasi-linear maps are again quasi-linear, the major part of the theorem is proved. The last remark about f is obvious.

Q

4.3. t6. If in 4.3.15, for a subpolyhedron K* of L^, h^K' is polyhedral, then we can arrange for f : L ^

y Lg of the theorem

.1 = h to be such that f | K'

I K. For, all we need to do is to assure that

has a

70

subpresentation covering Κ; then because h is linear on each simplex in K, the resultant 4.3.17. Corollary.

f is identical with h there. D

Links (resp. stars) of χ in Κ exist and

all are all polyhedrally equivalent. 4.3-1 β. Proposition. If

D

f : Ρ — ^ Q is a polyhedral equivalence, ,

then any link of χ in F is polyhedrally equivalent to any link of X in Q. For'triangulate are obviously isomorphic.

f, and look at the simplicial links; they Q

4.3.1?· allows to define the local dimension of polyhedron Κ• at x. This is defined to be the dimension of any star of χ in K. By c

4.3.17 this is well defined. It can be easily seen that (by 4.3.12) the closure of the set of points where the local dimension is ρ is a subpolyhedron of K, for any integer p. We will next consider links and stars in products and joins. Ex. 4.3.19. Let C(P) and C(Q) be cones with vertices ν and w. Let Ζ = (P

(Q)) U (C(P)% Q). Then

(a) C(P)

C(Q) = C(Z), the cone on Ζ with

vertex (v, w) (b) Ρ y. w and ν X. Q are joinable, and (P A w) * (v KQ) is a linkf of (v, w) in C(Z). Hence by straightening our the standard mistake^ we get a polyhedral equivalence Ρ -^ί· Q C^i Z, which extends the canonical maps Ρ

^

71

and Q — ^

νQ.

Hint; It is enough to look at the following 2-dimensional picture for arbitrary ρ t P> q G- QJ

ν

(υ.ν) ih^)

Ex, 4.3.20. Prove that Ρ * Q 4.3.8. If

φ : Ρ * Q — ^ [^0, Q

(C(p) >. Q O P ^ C(q)) utilising is the join of Ρ — >

the equivalence can be chosen so that P>

)

say

so that

)| - ό 6Γ * (Lk( ζ-, Λ )] ί'(Χ)Γ\Ν = 0. Therefore

^ 0. Let T| = min( ^ , € /2). Since Μ is ND (n),

we can obtain an ^

-approximation to

nonde gene rate, g is an ^

f', say g which is

-approximation to f and g(X) Γ\ Ν = 0,

88

g(X)r Μ, Therefore is

an

N D

g(X) C

)j . Hence I St(r ,Λ )|

(n)-spaGe.

Next we establish a sort of "general position" theorem for ND (n)-space. 5.2·. 5. Theorem. Let Μ be an ND (n)-space, Κ a subpolyhedron of Μ of dimension ^ k. Let

f :X

^M

be a map from a polyhedron X

of dimension. ^ η ^ k — 1. Then f csn be approxiniated. by a nicLp Λ

g !X

^ Μ such that g(X) ΓΝ Κ = 0.

Proof! Let D be a (k + 1 )-cell. Let

f' s D

^ Μ be the

composition of the projection D X X — > X and f, that is f'(a, x) = f(x) for a 'r D, X e X. By hypothesis, dim (D ><X) ^ n. I

Hence f

can be approximated by a map g

The dimension of g'-^(K) ^ k. Consider

which is nondegenerate. ΓΓ (g'~^(K)); (where Π" i®

the projection D A X — > D), this has dimension ^ k; hence it cannot be all of the (k + 1 )-dimensional D. Choose some a ^ D Then g' (aV.X)nK - 0 . X ζ; X. Since

iV(g'~^(K)).

We define g by, g(x) = g'(a, x), for

f(x) = f'(a, x), and g' can be chosen to be as close

to f' as we like, we can get a g as close to f as we like.

Ο

We can draw a few corollaries, by applsdng the earlier approximation theorems. Ex. 5.2.6.

If Μ is ND (n), Κ a subpolyhedron of Μ of dimension

^ k, then the pair (M, Μ - Κ) is (η - k - 1)-connected. /"Hint! It is enou^ to consider maps

f s (D, ^ D)

^ (M, Μ - Κ),

and show that such an f is homotopic to a map g by a homotopy which is fixed on ^ D, and with g(D)

Μ ~ K. First, by 5.2.5, one can

89

get a very close approximation g^ to f with since g^ ^^D and f j^D

Μ - K. Then

are very close, there will be a small

homotopy h (3.2.3) in a compact polyhedron in M - Κ with hQ = f

D, h^ = g^ 6 D, Expressing D as the identification space

of b D X. I and D, (a cell with

^ D^ = b D X l) at ^ D X 1 t and patching up h and the equivalent of g on D , we get a map

J g : D — ^ M, with g j^D = f

^ D, g(D)(2 Μ - Κ and g close

to f. Then there will a homotopy of f and g fixed on

DJ7.

As an application this and 5.2.4 we have: 5.2.7. Proposition. If ND (n)-spaGe and Σ- Ζ.

is a simplicial presentation of an , then |LK( ζ" , Λ )

IS (η - dim ^ -2)~

connected. Proof: For by 5.2.4,

IS

|pt(5-,/i)| is |st( ^ , Λ )| - ? j i

ND (η), and by 5.2.6,

(n - dim ΰ" - 1 )-connected, thus giving that St( t5" , ) - ^ (n - dim C - 2)-connected, But Lk( ) is a deformation retract of

st(

A) -

C

is

π

5.3. Characterisations of ND (n)-5paces. We now introduce two more properties: the first an inductively defined local property called A(n) and the second a property of simplicial presentations called B(n) and which is satisfied by the simplicial presentations of ND (n)-spaces.

It turns out

that if Μ is a polyhedron and / 4 a simplicial presentation of M, then Μ is A(n) if and ohly if

is B(n).

Finally, we complete

the circle by showing that A(n)-space have an approxima,tion property

90

which is somewhat stronger than that assumed for ND (n)-spaces. A(n) shows that ND (n) is a local property. B(n) is useful in checking whether a given polyhedron is ND (n) or not. Using these, some more descriptions and properties of ND (n)-spaces can be given.. 5.3.1. Definition

(The property A(n) for polyhedra).

Any polyhedron is A(0). If η ^ 1 , a polyhedron Μ

is A(n) if and only if the

link of every point in Μ is a (n - 2)-connected 5.3.2. Definition.

(The property B(n) for simplicial presentations),

A simplicial presentation every fT

Ji ,

A(n - l),

Lk( f ,

)

is B(n), if and only if for

is (n - dim tT - 2)-connected.

By 5.2.7i we have 5.3.3. Proposition. If Μ is ND (n), then every simplicial presentation

of Μ is B(n).

Q

The next to propositions show that A(n) and B(n) are equivalent (ignoring logical difficulties), /

5.3.4. Proposition. If Μ is A(n,),' then every simplicial presentation of Μ is B(n). Proof! The proof is by induction on n. For η = 0, the B(n) condition says that certain sets are (^ - 2)-conhected,.i.e, any X is B(0), agreeing with the fact that any Μ is A(0). Let n ^ 0, and assume the proposition for m Let|A|c)M,rt4and dim C

n. = k. f

If k = 0, then by the condition A(n), the link of the element of Q"" , which can be taken to be | Lk(

, A)

IS

91

(η - 2)-connected. If k ^ Ο, let X be any point of if . Then a link of χ in Μ is

^ (Γ ^|Lk(

)j / which is A(n-l) by hypothesis.

Hence by inductive hypothesis, it^ simplicial presentation ^ ^ L k ( ^ , y^j ) satisfies B(n-l). If face of ^

is any (k-1 )-dimensional

, = (LkCT , [ ό ψ Lk(

which is ((n-l) - (k-l) - 2)-connected ) is B(n-l). 5.3»

Proposition.

))

i.e. (n-k-2)-connected since

•

If a polyhedron Μ

has a simplicial

presen-

tation which is B(n), then Μ is A(n). Λ Proof: The proof is again by induction.

For η = 0, it is the same

as in the previous case. And assume the proposition to be true for all m < η > 0. Let X ξ Μ.

^hen χ belongs to some simplex ίΤ" of /f ,

and a link of χ in Μ is ^ ^ ^ Lk( ίΤ , Λ ) ^ . We must show that this is an (n - 2)-connected A(n-l), As per connectivity, we note (setting k = dim ^

is a (k - l)-sphere; and by B(n), Lk( tT"» Λ )

connected.

) that is (n-k-2)-

As the join with a (k-l)-sphere rises connectivity by

k, ό Γ * Lk( 4 ) is (n-2)-connected. To prove that ^ ^ * Lk( )1 I enough to show that

Lk(, Λ )=

is A n-l

it is

' say is B(n-1); for

then by induction it would follow that

' = ^

* Lk( €Γ ,Λ ) |

is A(n-l). Take a typical simplex oC

/!'· It is of the form

92

p>V

, β C-

,/

Lk( Γ Λ

possible. Now Lk( ^ ,

), with

Pi or V

^ Lk( p, , {5-Γ] ) ^ Lk( V , Lk(

Let a, b, c be the dimensions of pectively.

a

Lk( p, ,

= 0 being

>.

/J )).

, V' res-

b + c + 1. Remember that dim g" = k. Therefore )

is a (k - b - 2)-sphere.

Lk(V , Lk(tr, /5))| =(Lk( V ^

,

Now^

) I ; and

by B(n)

assumption, this is (n - (c + k + 1) - 2) connected. Hence the join of

Lk(p>,

)

Lk( Y' , Lk( (Γ , Λ )) 1 which is

and

Lk(X , Λ') IS

(n - (c + k + l) - 2) + (k - b -

2) +1

-connected

that is ((n - 1) - a - 2)-connected. Thus

is B(n-l), and therefore by induction )l

a link of χ in Μ is a (n-2)-connected

A(n-1). Hence Μ is A(n).

0

We need the follovdng proposition for the next theorem. 5.3.6. Proposition. Let ^

be a regular presentation of an

A(n)-space Μ and V| be any centering of ^ . Let' A_ be any element of Ip , and dim A ^ k. Then ^ A

is an (n - k - 2)-connected

is a contractible A h-k' Proof; We know that ^ A is the link of a k-simplex in and

Since d ^

satisfies B(n), If k = 0,

Xk

^A

^ ·

is (n-k-2)-connected.

is the link of a point and therefore

A(n-l), since Μ is A(n). k > 0, then ^ A * \/\ A

is a link of a point in M,

93

and so is A(n-1). Take a (k-1 )-simplex ^ of d ^ ^A

is Lk( ^ , d ^δ·^^·^

in

^ A; then

which (by induction on k), we know

to be' a presentation of an A((n-1) - (k-l) - l)-space. To prove that

| ί A ( is A(n-k), we prove that ο A is

B(n-k). Consider its vertex 7|A, then Lk( ^ A

is (n-k-2)-connected.

Lk( CTJ cf A)| simplex Ά

A, ^A) =' ^^ A, and

(t-l); and so

A)( is ((n-k-l) - (t-l) - 2)-connected, i.e.

Q

5.3»7. Theorem. Let Μ be an A(n)-3pace, Y

degenerate. f

k\ being

is ((n-k) - t - 2)-connected. This shows that ζ k

satisfies B(n-k).

dimension . ^

Λ A).

n, and f t X —— ^ Μ Given any

ξ

X polyhedra of

a map such that f Y is non-

0, there is an

^ -approximation g to

such that g is nondegenerate and g | Y = f Y.

Proof; The proof will be by induction on n. If η =

we take

g = f, since any map on a 0-dim.ensional polyhedron is nondegenerate. So assume η η to be true for all m

0, that the proposition with m instead of n.

Without loss of generality we can assume that hedral. Choose simplicial presentations ^ and Μ

Q

f is poly-

j'^TfX of Y?· X

such that f is simplicial with respect to

a n d j

such that the diameter of the star of each simplex in '•'llYC. is less than ζ

. Let β , 'Tj be centerings of

and^A^ with

and

94

f(Q

IT ) - η

εε

C ^ ^

every ^

(f-6") for all

6Λ

. Then clearly is less than ζ· for

and the diameter of \ 0 f .

Consider an arrangement A^

A^ of simplexes of '^Υΐΐ^

so that dim k^ - > dim A. Ι-ίΊ,, for 1 i < k. The crucial fact about such an arrangement is, for each i, (*) }\A, is the union of S A. for some j's less than i. We construct an inductive situation

such that

-1

(1)

...

U|iA.\)

(2) Y. . x.n Y (3) g, : X, - —^ M, a nondegenerate map (4) g. (f-'lcTAj) C l ^ A . j (5) g.

i-1

(6) g. Y. = f Y. 1

η

, , is the union of certain A^ s in the beginning, say A^ s

with i ^ i .

f-^^"") ch""

and

1 Λ " is 0-dimensional.

Hence f f (

···

we take this to be g

all the above properties are satisfied and we

have more than started the induction* that g.

) is already nondegenerate. If

Now let i > i, and suppose

is defined, that is we already have the situation

1-1

It follows from (4) and (5), that for j < i,

aJ ) into

I^A. .

^

.

95

Also this shows that if x ^ f

(^A.l) then both JI

and f(x) are in

, which has diameter 3\

\Sb

96

χ' is of dimension ^ n-1, and contains

f ( ^A^^ ). Let

Y' = f ( ^A^ ). Here the important point to notice is, that Y H x ' C y'. This is because so f(Y Γ\Χ') C

f(Ynx')

f Y is nondegenerate: Y Π x' C

It is also in

I al5A.j. We first extend

g.

\ξ A^

SC^,

and therefore

|;\aJ. Y

to X

and then by conical

extension to Ϊ \ ^ A^ ). g^ ^ maps γ' into ^ A^ and is nondegenerate on Y . Since

is (n-2)-connected, and dim X^n-1,

can be extended to a map f' of x' into \ is also A(n-l). Hence by the inductive hypothesis we can approximate f' by a nondegenerate map f" such that f" γ· . f' Y' =.g • Y'. 1-1 Hence f is nondegenerate and maps , 1 4 j 4s ^ B^ into I ^ · We extend this to a map h^ : ^B^ | — ^ ^ A^ by mapping B. to A. and taking the join . h. 3 1 J is clearly nondegenerate. Since lis I α \ dim Ρ ^n, f ; Ρ — ^ M, Μ

a ND (n)-space, and

if f X is in general position then f X

can be extended to a map

g ;Ρ — Μ

g closely approximates f.

5.4.18.

in general position such that

Conclusion.

Finally, it should be remarked, that the above

'general position' theorems, interesting though they arej are not delicate enough for many applications in manifolds.

For example, one

needs : If f ; X — ^ Μ a map of a polyhedron X .into a manifold, and Y (^M, the approximation g should be such that not only dim (g(X) f^Y) is minimal, but also should have S^XgJ intersect Y »

minimally e.g. if 2x + y

2n, S(g) should not intersect Y at

all. The above procedure does not seem to give such results. If for example we know that Y can be moved by an isotopy of Μ to make its intersections minimal with some subpolyhedra of M, then these delicate theorems can be proved. This is true in the case of manifolds, and we refer to Zeeman's notes for all those theorems.

115

Chapter VI Regular Neipjhbourhoods The theory of regular neighbourhoods in due to J.H.C. Whitehead, and it has proved to be a very important tool in the study of piecewise linear manifolds.

Some of the important features of regular

neighbourhoods, which have proved to be useful in practice can be stated roughly as follows:

^

(1) a second derived neighbourhood is regular

(2) equivalence of two

regular neighbourhoods of the same polyhedron

(3) a regular neighbour-

hood collapses to the polyhedron to which it is regular neighbo\irhood .(4) a regular neighbourhood can be characterised in terms of collapsing. Whitehead's theory as well as its improvement by Zeeman are stated only for manifolds.

Here we try to obtain a workable theory of regular

neighbourhoods in arbitrary polyhedra; our point of view was suggested by M. Cohen, If X is a subpolyhedron of a polyhedron K, we define a regular neighbourhood of X in Ρ to be any subpolyhedron of Κ which is the image of a second derived neighbourhood of X, under a polyhedral equivalence of Κ which is fixed on X. It turns out that this is a polyhedral invariant, and any two regi£Lar neighbourhoods of X in Κ are equivalent by an isotopy which fixed both X and the complement of a common neighbourhood of the two regular neighbourhoods.

To secure

(4) above, we introduce "homogeneous collapsing". Applications to manifolds are scattered over the chapter. These and similar theorems

116

are due especially to Newman, Alexander, Whitehead and Zeeman. 6.1. Isotopy. Let X be a polyhedron and I the standard 1-cell. 6.1.1. Definition.

An isotopy of X in itself is a polyhedral self-

equivalence of X ^ I , which preserves the I-coordinate. That is, if h is the polyhedral equivalence of X 'XI, writing h(x,t) = ^h^ (x,t), h^ (x,t)), we have h2 (x,t) = t. The map of X into itself which takes χ to h^ (x,t) is a polyhedral equivalence of X and we denote this by h . Thus we can write h as h(x,t) = (h^ (x), t). We usually say that

'h is an isotopy between hQ and h^ ' , or 'h^

is isotopic to h^' or

'h is an isotopy from h^ to h^ '. The

composition (as functions) of two isotopics is again an isotopy, and the composition of two functions isotopic to identity is again isotopic to identity, Now we describe a way of constructing isotopies, which is particularly useful in the theory of regiilar neighbourhoods. 6.1.2. Proposition. Let X be the cone on A. Let f : X — ^ X be a polyhedral equivalence, such that f A = id^. Then there is an isotopy h : X X I and

> X .)ς I, such that h \ ( X X O ) y A X I = identity

h^ = f.

Proof; Let X be the cone on A with vertex v, the interval I =[l,o3 is the cone on 1 with vertex 0. Therefore by 4.3.19, X X I is the cone on X X 1 U A Λ I with vertex (v,0). Define

117

by h (x, 1) = (f{x), 1) for X ζ: X h (a, t) = (a, t)

for a t A^ t ^ I. ^

•

Since h | A = id^, h is well defined and is clearly a polyhedral equivalence. . We have h defined on the base of the cone; we extend it radially, by mapping (v, O) to and Identity on (v, 0),

(v, O),, that is we take the join of h

Calling this extension also h, we see that

h is a polyhedral equivalence and is the identity on (A χ I)

(v, O).

Since X X O U A K i C ( A X I ) ^^ (v, O) , h is identity on X y. 0 U A X I, To show that h preserves the I-coordinate, it is enough to check on (X Κ l)

(v., O), and this can be seen for example'

by observing that the t(x, 1) + (l-t) (v,0) of X )( I with reference to the conical representation is the same as the point (t χ + (l-t) v,t) of X X I with reference to the product representation, and writing down the maps,

Q

If h is an isotopy of X in itself j A h A

X, and if

= Id(A Κ l) as in the above case, we say that h leaves A

fixed. And some times, if h is an isotopy between Id^ and h., Λ

we will just say that

I

'h is an isotopy of X', and then an arbitrary

isotopy will be referred to as 'an isotopy of X in itself.

Probably

this is not strictly adhered to in what follows; perhaps.lt will be clear from the context, which is which. From the above proposition, the following well known theorem of Alexander can be deduced!

118

6.1.3. Corollary.

A polyhedral automorphism of an n-cell which is the

identity on the boundary, is isotopic to the identity by an isotopy leaving the boundary fixed,

D

It should be remarked that we are dealing with I-isotopics and these can be generalised as follows: 6. l./t. Definition.

Let J be the cone on Κ with vertex 0, A

J-isotopy of X is a polyhedral equivalence of X X J

which preserves

the J-coordinate. The isotopy is said to be between the map X >> Κ, both induced by the equivalence of

XXJ.

And we can prove as above: 6.1.5. Proposition. Let X be the cone on A, and let f : X ) Κ — ^ X

Κ be a polyhedral equivalence preserving the K-co-

ordinate and such that

f A X Κ = Id. . Then f is isotopic to AK Κ ^

the identity map of X by a J-isotopy h ; X >C J — ^

X

that on A "A J and X X 0, h is the identity map.

•

J, such

This is in particular applicable when J is an n-cell. 6.2.

Centerings, Isotopies and Neighbourhoods of Subpolyhedra. Let ^

be a regular presentation of a polyhedron P, and

let Y , 0 be two centerings of ^

Vjc

Gc

. Then obviously the correspondence

,

c f(p

gives a simplicial isomorphism of d( φ , V^ ) and d(

, θ), which

gives a polyhedral equivalence of P. We denote this by ^QJ» (coming from the map and f Yj

Tk C ' Q 0 fQ

0 C). Clearly f ^ = (f ) • η, 0 θ , rj ΐ ^ where V-j , 0, are three

119

centerings of 6.2.1. Proposition.

The map f

described above is isotopic to

the identity through an isotopy h : P ) < . I — ^ ^ ^(p >

Ρ χ ΐ ί such that if for

and θ are the same on C and all D ^ (p with D < C

then h C X I is identity.^ Proof; First, let us consider the case when Y| and β a single cell A. Then ^ Q St(ηA, d

((p,R]

aj^d f ^^ η

differ only on

is identity except on

ΘΑ, d ((p, θ

))| = |st(

This is a cone,

is identity on its base; then by 6.1.2.· We obtain an

isotopy of 1 St( V^A, d ((p , V] )) , which fixes the base. Hence it will patch up with the identity isotopy of Κ - |(St(V|A, d( φ , The general ^

^

))|.

is the composition of finitely many

of these special cases, and we just compose the isotopies obtained as above in the special cases. For isotopies constructed this way, the second assertion is obvious. . D Let X be a subpolyhedron of a polyhedron P, and let ^ be a simplicial presentation of Ρ

containing a full subpresentation

TO) covering X. We have defined Νφ( Χ») (in 3.1·) as the full subpresentation of d ^

, whose vertices are Y^ C for C

C Π X / 0. This of course depends on a centering Y\ of ^

with , and to

make this explicit we denote it by is usually called a 'second derived neighbourhood of X'. We know that (XJ , V^ )

is a neighbourhood of X, and that X is a deformation

retract of ^ Ν ^ ( Xj , ^ ) | (see 3.1). Our next aim is to show that any two second derived neighbourhoods of X in Ρ are equivalent by

120

an isotopy of Ρ leaving X, and a complement of a neighbourhood of both fixed. We go through a few preliminaries first. Ex. 6.2.2. With the same notation as above. Let V^ and ^ centerings of ^ C m

such that for every C

be two

> with

^ 0, η C = θ C. Then ί Ν φ ( ? ς , η )( =

(Λ.

) .

/"Hint! This can be seen for example by taking subdivisions of which are almost the same as d((p ,

) and d( (p , θ ), but leave

unalteredJ7. 6.2.3. Proposition. two centerings of φ

With X, P,}Cj , (p as above, let , and XJ

and 6 be

the union of all elements of ^

, whose

closure intersects X. Then there is an isotopy h of Ρ fixed on X and P - U

, such that h^ ( | Ν φ ( X,, Vj ) |) = j N(p ( Χ , 6 ) |

Proof; We first observe that P - U is a full subpresentation (J\. of ^

is a subpolyhedron of Ρ and there which covers P - U , namely, C

if and only if G Π X = 0. By 6.2.2 we can change Yj and and Q l without altering

Νρ (

may assume that V| and θ h

; Y| )

and

are the same on

of proposition 6.2.1 with the new f

desired properties,

.

on

Ql. ^

© ) . So we and (FL . The isotopy

^^ in the hypothesis has the

θ,η

D

With X, P, ")(Q Ί (P

as above, let CD t Ρ — ^

"ο, ϊ"

be

map given by: if ν is a ^-vertex φ ί ν ) = 0, if ν is a (φ _ X, )vertex- Cip(v) = 1, and CD is linear on the closures of ^ Then

cp~\o) = X, since ^

( p - %^ , then

X

is full in (p i If

0, if and only if

-simplexes.

is a simplex of

φ ( 1Γ) = (0, 1). If

121

^ s a simplex of (Jt ( CFl- as in the proof of proposition 2.3) then φ ( 6") = I· Roughly, the map ( χ ' , η ' ) | , and i f therefore

f . ρ.

χ'6 f(X),

) = f. ρ (f"Vx')) = f ·

) tX,

) = x'.

Q

6.3.4. Notation. If A is a subset of a polyhedron P, we will denote by int ρ Μ and bd ^ Ν the interior and the boundary of Ν in the (unique) topology of P. Ex. 6.3.6. If Ν is a regular neighbourhood of X in P, and Β = bd ρ N, then Χ ζ ^ Ν - Β .

Q

Ex. 6.3.7. Let X C n C ^ Q ^ P

be polyhedra, vdth Ν ζ"int ρ Q. Then

Ν . is a regular neighbourhood of X in Q if and only if Ν is a regular neighbourhood of X in P. Β Ex. 6.3.8. Let Χζ_Ρ

be polyhedra.

If A is any subpolyhedron of

P, let A' denote the polyhedron A - intp X. Then Ν is a regular neighbourhood of X in Ρ if and only if N' is a regular neighbourhood of X' in P'. α Ex. 6.3.9. Let A be any polyhedron, and I the standard 1-cell. Let 0 < eC < p) < y


[_0, Q Let ^

both in ^

covering X; and

be the usual map. Take Ν -

denote all the simplexes of (p having vertices

and ( φ - % ) .

number of elements of ^

We prove Ν ^s/X by induction on the

. If

χ

= 0, then Ν = X, and there is

nothing to do. Hence we can start the induction. LetJft

=X

0

υ

ι ^

Then "Tt. is a regular presentation of N. If of maximal dimension, attaching membrane

^ Π φ ς;"·

principal simplex of φ

Vg)

is an element of

is a free edge of "Tt with

((O,

(Mote that CT is a

i.e. not the face of any other simplex).

After doing the elementary collapse we are left with

-OHC'· Now

(p' is a regular presentation containing the corresponding so, -Tt

^ .

= ^

-

, and

^ . Hence inductively

α

Ex. 6.6.9. Let N' be a neighbourhood of X in P, (all polyhedra). If Ν

V X, then there is a regular neighbourhood

M([_Int ρ n' such that N. 6.7. Homogeneous Collapsing. Let ^

Ν of X in P,

D

be a regular presentation, with E, A ^ffi , Ε < A

and dim A = dim Ε + 1. Recall the definition of ^

φ) E. This is defined.

145^

relative to some centering

of ^

, to be the fiill subpresentation

of d (p whose vertices are 6.7.1.

\® ^^

Definition. Let E, A, ^ ; be as above and

V^ be a

centering of (p . (E, A) is said to be homogenous in (p , if there is a polyhedron X and a polyhedral equivalence a suspension of X, such that

f(

f:

^X

A) = w.

It is easily seen that if this is true for one centering of

, then it is true for any other centering of

"(E, A) is homogeneous in ^

; hence

" is well defined.

6.7.2. Definition. Let % C

subpresentationsof ^

. We say

that "ATT collapses to 'Xa homogeneously (comblnatorially) in (j^ , if there is a finite sequence of subpresentations of (p ,

and pairs of cells (E^, A ^

, E., A. ^ ^ t ·

such that 1) Jit, =

, -^rt =

' 2)

. k is obtained from

i+1

by an elementary i

collapse at E^^ a free edge of "tc . with attaching membrane A^, for i = 1, and

3) (E^, A^) is homogeneous in φ) , for i = 1,.., k-lj

6.7.3. Proposition.

If

(p*is obtained from

cell C by a bisection of space with

obtained from

,

^

by bisecting a

(L; H+ , H-); and if X C ^ ^ CL(p >

by an elementary collapse at a free edge

Ε with attaching membrane A, where and if

k-1

(E^ A) is homogenous in ("b ;

are the subpresentations of

^

covering

146

and

; then

homogeneously in

^

.

Proof; If the bisection in trivial there is nothing to prove. If it is not trivial, there are three cases as in the proof of proposition 6.6^3· Case 1; C is neither Ε nor Α., In this case the only problem is to show that (E, A) is homogeneous in (p

. Let us suppose that

everything is occuring in a vector space V of dim n; and let dim Ε = k. Then there is an orthogonal linear manifold Μ of dimension (n-k), intersecting Ε in a single point

(E) = e, say..

It is fairly easy to verify that in such a situation if

If we now choose centerings of Q

D,. then

and

so that

whenever D fV Μ / 0,. we have the center of D belonging to M, then defining D A M D r^M ^ 0, D and

and

similarly with respect to

,

^

·

, we will have:

are regular presentations of

λ(ρ (Ε)

and

ί\|ρΐ(Ε)| are links of e in

hence polyhedrally equivalent

Γ\ M. Hence both (p Π

and

(by an approximation to the standard

mistake); if we choose a center of A the same in both case, we get a polyhedral equivalence taking hypothesis

/\Π(Ε)

A to

Y^ A.

Finally, by

is equivalent to a suspension with Y^ A as

a pole; and so

Λφ'(Ε)

homogenous in

'

has the same property, and (E, A) is

\hl

Case 2; C = Ε ; we define E^, E^, F as in the proof of 6.6.3· We have to show that

(E^, Λ) and

(F, E^) are homogeneous in

(p '·

I That fact

(Ε)

any D ^E^ ^

(E^, A) is homogeneous in

^

follows from the

appropriate centerings) because φ

is an element of Q) which is

and hence,

being regular ^ E. That

(F, E^) . is homogeneous in

, we see by the

formula:

(calling the appropriate centering of Case 3: C = A ; we define have to show that

^ ' also Y ).

A^, A^, Β as in the proof of 6.6.3· We

(E, A^) and

(B, A^) are homogenous.

There is a simplicial isomorphism taking Y" (A) onto we have

Γ|(Α^).

vAp' (E)

And as (E, A) is homogeneous in φ ,

(E, A^) is homogeneous in That

^^^

^

.

(B, k^) is homogeneous in

^

we see by a formula

like that in case 2:

(Β) - /\(p(A) 6.7· 4. Proposition.

If --ίγ^ ^ ^

k^, η A^

α

homogeneously in (p , and

Ο

is obtained from ^^ by a finite sequence of bisections of space, and are the subpresentations of then

homogeneously in

^

(p

covering (vJK | and \ Xj ,

.

This follows from 6.7.3, as 6,6.4 from 6.6.3· 6.7.5.

Definition.

O-

Let Ρ be a polyhedron, and X, Ν subpolyhedra

148

of P. Ν is said to collapse homogeneously (geometrically) to X in P, if there are regular presentation

C. "Tt.

covering

X, Ν and Ρ respectively such that ^ ^ collapses homogeneously to combinatorially in ^^ We write

X

homogeneously in P. This definition is

again polyhedrally invariant: 6.-7.6, Proposition.. If Ν

X homogeneously in P, and dC '· Ρ — ^ Q

is a polyhedral equivalence, then

(N) — ^

homogeneously in

Q. This follows from 6.7Λ as 6.6.6 from 6.6.4. 6.7.7. Proposition. If Ν is a regular neighbourhood of X in P, then Ν

X homogeneously in P.

Proof; As in 6.6.8j we start with a simplicial presentation φ Ρ

of

in which a full subpresentation 'Xj , covers X, and take

Ν

where C^ : Ρ — ^ [o, 1

is the. usual map. By .

virtue of 6.7.6, and the definition of regular neighbourhood, it is enough to prove that this Ν V X homogeneously. Let "-^IX be the regular presentation of Ν consisting of cells of the form: simplexes of k)), for ···' ^k ' ^

>

k that t^ ^ 0, 2 Γ t i =1, g 4

and the point is; k ^

1

i, 1 is uniquely determined by

· 1

4

153

Γ On the otherhand, a simplex of of a simplex determined by some L vdtha=ηQ,

(a)

Q

( )

.

is a face

between 0 and k, and vertices

η .

< Γ 2 ···

A typical point in the closure of such a simplex is uniquely determined by ϊ^Ο

'

'•^here r^, s. ^ 0 and

k Σ1 ° («0

I'i + A

S. '

t

"i η

1. The point is k

1 ·

Ί1

(έ η ^ - i V).

Comparing coefficients in (*) and

we find that these

points coincide if: (A)

^

- 1- έ ^

3

Τ t, -li «c t;

, i < L.

= ''i·' h

tj - έ !j. (B)

r.

'I

Sj

> j>

It., i y 0, Q

Remark»

If

is a set of verticer of (p , we have as above,

c^^CMpi ^

^^

%

( r ) X έ, 1

Now let Ρ = X * u, w h be a suspension. We define the lower hemisphere L of Ρ to be X

u

j it should be, remarked

that L is a regular neighboiorhood of u in P. 6.8.4. Proposition. With P, X, L as above there is a polyhedral equivalence h : C(P)

h fLY,

(T

1 ] U

^ C(P)

with h Ρ ^ idp , such that

Ci (P)) = L A [4, l].

Proof; We, can draw a picture which is a "cross section" through any particular point χ in X:

155

/~The picture is actually the union of the two triangles Qc, V,

v, w and

in C(P), which we have

flattened out to put in a planar picture. The vertically shaded part is the portion of Ly,^^,

in

the cross section and the horizontally shaded part is the part of Ci (P) ' in the cross section. We have to push the union of these two into the vertically shaded portion, and this uniformly over all cross sections_7. From this picture we may see the following: (^P) is the union of A=

[έ,

Β = I^X And

A Π Β = ^X

*

wj , and

s]·^^ J where J = ν ^^ [u, wj . |^ *

[

'

Now J is just, polyhedrally, an interval, and so there is obviously a polyhedral equivalence

f ;J— ^ J

such that

f (u) = u, f (w) = w f(|v+|w)=|v+|u. Such an f will take the part j^u, vj ^ j^v, g ν + g w"^ onto [u, i u + έ vj. Let g ,ί Β — ^ Β be the join of f on J and identity on Χ Χ 2. It is clear that g A Γ\ Β is the identity map, and so by extending by identity on A, we get a polyhedral equivalence

156

say h : C(P)

^ C(P).

It should be pictorially evident that h has the desired properties.

Q.

Putting all these together we get the proposition which we need! 6.8.5. Proposition. Hypotheses;

(l) φ) is a simplicial presentation of P, C(P) the

cone over Ρ with vertex v, C((p) = (p set of vertices of (p

{[v}} , and

χ;

a

.

(2) There is a polyhedron X and a polyhedral equivalence h ; Ρ

> X ^^ -^u,

such that h(

)= Y

u, for

some Y C (3)

and

with reference to some centering'^"

(P ),ίΣ)

of C( (T^ ).

Conclusion; There is a polyhedral equivalence such that

X

Ρ = id ρ

and

are constructed

; C(P)

maps C((p )

onto

( i C(

be the centering of C( (p ) described in proposition f= f ^

onto d(C(^

X

simplicial isomorphism of d(C((p

/> )

).

Let h^ : C(P) h JΡ

υ (ί.

.

Proof: Let 6.8.3. Let

^ C(P)

^ . C(X *

wj) be the join of

f u, w]· and the map vertex to vertex. _J in

Now

Οφ^"^'

P, and therefore h f( h f(

|(pj. I

) in

is a regular neighbourhood of ) is a reg-ular neighbourhood - in w}. But f( )= iP^

fact- f, maps -every ^ -simplex onto itself - and h( ^(p^ ) - Y * u.

157

Thus h f( (^φ('Σΐ) ) is a regular neighbourhood of Y X ^^ ^u, wj . Therefore by 6.8.2, h f( neighbourhood of u

u in

) is a regular

in X * -^u, w j > But so is X * u. Hence there

is a polyhedral equivalence ^

:X

' u, w^ — ^

X

•

such

that Ρ Let

(h f(

j2> ^ : C(X * ^u, w^ )

)) = X ^^ u. C(X

{u, w^ ) be the join of

and identity map of the vertex of the cone. Now f is such that fi C(x ^ {u, wj ) with y j X * [u, wj =

identity and

y

^ ( X ^^ u )

l] = (X

U

c ^ (X ^ [ u , w ]

u)

Q .

The desired map J^ is now, Χ

=

0 h"^ 0

ογ

0

0 h^ 0 f

.D

We v/ill now write down two specific corollaries of proposition 6.8.5, which will immediately give the regular neighbourhood theorem. First we recall the notation at the end of section

3 (6.3.10). If ^

is regular presentation, given a centering "Tj of^p

and a centering of

,Υγ ), we defined c·»^ =

and, If

of and in

, for any C

= U -[ c* C

for any subset

of (p .

is a subpresentation of J) , then d(-Jlft , η ) (where Y] Jy^

is again denoted by ,

0 d(p

) is full in d( φ , rj ). Writing

=

rj )

(p' = ά(φ

(p'

of the form

i(p'(

Σ ) = /It'

η

), and £

C, for

, which

is

C

as the set of vertices

, we see that

a regular

neighbourhood

^ of

| ^^

j

IP

6.8.6. Corollary. Let ^ presentation

be a regular presentation with a sub-

, Ε a free edge of -^f^ with: attaching membrane A

such that (E, A) is homogeneous in φ) . Then there is a polyhedral equivalence h ·= \(p Ε * \λ^Ε

— > |(p

and which takes

which is identity outside of onto ( ^

·τ

.

/~Note; It is understood that there is a centering V^ a centering of d{(p

,Y| )J7.

Proof; Look at St(

E, d

of Ε * the form

i) Ε

A (P Ε

Let ^

); this is a presentation say

λ

(Ρ

(P

denote the set of vertices of d (p of

F for F < Ε and

to ^ A. Since

and

tj A.

Then

is the join of

is equivalent to a·suspension

'

159

(homogenity of (Ε, A)) with

\λφ Ε

A going to a pole,, we see that

is equivalent to a suspension with

going to a subcone. And Ε

Ε

Ρ

And consider the centering of

(p

IS a cone over

0Ε

Ε

•if·

coming from that of d( (p, η ).

'Thus we have the situation of 6.8.5, and making the necessarysubstitutions in 6.8. 5, we get a polyhedral equivalence ζ/^ of = 0(p'( Σ

[ λ φ Ε I taking ).

ίφ ( ^ >) j

0

(£

) onto

^^φ'ί ^ E)j is just E*. Now observe that the

set of centres of elements of 'iy)· in

φ'

is

Ε "

(This is where we use the fact that Ε is a free edge). Therefore ^ ^

Λ|(Ρ Ί =

U (Γ(Ρ'(Ζ) and (-iyt-lEj)^

So (j^ takes the part of (Jy^"^^) in ('"Vt^ - I E }

in

.

φ'

by patching up with identity outside onto .= Ε

·

Qorollary 6,8t7i

to an. equivalence h of φ>

we see that h

and is identity outside Β

In the same situation, there is a polyhedral

equivalence h' : | (p ^ — ^ A*

onto the part of

is Identity on the base of the cone,

and E'"^ C |(P Ί · Therefore extending

takes

=

(p | which is identity outside

λ ^ Aj , which takes

{e] ^^ onto

- {e, aJ

/ffor this corollary we need only that Ε is a free edge of A, and A is the attaching membrane. Homogenity of (E, A) is not necessary_7. I Proof; This time we call

φ

= St(

A, d

of vertices Yj F, F < A and' F / E. Then is equivalent to a suspension^ with

and

the set

(p^ = "^A - E.

0Α

^ A ~ Ε as the lower hemisphere.

160

Hence

^Α -!ί·

is equivalent to a suspension with

(p^

mapping onto a subcone. J Appljring 6.8.5, we get a polyhedral equivalence

of A ^^

^

1

on itself, which is identity on

^akes

έφ'( Γ ) U A"" onto

X).

Ε is a free edge and A is principal in·-^,

Σΐ) is just

the part of (At - [E, A])'' in A ^^

L^Xp A , and

13 the part of ( - T i l - i n

Ι ^ φ A . Extending

an equivalence h' of |(p A -ίί-

Ti *

to

by patching up with identity outside

^ ) since A'^ is contained in A •«·

h' takes

Since

onto

^ ^ A^

we see that

-[E, A^ )* and is identity outside

Q Thus in the situation of

6 .8.6,

if we take the composition

h' h of the equivalences given by 6.8.6 and 6.8.7, h' ο h takes onto (-^-{E, A]·)^ of h C ®

Λ(Ρ AJ )

Support of h'C ^ ^^^

hence h' ο h fixes, the polyhedron

. This at once gives,

Proposition. If

homogeneously in ^

a polyhedral equivalence of φ takes J ^ * 6.8.9.

support

onto

Corollary;

, then there is

, which is identity on | X^

and

Q If Ν

X homogeneously in P, then any regular

neighbourhood of Ν in Ρ is a regular neighbourhood of X in P. CV Proof of the regular neighbourhood theorem 6.8.1. By 6.8,9 any regular neighbourhood say N' of N is a regular neighbourhood of X. Since Ν is bicoUared in P, there is a polyhedral equivalence h of Ρ taking Ν onto N'. Since XQintp N, h can be chosen to be fixed on X (see 6.4.8). Therefore Ν is a regular neighbourhood of X.

D

161

6.9. Some applications and remarks. In this section we make a few observations about the previous concepts in the context of PL-manifolds 6.9.1. Let Μ be a PL-manifold, ^ Μ its boundary, (p a regular presentation of M. Let E,

ζ (p

(pj^^

, combinatorially, and

1 < i

^ D = D Λ Ji

a locally un-knotted pair

- . . Then if

Δ

\

^

point,

the pair (/i^, D) is an absolute regular neighbourhood of a point (relative to ( b A equivalent to (S ( Δ.»

. ^ D)

and so ( A j D) is polyhedrally

D, D) where S is a (b-a-1)-sphere, i.e.

is un-knotted. /"This is a key lemma for Zeeman's theorem, that

(b - a)

^^^^^^ ( Δ 3

is un-knotted.

combinatorial Topology", Chapter IV, pp.

See Zemman "Seminar on

168

Chapter VII Regular collapsing and applications 7- 1. Let

be a simplicial presentation.

outer edge of then

^

, if there is a

, and dim Δ

)> dim

having ^

. In this case =f

, η;-

are of the form

in otherwords they consist of

is a subpresentation of ^

Let dim

,

·

, and

n. Then, we say that

is obtained from and

«

If /Λ =

/ from

τ' ^ Τ •

CT

by an elementary regular collapse (n) with outer edge ma.ior simplex

ί ^^L·

0. The elements

Cf' τ''

^

is an

is uniquely

as a face are exactly of the form

The remaining faces of

(Γ

Z^ /f , such that if cT^

fixed by ς— , and is of the form of

We say that

=

J and

is obtained

X _ by an elementary reg\ilar collapse (n), we say that /o

regularly collapses (n) to ^ ^ . The elements of the theory of regular collapsing can be approached from the point of view of "steller subdivisions" ( c.C«f. Section 13 of "simplicial spaces, nuclei and mr-groups" or the first few pages of Zeeman's "unknotting spheres", Annals of Mathematics, 72, (1960) 350-361), but for the sake of novelty we shall do something else.

169

7.1.1. Recalling: .Notations. simplexes. (5r ί Τ and ^ ζ- >

^

^

, usually denote open

' · · denote their closures (closed simplexes),

J .. their boundaries. The simplicial presentation,

of ζΤ consisting of ^ of

>^

and its faces is denoted by

consisting of faces of C

by' ^C~|(see 4.0·

for the join of the two open simplexes defined.

If

and

and that Τ stands

, when the join is

is a 0-simplex and χ is the unique point of

we will write jx^

^

for ς- C^ . On the other hand the join of two

polyhedra Ρ and Q when it is defined is denoted by Ρ ^^ Q. Similarly the join of two simplicial presentations φ ) and it is defined is denoted by (p * (3^ . For example if g" defined, then

when is

is the canonical simplicial presentation

of the polyhedron

^ 5~

Τ . If Ρ is a polyhedron consisting of

a single point x, we will sometimes write χ * Q instead of Ρ * Q. With this notation {^Χ^ΙΓ" and χ Let A

^

be an (n-1 )-simplex, I = [o, l"^, and let J^ be

a simplicial presentation of Ρ t

are the same.

such that the projection

— ^ -Δ i® simplicial with reference to The n-simplexes of

face

(\

That is Δ

intersects the

^ is a face of T^ ^, Ή ^ has another

which maps onto Δ. ,

is a face of "P j · • · · j ^ . 1 2 1—1 has another face that maps onto

» is a face of

Λ,

, but T^ i i call it Δ ^ and so on. We start with with Λ k =

.

can be ordered as follows:

T ^ 1 ^•••'"P^ J so that if Χζ-Δ» X in order.

and " Δ

Δ

1.

= Δ

° ^^^

^P

170

Let us write Τ = Λ Κ Ο y ( ^ be just ^

= ς- ^ in some way. Let ^ ) % 1. (If

=

should be taken to

7\ 0). Then there is a subpresentation ^ ^

of

which

covers T, 7.1.2, Lemma. (With the above hypotheses and notation) collapses (n) to ^

regularly

.

Proof; We in fact show that there is a sequence of regular collapses with major simplexes

XT' , We must then define 1

k S

.... u i r , ^

and find some outer edge lying on ^ .j so that the corresponding regiilar collapse results in / ^ i-1 Now

^ is an n-simplex and its projection

is an

(n-1)-simplex, therefore there are two vertices v^ and v^ of

^

(choose v^ , V2 so that the I-co-ordinate of v^ is

the

I-co-ordinate of v^) which map into one vertex ν of Δ

. Now

l\ = ^ ^ and so ν - is a vertex of either ζ"" or "T" , Case 1; ν is a vertex of

be the faces of

. lying over

and the two faces of r and Define

, Write

and

. Then

| ' . which are mapped onto Δ ,

^v^j

tT^ = [v^J

= {v\ ζ"^, Let ^J

I --vf

,

^

- Aj^

·

. Τ^^Ι^,^

%

are

and^

171

It is claimed that if we take ^

_ as an outer edge then the result

of the elementary regular collapse with major simplex

^CT

^ in

cannot be in ^

)< I which is in ^

a vertex of

C., 1

; because the only (dim

is

j ^ind

/

since v^ is

Also Ίρ . is the only simplex among 1 *

which contains v^ as a vertex. Hence if We then have to show that

The first term here is

"p.

^ p^j then A/'^il

)-simplex

ί · · j T^. I 1 ΤΓΛ ^^ ^ * i'

~

. > which is where "P. intersects Ρ

υ., or,.

172

The second term written slightly differently is

v^ v^

ψ

to which we may add a part of the first term namely ("^ΓΓ^Γ*) to obtain all faces of J*^ ^ which map to == ^ {[v] r M

Τ

- ( ^ ^ ^) υ

Τ L^

^

* v^

In other words, this This shows that

and so edge

i

^^

i 1

elementary regular collapse with outer

and major simplex t

Case 2; ν is a vertex of ff . Write to be faces of "P ^ lying over ^

ν

and

, define In this case

and the two faces of "P ^ which are mapped on Δ

and [v^} r

Τ

-

^

are

Δ -

We now define

and make computations as before. cl ^ i ^^ T i ^^ —

i

U K

- Δι., υ

w

-

( V2 ^^ i Γ

/

δ Γ - { j , - Ο *

χΛ

^^ T7

'Τ

1

'

'^t , 'X'

173

and

^ [y^ V2I ''

^

^

[( Sr^i-'T)

TU ^ ^ τ

jjJJJ^^ \s>

r Ίβ

And this shows that if we perform an elementary regular collapse (n) on Λ we get / %

g" . and major simplex "P.}

. Hence Define

and Τ = iC

. with outer edge

regularly collapses (n) = I,

to Κ I, T^ = 0 C

=

0) U (T X iC"· t

D Q

.

α

k

I t is easy to see that T^^ is a (k-l)-cell in Q I , k and is the set o f points of is zero.

I

at least one co-ordinate of which

^et

^ :

=

V

Τ —>

^ b^- the projection.

^.1.3. LcHFia. L't Jj^ ' /f

' · · · ' /q simplicial presenΊ >•··, I vith rf^sp'ct to which all the maps

tations of l", l''^ c/j

ί. . . , Χοί

% n

> ^ n - 1 '··' 2JI

and such that

firr simolicial.

Then there exist subpresentations

covering T^,

T, respectively,

^ regularly collapses (i) to ^

Proof; The proof is by induction.

. for all i.

It is easily verified that

^

collapses (1) to ^ ^, So, inductively, we know that for i < n-1. Now ^ ^ ^ covering y^ 0 ^

. collapses (i) to ^ ^ ,

is just the subpresentation of

η I Vs I = Τ . n-1 1 η Let the collapsing of X to ^ occur along the / 0 n_ 1 n-1 major simplexes 3 · · • 3 Δ. · Then we define ^ k

and write

^ ' where ^ is the outer edge of i i i i the regalar collapse (n-l) from to Q ^ . Then Ai

nlOVi.il Define 0 plus ^

=

• ^ = the subpresentation of ^ 1 ). Thus β

,- ^

covering and

= ^

We mil shovr that (j^ _ regularly collapses (n) to (j^

, stringing these together, then

regularly collapses (n)

175

To show that (J^ . rei?;ularly coll-'^pses (n) to (j^ it i ^ i +1 is enoufih to look at the part of

. covering J^

Δ,χι. A. X I and cL ^

( Δ . ) i·®.

Λ,Α 0 U[c 6 T i - T . ^ i ]

Aj_ X ^

^^^ projection

Δ .X ^ — ^

is simplicial with reference to the subpresentation of ^ ^ ^ I and

·^·

for this situation.

Δ^

which

^ covering

leama 7.1.2 is especially tailored

Q

7-1.4. Theorem. Let A be a n-cell, Β an n-cell in

A, and |p

a regular presentation of A. Then there is a simplicial presentation refinin^:^ ^

, with a subpresentation ^

^regularly collapses (n) to ^

covering B, such that

.

Proof; There is a polyhedral equivalence

h ίA

^ l", with

h (B) ^ t". Then h is simplicial with reference to some φ ^ and , where ^ η

Χ

I

can be assumed to refine

η

, n-1

> 1

. The diagram



this is the

^ which 'could contain points in ^ E^. Let s and t

be the centres of

^^

and

C^ ., the line segment

prolonged a little bit (here we use the fact that A

[s, t^

can be

is standard) to

V and w in ί\ , so that

(^[v, w ]

^ t''

( i)

a Τ J

"^ ^

δ

Γλ X c ^

^^

(Λ

c)T.

boundary

(here we use the fact th 't if L C\ (c" ii- K^C L Γ\Κ, where

is a I

simplex and K, L arf polyhedra, then there is a stretching i.r.

containing- ^

we have in order f of

s, t,

w^ , taking ν to

to the identity on outside of

such that L Γ\

jV, w^

^

^ δ ^^ ^^

^^ K ^ Q K Γ\ΐ). Thus

and there is a polyhedral equivalence v, s to t and v; to w. Join f c) Tj " Σ. "

extend by identity 5

to the the unper ofisotopy and takes th- upper boundary of is resultboundary of a nice

^i .

179

The composition of the h^, will then take the upper boundary of

= E^ to the uppT boundary of

7.1.7· Kem?rk; In theorem 7.1.0, PL-manifold. Of course Ex. 7. '.8»

^ = E^ . Q

can be replaced by any

D should be in the interior. Q

If Ν and Μ are two PL-manifolds and f: Ν > that is

Hence by induction

^ and

^ ^ coincide for all ±, and ^

is well

defined.

k.U Proof: Suppose Either

6

T. C

2|)·

There are four possibilities:

Β

or

(2)

or

(3) T i C Β - ^B

or

(Z.) T i C A - ^ A.

Q^ Β

Φ

By A.2 in cases (2) and (3)

•

) is obtained from

'^y perforations and regular collapses of dimension (n-dim

- l). In cases (l) and ω

notation of A.2) is obtained from M X , Λ

{Ac^.^) (with the ) by perforations and

regular collapses of dimension (n-dim ^ ^ - 0·

By A. 3 and remarks

thereafter, there cannot be any perforations in cases (l) and (2) and there is exactly one perforation in cases (3) and (4). By A.I; b) in the collapse at outer edge ^ ^ simplex ^ ^ =

^ . , what happens to Lk( T.

and major exactly a

189

perforation of dLmension (n-dim TiC

- Β or T ^

- 0·

So straightaway -WP. have

is impossible.

So, the only possibilities that remain are (3) and {h). Let us consider case (4) first.

We claim that if

the one perforation on LkC"^^,

) is already made. Since

is a sphere, any (n-1)-simplex of face of two n-simplexes. So For the same reason, ^ Lk( T i , Ji^) =

^ ([_

having 'X . as a face must be the

. cannot be in

^ cannot be in any ^ ., /i^). Thus

Lk( Ti'/^i) ^ i'^'^f i

with

Qd

implies

Suppose Lk(

) is changed for the

first time in the k.^*^ collapse, k. 1

that is Lk(

1

,^ ) = ^ i

is a sphere; this operation from

M^.,

yi) = MX.,

i 1 is necessarily a perforation.

Lk(

A, ) to i k^ So the one perforation

i on L k C ' ^ ^ , i s already made. But in the i^^ collapse also what happens to Lk(

j^) is a perforation since

A. 1.b). Since this is impossible

^^

~^ ^ ^

cannot be in A _ ^ A.

Let us consider the remaining possibility (3), ^ ^ C ^ -"bB. ' LkCc^^,/^)^ is an i-cell with boundary Τ iC

Lk(IT' k k k-1

g.nd bases of these groups. This is ionr in

)

k-1 k-1 k-2

·~.'-ί.

(3) Inserting; cancelling p-^irs of handles, the opposite to ( 1 ) is sometim'^'s necessary in order to simplify th·· algebrai c structure; this occurs in δ. 6. This, togctaer ivith (l) and (2), alloviS us to prove Theorem A for J?, = 1 , -t the -χρ^η?' of "xtra 3-handles, Once we have done this, there are no more Knotty rroup-theoretic difficulties, and the universal covering soaces of the Λ. s are all emb-dded in 1

each other. Then vre can take a closer look at: {U) The algebraic structure. This consists of the bound^.ry m'-^ps ΓΓ (A ,Α j Γ Γ ,A ). k k k-1 k-Γ k-Γ k-1

Whp-n there ar·^ no 1-handles,

these groups are free modules over the fundamental-croup-ring, with bas^s determined, upto mulLiplyine by ± FT, by the handles. We can

202

change bases in certain prescribed ways by inserting and csncelling pairs of hcndles.

This allows us to set up a situation vrhere a

(k-O-handle and a k-handle algebraically cancel. We discuss this in 8,9.

'-^d now, both Theorems

and 3 follow if we can get algebraically

cancelling handles to cancel in the real ^eoinetric sense. This amounts to getting an isotopy out of a homotopy of attaching spheres; this is, of course, the whole point; all the oth'^r techniques are a simple translation to handle piOsentations of the theory of simple homotopy types of J.H. C. Vi/hitehead. (5) The isotopy lemma. This is the point where all dimensional restrictions really maiiP themselves felt. The delicate case, which applies to (n-3)- and (n-i!i)-handles, just barely squeaks by. 8.3·4. Thr s-cobordism thforem. By an s-cobordism is meant a triple (M;A,B), where Μ is an n-manifold:

and Ε are disjoint

(n-1 )-6ubmanifolds of ^ M; jj i^i - Ay Β is polyhedrally eauivalent to ^ A

I in such a way that δ A corresponds to

course, ^ B and

to ^A

l);

A

Μ

A X 0 (and, of

and Β C^ Μ are homotopy equivalences;

'Τ(Μ,Α) - 0. A trivial cobordism is

triple

(M;A,B) equivalent to

(A X I; A V. 0, A X 1). Theorem.

If (M;n,B) is an s-cobord'.sm, and dim Μ ^ 6, then

(M;A,B) is a trivial cobordism. Proof! This follows from theorem C. The rair (M,A) is a relative manifold, sp^cicU. cas·-.; and -iill the hyiDotheses of Theorem C are clearly valid; in particular, Π (Β) Γί I II M-A) ~ Π" (M) , since Β ς Μ

is a

203

homotopy· equivalence. Hence, by thporem C, (Μ,^'Ι·) is equivalent to (Αχ I, A XO).

Me know, by assumption, ta^t ^ Μ - A \J Β is a

regular neighbourhood of

^ ·'>· in ^ Μ - a; anl clearly c) A )C I is a

regular neighbourhood of ^ A X 0 in

5 C^»· X I) -

X

Thus we can

fix up the eqioivalence of (Μ,η) to (Αχ I, A X 0) to take 3 M-^ \J Β onto ^ A X I; this leaves Β to map onto A

i, which shows the

cobordism is trivial. D "We remark that if VC^ik) is trivial, then T(M,A) = 0 automatically. It is with this hypothesis that Smale originally proved his theorem; various people (Mazur and Bardon) noticed that the hypothesis needed in th-^ non-simply-connected case, was just that A Q Μ he a simple homotopy equivalence (vfhence the "ε"); i.e. ^(M,A) = 0. 8.3.5. Zeeman's unknotting theorem. We have already described the notion of an unknotted sphere. Theorem. If A(3. B, where A is a k-sphere, Β an n-sphere, and k ^ n-3, then A is unknotted in 3. Proof; By induction on n. For η < 5, th^^ c'.ses are all quite trivial, except for k = 2, η ρ 5, which has been treated earlier.

For η ^ β

•'ffz will show that tlie pair (B, A) is equivalent to the suspension of (B , A ) where Β

is an (n-1)-sphere and A

a (k—1)—sphere; and

clearly the suspension of an unknotted pair of spheres is unknotted. To desuspend, for η ^ 6, v;e proceed thus: If χζΑ,

then th- link of χ in (3,A) is a pair of

spheres which is unknotted, by the inductive hypothesis. That is to say, A is locally unknott- d in B. In particular, we can find an n-cell

204

Ε (2 Β, such that S Γ\ Λ is a k-cell unknotted in E; and so that E, ^(ΕΠΑ)) is bicollared in (B,A); ir fact, this coald have been done whether or not A were locally unknotted. Define

F = Β - E. B\/ earlier results ^ is an n--cell.

f Consider the relative manifold

(F, ΡΓ\Α).

It

easily seen that

this pair is algebraically trivial; be caure of codiiTiensiori \ 3 all fundamental groups are trivial, and so Wliitehead torsio.i is no problem; the homology situation is an easy exercize in Alexander duality. Hence, by Theorem. C, F collapses tc

FT\ A ^ 3jnco

FP. Λ·

is a k-cellj it collapses to a point; putting these together, (F, FnA; collapses to

(pt, pt) in the category of pairs; these co^lapf-es are

homogeneous in the pair (BjA) because of local unkncttednefs. We started vdth

(F, FT,A) bicollared, and hence, by the regular neigh-

bourhood theorem, suitably stated for pairs, (Γ, F

is a regular

neighbourhood of χ t A in (θ,A), whicn is an unknotted cell pair (again using local unknottedness). Thus (B,A) is the unioa of two un:-Qiot,ted cell oaxrs (E, ΕΠΑ)

and

(Γ, ϊΤ\Α), which sno-./s it

a pcxyhfdrally equivalent

to the suspension of ( b F, c^ (Σ Π A;). Ζ. Remark

1. This is just Zeeman's proof, except that we use our

Theorem C where Zeeman uses the cunberson·.'^ technique of "sunny collapsing". Remark

2. Lickorish has a theorem for desuspending general suspen--

sions embedded in S

in ccdinension

3· It is possible, by a

similar argument, to reolace "sunny collapsing" b:/" Theorem

T'^e

205

case π = 5 can be treated by a very simple case of sunny collapsing. Remark

3. If ΑζΒ,

where

A is an (n-2)-sphere locally unknotted

in the n-sphere 3, and η ^ D, and 3 - λ has the homotopy type of a 1-sphere, then η is lonknotted in B. Proof exactly as in the cociimension 3 case : we need to know that V/hitehead torsion is OK, v.'hich it is since the fundamental group of the 1-sphere is infinite cyclic; and this proup has zero \\"hitehead torsion, by a result of Graham Higman (units of group-rines). Remark

h. It has been a folk result for quite a while that the

•^nknottinK Theorem followed from the prooer statement of the h-cobordism theorem. 8.3-6. 'rtMtchead torsion.

For any group {"C' , there is defined a

commutative group Vi/h( ΓΤ). Elements of V/h( \T ) fi^e represented by square, invertible matrixes over the intrtrcr group ring Ζ ΓΓ . Two matrices

η and Β represent the same element in VJh( iT ), if and

only if there are identity matrices I,^ and Ij^ , and a product Ε of elementary matrices, so that

Ij^ ^ Έ. (B^i^J. Here

denotes the kxk

identity matrix.

Λη elementary matrix is one of the follov.rini: (a) I + e. ., where e. . is the nxn whose entries bv zero except for the ij — , (b)

k), which is tht· nxn

which is

1; and i / j.

matrix ecual to the

identity matrix, except that the kk'^'' entry is ^ to be an element of 1 ΓΓ ·

matrix all of

; we restrict oC

206

3y clevfrly composing matrices of this sort, we can obtain ^n

^^^ Λ ^ Ζ ΓΓ , for instance. Addition in Wh( V\" ) is induced from

, or, eauivalently,

from matrix multiplication. The geometric significance of Wh( !T ) is that a homotopy equivalenc-. f : Κ — L

between finite

element of Wh(rV ), ίΥ= Π'^(Κ), torsion is zero, wonderful things

GW-complrxes determines an

called the torsion of

f. If the.

(e.g. 3-cobordisra) happen.

If ΓΓ is the trivial group, then Wh( Γ\ ) = 0, basically because

Ζ ΓΓ is then a Euclidean domain. If (T is infinite cyclic, then Wh( iT) = 0, by Higman.

His algebraic argument is easily understood; it is, in some mystical sense, the analogue of breakinr something the homotopy tvpe of a circle into two contractible pieces on which we us^- the result for the trivial group. If (T has order 5, then V/h( iT )

0.

In fact, recent

computations show Wh( fT) to oe infinit·": cyclic. Various facts about V/h can be found in Milnor's paper. /'^Whitehe ad Torsion" Bulletin of A. M.S , Vol.72, No.3,

1966_7. In

particular, the torsion of an h-cobordism can be computed (in a straight-forward, may be obvious, vay) from any handle presentation. There is another remark about matrices that is useful. Let A be an nxk

matrix over Ζ ΓΓ^ such that any k-rov.t-vector /"i.e.

1 A k matrix_7 is some left linear combination, with coefficients in Ζ (Τ, of the rows of A; in other words, Λ corresponds to a

207

sur.lection of a free Ζ (Γ-module with η basis elements onto one vri.th k basis elements. Let

denote thf k A k zero matrix.

Then there is a product of elementary so that E. ( ^ ^ = /'^k \ V^kJ ^"nxk/

(n+k)>>(n+k) matrices, E,

This is an easy exercise.

8.3.7. In homotopy theory we shall use such devices as univeral covering spaces, the relative Hurewicz th?orΓΓ (N) is an isomorphism; this implies that wp can imagine not only that MCN, '-'Λ* « but that M C where"/*»/ denotes universal covering space. Call rr^(M). Then the homology groups

are left Ζ ΓΓ-modules.

More explicitly, H.(Tl,K) if i / k; and Η (N,M) is a free Ζ [T-modulp with basis ^ ^^^ ,·•', 1 · Whatκdoes 1 mean? We take any lifting of (H,T) to -- handl-- (Η',Τ') in N; J I

we pick cither generator of

1

,T ), and map into Η^^(Ν,Μ) by

inclusion; the ri=sult is

The ambieuity in defining J

is

208

simply st^tpd: If we mike mother choice, then instead of

^

we

λ > wh'^^'^ X e - ΓΓ · When k

2, we c m go further and say that, by the relative

Hurcwicz theorem, |\ (N,M) ^ii h (N,j'i) ~ ΓΓ (Μ,Μ). κ k k

And tnus we hive a

fairly well-defined basis of ΙΤ^^(Ν,Μ) as a Ζ ΓΓ -mod'ole, dependent on the handles

..... i . Ρ This shows, by thr way, that (Ν,'ΐ) is (k-1 )-connected. We mieht have expected this, sinc£, homotopic-illy, Μ is obtained from Μ by attaching k-cells. Anothfr thing is a version of lefschetz duality as follows; If Μ is an ori^^nted manifold, .and X then

Μ v^lth X a polyhedron,

δ M-X) . Since universal covering spaces can all

be oriented, this vrarks th-re.

In particular, if Χ

Μ is a homotopy

equiv.alence, then H-'CM, X) . 0 for ill i, and so H^(M, jj M-X) - 0 for all i. ^"hen

c) M-X is the univ- rsal covring spacf of

that is, when fT, (

M-X)

^ M-X,

ΓΓ, (Mj, then the relative Hurewicz theorem

will show that Β M-XC^ Μ is a homotoDy equiv'jLenc". 8.3.8·

Infinite polyhedra. .'m infinite polyhc dron Ρ is a locally

compact subset of some finite-dimf^'nsicn-ir. real vector space, such that for every xfeP, therr. is an ordinary polyh'^dron Q, C P, such that χ is conti=ined in th^- topoloiTicil int'--rior of Q in P.

polyhedral

map f : Ρ — ^ Pp, bi^'tw:'-n infinite poly^^-'dr^i 5.s ·> function, such that for every ordin^^ry pol/hear^n

Q

T^ (f | Q) is an

ordinary polyhedron. Th^ category of infinite polyhf-~dri includes ordinary

209

polyhedra; and in addition, every open subset of an infinite polyhedron is an infinite polyhedron. The link of a point in an infinite polyhedron is easily defined; it turns out to be. a polyhedr.al equivalence class of ordinary polyhedra. Hence the notions of manifold and boundary in this setting are easily defined. If Μ is an infinite n-raanifold, then any coniOact subset Χζ,Μ is contained in the topological interior of some ordinary n-m.anifold

M. As for isotopies, we restrict ourselves to isotopies which

are the identity outside some compact set; such are the isotopies obtained from finitely many cellular moves. Any such isotopy on the bomdary of Μ

can be·, extended to an isotopy of this sort on M.

We can talk of regular neighbourhoods of ordinary (= compact) subpolyhedra in an infinite polyhedron, and th- same theorems (including isotopy, in this sense) hold. These concepts are useful here because if (M,X) is a relative n-manifold, then ^(M,X) is r,n infinite n-manifold. And now, any isotopy of ^(M,X) extends to an isotopy of M, leaving a neighbourhood of X fixed. In other words, this is convenient language for dealing with relative manifolds. This is the only' situation where we shall speak of infinite polyhedra; it is, of course, obvious that infinite polyhedra can be of use in many other cases which are not discussed in these notes (in particular, in topological applications of the "Engulfing Theorem").

210

Modification of handle presentqtions. If

=

n^) and ^t^ =

B^^) are

handle presentations of the relative n-manifolds

(M,X) and

respectively, an isomorphism betv/een ^ ^ ^nd equivalence h 5 Μ - · > Ν taking X

·

(M^X^)

is a polyhedral

onto X'

?nd

A^ onto B^ for

all i. Such an isomorphism eives a 1-1 function between handles and preserves various other structuresi Let ^

-

)·•··, A^) be a handle presentation of the

relative n-manifold

(M,X) and let

f :A

A

equivalence taking presentation

X onto itself.

Then by

be a polyhedral κ

Κ

^ is meant the handle

}···, B^) of (M,X), where B^ = f(A^) for i < k

B^ =

for i > k.

It is clear that

^ is a handle presentation of (M,X),

the handles of index > k are equal to those of "^l^^while a handle of of .index ^ k will correspond via

f to a handle of ^v.^·

There is another way upto isomorphism of looking at ^ ^ Suppose

f :A

^ A

Κ

a

is as before.

(k+1 )-handle of the presentation

by the inclusion of Τ in attaching all an equivalence'

Let

(H, Τ) be

^

. ^^ttacb

^

but by

(H,T) to

f|T, In this way

(k + 1 )-han'il.es we get a relative manifold extending

to Bj^^^ one for each

not

Similarly attach the

(B , X) and kI (k+2)-handles

(k+2)-handle of

restricted; and so on. In this way we get a relative manifold

(B^, X)

21 1

and a handle presentation

(A , ,. . .., ^i·, , Β ' κ κ 1

,.. , , Β ) of (Β ,Χ). η ^^

-ρ

This will be denotid by ^ ^ . f (B^, X) and an. isomorphism of

givf^s an equivalence of (M,X) vdtb Λf vd.th ^ ^ .

Thp main use in this chapter of the above modifications is for simplifjdng handle presentations, that is to obtain presentations vd-th as few handles as possible, or vathout any handles or vdthout handles upto certain index usin.T thefi;ivenalgebr aic data about It should be noted that (l) f

^ n^ei not be isomorphic to ^ ^ and

(2)

is not a handle orfsent^tion of (M,X). (2) is not a serious ^ 1 drawback, since isomorphic vi.i f~ and so whatever simplification one· can do foi·

can be done also

which

is a handle presentation of (Κ,Χ) or we Cc'^n fii^st '""o the sinplificationi f -1 and pull the new handle presentation to one of (M,X) by f^ . We will adopt tlie 'irocedure which is convenient in the particular case. If f : k^·

^ A^ is isotopic to thp identity leaving X fixed,

(and this ivill be usually th^ case), th-=n QV and

^ 'rfilj have many

homotopy properties in common; but more of this later. The most frequently used vfays of modifications are catalo^ueii below: 8.4. 1. Let (H,T) be a k-handle of the present aticn ^ = of (M,X). Then if ό . is a transvers'^ sphere and "i - ζ/Η-Τ the transverse tube, we have Ν a regular neighbourhood of ^ in If N' is any other r^^g^ular neighbourhood of 3 in ^ (Α^^,,Χ), there is an isotopy of ^ (A ,X) relating Ν and n' and this can b- extended

Κ

212

to Aj^, to give an end result

f^, with f,(N) = N'. Then ^

^

has its new handle (f^ H, f^ T) whose transverse tube is n'. D $>.k.2. Let (H^, T^) be a (k+1 )-handle of

, with an attaching

sphere^ . Then T^ is a regular neighbourhood of

dC-'^j-^)·

If T^' is any other regular neighbourhood of ^ in ^ i-'^,^) we can obtain a polyhedral equivalence

f^ of

which isotopic to 1

fixing X, such that fj (T^) ^ T^'. Then

2 (vMch is isomorphic

will have its (k+1 )-hanclf. corresponding to . (H , Τ ) 2 to have attaching tube

T^', and handles of index ^ k will be un-

changed. Q. Combining 8.4.1 and 8 4.2, vie have 8.4. β. Proposition. Let.^= of the relative n-manifold

Λ^) be a handle presentation

(M.X); let

be a k-handle and

(k+l)-handle with a transverse sphere of ^ sphere of ^

being ^

of 3 and ^

being S and an attaching

. Let Ν and Τ be regular neighbourhoods

in ^ (Λ^^,Χ). Then there is a handle presentation^^

of (M'j X') is equivalent to (M,X) with for some f ί I·^ t fixed; so that in,^^ '/ζ and

being isomorphic to

^ A^^ isotopic to the identity leaving X ρ^ // , the handles

and

corresponding to

are such that: the transverse tube of ^ the attaching tube of

Proofi

a

is N, ' is T, and

the k^*^ levl

/i^' of ^

k^h level

of^.

Using the equivalences f^

is equal to the

and f^ given by 8.4.1 and 8.4.2,

213

Qi^f^^ 2 is the required presentation. presentation ^ ^ ^ ^ - I f )

It is isonorphic to the

· Since both f^

and f^ are

isotopic to the identity lea\dng X fixed, ΐ^'^ property.

The last point is obvious, L1

8,k.U. L e t ^ = (H,T) be a (k + l)-handle of^^^ , and S an attaching sphere of ^

. S is in ^ (Α^^,Χ), Suppose that ε' is another

k-sphere in ,X) and that there is an equivalence taking; X onto'itself and such that f(S) = s\ handle-^ ^ correspondinp; to

f of A,

Then in ^

the

will have S^ as an attaching sphere.

If, for example, we can go from S to S^ by cellular moves, the we can obtain an equivalence f of Aj^ isotopic to and with f(S) = s\

1 leaving X fixed

This will also be used in cancellation of handles,

where it is more convenient to have certain spheres as attaching spheres

than the given ones. D 8. k. 5. Let ^ be a k-handle in a handle presentation = ' · · · ^^"n of a relative n-manifold. Then if k n-2, there is h isotopic to the identity, h ;/i^ of

^ 'V

corresponding to

^ fixed, such that the handle has a boundary core in

^

»/^-^here

(Keader, have faith that this ic useful.') We prove this by choosing attaching spheres for all the (k+1)-handles, finding a transverse sphere for ^

that intersects all

the attaching spheres only finitely, notin.e that t^ie transverse sphere, contains other points, and then shrinking the attaching tubes and the

214

transverse tube conveniently. More explicitly, Let

be the (k-1 )-handles of ^

attaching spheres S^

, vdth

S^. Let Ν be the transverse tube of^L ;

there is a polyhedral equivalence any x^ int D^,

f : Ν s: d'^

Δ

so that for

is a transverse sphere to^h. Now,

(S^U... \JSp)f\N is

k)-dimensional, and so, triangulating

projjjk * f so as to be simplicial and picking χ in the interior of a k-simplex of D (see 4.2.14), we have found a transverse sphere to finite. Now,

, such that ^

pS (S^vj ... US^) is

is an (n-k-1 )-sphere; and since k ^ n-2,

contains infinitely many points; there is y t Σ -

^p)·

Now then, if we teke very thin regular neighbourhoods of ^

, Sj

in ^(Aj^jX'^the regular neighbourhood of ^

will

intersect those of S^

S^ in only small cells near each point

of intersection of

···

Γ\(^i

hence there will be a cross-

section of the 21 -neighbourhood /"i.e., corresponding to D^aS,

,z

(X,2) = f(y)_7, through y, not meeting any of

the S^ neighbourhoods. transverse tube of

We make these regular neighbourhoods the

and the attaching tubes oi A

, by 1

changing ^

to [

where g^ and

Aj^—^Aj^ isotopic to the identity, fixing X. In

ρ

equivalences ^ '2

^e have

/ misses all a boundary core of the handle corresponding to i11 Vwhich the attaching tubes of the through y"). We define

(k+1)-handles (this is that "cross-section

h = g^"' g^ ; and since

is isomorphic

to {^^g^^ ^ , we have some boundary core of 'J^^ where

^^

is

215

the handle corresponding to ^

which does not intersect the attaching

tubes of all the (k+1 )-hsndles, and is therefore in

K)·

S. 5. Cancellation of handles. Convention; Let us make the convention that a submanifold of another manifold should mean this; If A

Β , A and Β are PL-manifolds, we call A a

submanifold of B, if and only if, An^ Β is (dim A - 1 )-subraanifold of ^A. We are usually in this section interested only in the case dim A = dim B. vJith this convention, if A is a submanifold of B, then Β - A

is a submanifold of B, and bd J k ) = bd „(B - A) = ό A - dB. ti D

If C C Β c A all PL-nanifolds such that each is a submanifold of the next, then A _ (B - C) = A - Β U C. We may therefore be justified somewhat in writing A _ Β for A - B. Thus, hereafter, Η is a submanifold of Β means that A is a submanifold of Β in the above sense, and in that case Β - A stands for Β - A. Let

= ^^ 1 )···')

be a handle presentation of a

relative n-manifold (M,X). Let "fu = (H, 3 h - N ) with transverse tube N, and

be a k-handle

fi = (Κ,Τ) be a (k+l)-handle with

attaching tube T. Mote that KU Τ C 0 (Aj^j-"^) with the above conventions we can write - fi^ = {{I^

) U K) U H.

216

a transverse sphere of

an attachi sphere of

A iis attached

to A^^ by Τ Ν

217

8.5.1. Definition, 1) ΝΛΤ

say that A

and

can be cancelled if

is a submanifold of both Ν and Τ

2) Ν - (NOT) and Τ - (νΛΤ) are both (n-l)-cells. . Suppose A Assertion 1. (,Α^

and

can be cancelled. Then

)Π Κ is an (n-l)-cell contained in ^(A^-^ ,X)

and in 0 Κ. In fact, {ky. -A )Γ\Κ = Τ - (ΝΠ Τ), which we assumed to be an (n-l)-cell. Assertion 2.

(_(A^^-/C ) U κ)Π Η is an (n-l)-cell contained in

^ ((A^-fi.)UK,/.')and in 0Η.

For

((Aj^-·^) ΟΚ)ηΗ - attaching tube of =

plus NPvT

Η - N) U (NOT)

= ό Η - (N - ΝΠΤ) and this is an (n-l)-cell, since

is an (n-l)-sphere and (N - ΝΠΤ)

is an (n-l)-cell in ό H.

.

•

'

*

Combining these two assertions with proposition 8.2,1, we have

,

.

8. 2. Proposition.

Supoose

,.·.,

is a handle'presen-

tation of a relative n-manifold (M,X); and there are a k-handle, and

^ = (H, d H-N)

= (Κ,Τ) a (k+l)-handle that can be cancelled.

Let IjL be any neighbourhood of Μ Π Τ in Aj^. Then there is a polyhedral equivalence f : (Aj^-^ , X)

(Aj^ ^ k , Σ)

which is identity outside 'LiU· . • This being so, we construct a new-handle presentation

«

218

(B_i

B^) of (M,X), vihich we denote b y ^ -

,'k ) as

follows: B^ = f (A^) for i< k

\

-f

-

^^

B^ = A^ for i>k. This of course depends on f somewhat· observe that, since the attaching tubes of the (k+1)-handles are disjoint, the attaching tubes of (k+1 )-handles other than ^ that

-

are in

h

X) - Τ ζ ^(A^

is a genuine handle-presentation.

8.5.3. (Description of

- (^ , ^ )).

The number of i-handles in

- {'K, L·) is the same as the number of i-handles of i ^

so

for

k, k+1. For i > k, each i-handle o f ^ i s a i-handle of -

A ) with the single exception of 'h ; and conversely. For

i 4 k, each i-handle of.^^ e x c e p t , say "C , corresponds to the i-handle ^

) of ^

-

and conversely each i-handle of

- (A,"^) .is of this form. If the attaching tube of

intersect some k-handle that

itself occurs in

, we can arrange f

does not

to be identity, so

~ (^J"^)· Q

The conditions for

and

to cancel are somewhat

stringent. We now proceed to obtain a sufficient condition on ^ and which will enable us to cancel the handles corresponding to

and ^

in some This requires some preliminaries. Suppose A, B, C are three PL-manifolds, AUBCC - C) C. dim A = p, dim Β = q and dim C = p+q, ^A ^ δβ = ^ . Let χfcAfVB.

,

219

8 5.4. Definition.

A and Β are said to intersect transversally at

X in C , if there is a neighbourhood hedral equivalence ^

f :F

F of χ in C and a poly-

S * 2Γ-ii- ν

v/here 3 is a (p-1 )-sphere,

a (q-1)-spherej such that 1) f(x) = V 2) f(A OF) - S * V 3) f(B fVF) = Σ:-"^ V

8.5.5. Proposition. Let S and ^ be

.

(p-l)- and (q-1)-spheres

respectively and Ε = S * X * ν . Let D = S * ν , Δ^Σ.^·· ^ · Suppose ^

is any simplicial presentation of Ε containing full sub-

presentations J^ and ^ and Q = |N 0

covering D and A . Let Ρ = ^

^

Π

. Then PHQ

is a submanifold both of Ρ and Q and

is contained in the interior of Ε (P, Q, and POQ

are all (p+q)-raanifolds)

2) Ρ - paQ Ν

Q - PHQ

Ρ A3E

^ Q Π DE.

Proof; First observe that, if the proposition is true for some centering: of ^ , then it is true for any centering of ζ

. Next, if

some other presentation of Ε such that D and ^

are covered by full

subpresentations, it is possible to choose centerings of ^ so that Ρ = Ρ and Q = Q . ' (P , Q and Q with reference to

, Novr wc choose

and

^

denoting the analogoues of Ρ

Thus it is enough to prove the

proposition for some suitable presentation centering of ^

is

'ζ

of Ε and a suitable

to be a join presentation of

220

Ε =S

^

ν

and choose the centering so that

(see 6.8.3 and

the remark thereafter) P' Q' = lst(v, d

= Ci (S ^^Σ ),

PLP'HQ' =

and

QLPVQ' -

il .

And in this case (l) and (2) are obvious. 8 5.6. Proposition.

Suppose

Q

A and Β are spheres of dimensions

ρ

and q respectively, contained in the interior of a (ρ+q)-manifold

C

and that A and Β intersect at a single point χ transversally

in C. Then there are regular neighbourhoods Ν and Τ of A and Β in C, such that 1) NOT

is a submanifold of both Ν

and Τ

2) Ν - (ΝΓ\Τ) and Τ - (ΝΓλΤ) are both (p+q)-cells. Proof; Let F be the nice neighbourhood of χ in C given by 8.5.4 i.e. there is a polyhedral equivalence where S is a (p-l)-sphere and

f:Fc;i

^ and

of 4

f is siiriDlicial vdth

WF can assume

presentations covering S * ν J

ν , Then (A - fJ

(B - F) is a (q-l)-cell. Let /f^ and

be triangulations of F and Ε such that reference to

ν

a (q-1)-sphere, such that

f(x} = v, f(AaF) ^ S •«• V , and f(BnF) = is a (p-O-cell and

Ε=3*ί.*

and 21 * ν

^ contains full sub. Now some refinement

can be extended to a neighbourhood of A (j B, denote it

by /f 5 it can be supposed that

contains full suboresentations

(J^ J G3 covering A, Β respectively.

Let "Π be a centering of

221

Denote by ζ by

the triangulation of Ε corresponding to

the centering of ^ corresponding to ^

and Τ =

by f, and

. Choose Ν

( (β )| ; and let P, Q be as in 9.5.5. If

P, = f"UP), Q, = f"^·':), then P^ - (Ρ,Π

) ^ Ρ,Π ()f and

Q^ - (P, HQ,) V Q, ^θΡ.

Clearly P^CN, Q,C Τ are submanifolds and

ΝΠΤ = P^nQ^.

is a submanifold of both Ν and T.

Thus imx

Ν - (Ν η Τ) = Ν - (Ρ,Λ Q^) = (Ν - Ρ,) υ (Ρ^- (Ρ,Π Q^))" collapses to (Ν - Ρ^) since Ρ^-(Ρ^Λ ς^) collapses Ρ, r^ -l)-cell. Thus Ν - (NOT) \ N - P^ V Thus Ν - (MfVT) is a collapsible

- F which is collapsible.

(p+q)-manifold, hence a (p+q)-cell.

Similarly Τ-(ΝΓ\Τ) is a (p+q)-cell. • 8.5.6. Definition. Let ^^ =

,···,

of a relative n-manifold (M,X). Let (k+l)-handle of

be a k-handle and ^ be a

. We say that

there is a transverse sphere S of

tie a handle presentation

can be nearly cancelled if and an attaching sohere

which intersect a single point transversally in ^ 8 5.7»

Proposition.

n-manifold and

(M,X)

of ^

X).

Suppose^ is a handle presentation of relative a k-handle and il a (k+1 )-ha.ndle in ae.

ifi

can be nearly cancelled, then there is a polyhedral equivalence

f : jL^—^ A^ isotopic to th^^ identity leaving X fixed such that, in

^ the handles 'fi

and

corresponding to ^ and ^ can

be cancelled. Proof; Follows from 8.4.3 and 8.5.6.

Q

222

8.6. Insertion of cancelling pairs of handles. In this section we discuss the insertion of cancelling pairs of handles and two applications which are used in the following sections. First we form a standard trivial pair as follows:

Let D be a k-cell, I = (_0, l3 and Δ an (n-k-1 )-cell. Then Ε = 0 Χ Ι Χ Δ

is an n-cell. Let Δ

H. = D

Clearly

= (H,, T^) is a handle of index k. Next, let H2 =

= {(D X 0) υ (D)Ci) (J ( Clearly, A - (HJ, T2) is a handle of index (k+l). Finally, let F denote

(D ).··> B^) of

(M,X) as follows: Bi = Bj^ = Bi = A^

, for ik.

Next we consider the problem of attaching a cancelling pair of k- and (k+1 )-handles ("^,.Α) to

A^^, with ^ having a prescribed

attaching sphere. We recall from Chapter VII (7.2) that a sphere in the interior of a PL-manifold Ν is unknotted (by definition) if it bounds a k-cell. In such a case it bounds an unknotted cell (again in the sense of 7.2). If S and S' are two unknotted k-spheres in the same component of Ν - ^ N, then there is an isotopy h^ of Ν leaving

Ν fixed such that h^(S) = S', Similarly if D and D'

are two \inknotted k-cells in the same component of Ν - 3Ν, there is an isotopy of Ν taking D onto D'.' Similar remakrs apply in the

224

case of relative manifolds also. Now consider just Aj^, let F' be any (n-O-cell in ^ (Aj^jX) and form A^^UE by an equivalence

F

3 = ^ 0 ye,C.

^ e

•= C - (a regular neighbourhood of S in C). Finally C is unknotted in F. The result of all this is, if A is a (k-l)-sphere bounding an mknotted

k-cell Β in

then we can attach a cancelling

225

pair of k- and (k+1 )-handles

,'K) such that A is an attaching

sphere of Tl and an attaching sphere of α

intersects

Β - (a given a regular neighbourhood of A in B). This can also be seen as follows: Let L be an (n-l)-cellj A a (k-l)-sphere in L bounding an unknotted k-cell Β in the interior of L. Let Μ be an n-cell containing L in its boundary. We may join A and Β to an interior point ν of

and take second derived neighbourhoods.

Let Η be a second derived neighbourhood of A ·»· ν and Κ be the closure of [second derived neighbourhood of Β * ν - Η

. Then

(Η, HAL) is a k-handle, ,and (Κ, (ΚΓιΗ) u(Kf>L)) is a (k+1 )-handle. The k-handle has A as an attaching sphere, and an attaching sphere of the (k+l)-handle intersects L in (B - a regular neighbourhood of A in B). Thus we have, 8.6. 1. Let

=

,"·', A^) be a handle presentation of a

relative n-manifold (M,X) and let 3 C which bounds an unknotted cell Τ in

be a (k-l)-sphere Then there are a

k-handle ^ and a (k+l)-handle ^ , such that (1) S is an attaching sphere of (2) There is an attaching sphere ^ ^ Π. \

of

very closed to T, th^.t is

with Ο A^

can be assumed to be (T- R prescribed regular neighbourhood of S in T). exists and is polyhedrallyequivalent to (Α^,Χ) by an equivalence which

226

is identity outside a given neighbourhood of Τ in If S is in 5

^

^

w

have its attaching tube in ^

e

can choose Κ

to

that there is an obvious

handle presentation of ((V +

Λ

, X). V»'e sive below two appli-

cations of this construction. 8.6.2. Trading handles. Let

».·-·>

be a handle

presentation of a relative n-manifold (M,X). Let p^ be the number of i-handles in ^

. i^ppose that there is a (k-l)-handle L

(2 < k ^ n-l) with a transverse sphere ^ , and that there is a (k-l)-sphere S in

X) Π h

such that (l) 3 is

unknotted in b (Α^,Χ), (2) S intersects ^ one point in ^ (A^^

transversally at exactly

X). Then there is a procedure by which we can

obtain another handle presentation

of (M,X), such that (a)

for i / k-1 or k+1, the number of i-handles in Qf^ is equal to the number of i-handles in^^^'j

the nurabfr of (k-1 )-handles in ^

is p^^ ^ J- 1 (c) the number of (k+1 )-handles in

is

This is done as follows: First consider only A^. -Applying 3.6,1, we can add to A^^ a cancelling pair of k- and

such that S

is an attaching sphere of 'Ki , and the attaching tube of S is in )

= B. Then th^ relative manifold

(B,X) has the obvious handle presentation where B^

for i 4 k-1

- (B ^ ,'··•,

227

In

J •the handles

Hence for some equivalence

f of

leaving X fixed, in ponding to 'C and ^ ^^

isotopic to identity and

the handles

^

p.nd

) corres-

can be cancelled. Let ^J-^ = ^ ^ -(

)·

is a handle presentation of (B,X); the number of i-handle in for i

has at least

is an isomorphism,

we have a contradiction to the assumption that Μ is connected. D For the neact stage, we need a lemma: 8.7.2. Lemma. A null homotopic

1-sphere in the interior of a

PL-manifold

Μ of dimension ^ k is unknotted.

Proof: Let

S be a null homotopic

We have to show that and X

1-sphere in the interior of M.

S bounds a 2-cell in ^L

Let

D .be a 2-cell,

an equivalence of δ D • with· 3. Since S is null homotopic

Ji extends to D, Approximate «C t)y a map

in general position

such that JS j d D = (L (0 D, and (i (D) C int M. The singular set S2 ( p ) of Ρ

consists of finite number of points and

are all empty. So we can partition ^P,

, P^^ ,

β» ) etc.

p ) into two sets

Qjj^^ such that β (p^) = Ρ (q^), 1 $

and there are no other identifications.

m

Choose some point ρ on ό D

and join^p, p^ ,..., ρ I by an embedded arc V'which does not meet any L ' ®5

233

of the q^'s. Let Ν be a regular neighbourhood of / i n

D, which

does not contain any of the q^'s. Ν is a 2-cell, Let Ν A D = 'bN η ^D = L, ^ Ν - L = K, and D - Ν = d'. Since L is a 1-cell, Κ is also And P | N as vrell as

1-cell, and D' is a 2--cell.

D' are embeddings.

So

( d D') is un-

knotted in M. But by 7.1.6, there is an isotopy carrying Ρ (Κ) and leaving S

onto

^(L) to

J5(C)d'- K) fixed, that is, the isotopy carries

|3(δθ'). Hence S is also unknotted. D

Remarks;

(l) The same proof works in the case of a niill homotopic

n-sphere in the interior of a

^ (2n + 2)-dimensional manifold.

(2) The corresponding lemma is true in the case of relative manifolds also. (3) If S is in

^ M, then the result is not known.

It is conjectured"· by Zeeman, that the lemma in this case is in general false (e.g. in the case of contractible 4 dimensional manifolds of Mazur). 8.7.3. Proposition. Let

A^) be a handle presentation

without 0-handles of relative n-manifold

(M,X) and let ΓΓ^(Μ,Α

« 0.

Then by admissible changes involving the insertion of 2- and 3-handles and the cancelling of

1- and 2-handles, we can obtain from ^

presentation of (M,X) without Proof; Let

a handle

0- or 1-handles, provided

be a 1-handle of ^

. By 8.4.5, we can assume that

there is a surface core of 71 in ^(AG,

Because

X).

= 0, then

homotopy exact sequence of the triple

(M, A^, A

^ ° and so. C is

homotopic leaving its end points fixed to a path in is a horaotopy equivalence

(we are confining ourselves to the special

case after 8.3). So we get a map, where D is a 2-cell f : D — L · ^ - int with ^D :)C, such that

f(0 D - C)C

f | C - Id^.

Now in the (> 4 ) - m a n i f o l d r e m o v a l tubes of the

of the attaching

1-handles does not disturb any homotopy of dimension^2, so

that, we can arrange for . f(ilD - C) C

- (attaching tubes of 1-handles)

C

,X)

Likewise in ^(Α^,Χ),

(since

- A^).

the removal of the attaching tubes of

2-handles can be ignored as far as one-dimensional things go, so that we can assume f(5D - C ) C ^ (Α^,Χ), and that

f jdO

intersect ^

is an embedding.

Also, we can arrange

f(^ D) to

precisely along C.

Finally, then we have f JD

^ A^

with f(0D)c 'd (Α^,Χ) η ύ(Α^,Χ) fjc = Idp , and this is the only place vrtiere f(^D) intersects^ . Hence f( δ D) intersects a transverse sphere of ^

at exactly one point transversally. Now, upto homotopy,

attaching cells of dimensions

k^ is obtained from 0(Α2,Χ) by (n-2) and (n-l) /~cf. duality 8.8j7.

235

Since

(n-2)^3, ^

deformed into f

^

"

J (A2,X) leaving f|6D

is null homotopic in

^^us the map

f can be

fixed. Thus the 1-sphere hence by Lemma 8.7.2 it is

unknotted in d (/i2,X). Now we can apply 8.6.3 to trade ff

for a

3-handle. We can apply this procedure successively until all' the 1-handles are eliminated.

Since in this procedure, only the number of

1-handles and 3-handles is changed, in the final handle presentation of (M,X) there will be no 0-handles either. D Remarki

If (M,X) is i, -connected and 2l+ 3^n, we can adopt the

above procedure to get a handle presentation of (M,X) without handles of index $ ί. 8.8. Dualisation. In this section, we discuss a sort of dualization, which is useful in getting rid of the very high dimensional handles. Let

(M,X) be a relative n-manifold (remember that we are

dealing with the special case; X amand let ^

(n-1)-submanifold of d M),

be a handle presentation of (M,X).

Consider the manifold

M"·· obtained from Μ by attaching a collar over ^ (M,X) (= ^M-X by the notation of 8.5)5 · M^ = I^M U ( δΜ - X)

[pj l} ^

identifying χ with (x,o) for x t 0 M - X. Let

M* = M+X* = [(^M-X))*. IJ Vj[c)(

) is an isomorphism and if, (A,) —

is a surjection. We identity all these groups and call it fT . Now, the groups C^ =

rr^

(Aj^,

are identified vath H^ (A^,

They are free modules over Ζ ίΤ

with bases correspondine to handles.OJ i-handles, the basis of C^ is denoted by

^ ^^^ ^

are the

... , ^^^^^^

and

the elements of this basis are well defined upto multiplying by elements

±rr. . If f ! Aj^ — > Aj^ is a polyhedral equivalence isotopic to the identity, it is easily seen that the algebraic structures already described for ^

and ^ ^

may be identified.

In addition, we have a map

I

vrtiich is the boundsiry map of the triple (A^, also unchanged by changing ^

to Q^^·

If there are no handles of index < k-2 and ΓΓ

(M,X) = 0, k"· 1

we see; First, n~ (A, , A „) = 0, and hence from the exact ' k-1 k-2 sequence of the triple

{k^,

A^_2) the map S ^ '

is surjective. Ihially, if there are no handles of index y k and we have

Vl

- 0,

240

Κ

to be infective.

·

Now, the boundary map ^

plus the bases of C and C k κ k— 1 determine a matrix B^^ in the usual way. That is, if

matrix-

then Bj^ is the

(k-1) ^

2,1

^^2,2

· ··

·

P(k-1)

· ·

If we choose a different orientation of coret^^^^^ , then is replaced by - ^ ^

^^

(in the basis of Cj^) so that the i^'^

row of B^ is multiplied by -1. If we extend the handle ^ ^ ^ along a path representing^ CT , then X

>

^^ is replaced by

^^^^ ^^^

multiplied by cC .

Thus, by different choices of orientations of cores and paths to the "base point", we can change Bj^ somewhat. of modification which we can do on

There is another t3φe

that is adding a row of Bj^

to another row of Bj^. This is done by using 8.6.3 as follows: Consider two k-handles ^^^^

and

> and

let 24.k ^ n-2. We now apply 8.6,3, (Remark 2), vith A^ ^ = N, = n', \

= ^^ ^^^^^ =

away from n , ··

· T^s gives a new handle

'fLi J such that

(k) with proper choices, now,represents

and

(sign prescribed), in JT ^yi

^

θ

[in

Also, we can assume ,that

is away from the attaching tubes of the other handles, so that Β = (A + K— 1

+ (other ρ-2 k-handles of ^ k-

J

)

% and

can be assumed to be identity on X. Now (B,X) has an.obvious handle presentation =

The k

B^), where = \ iik-1

boundary map of^^ , with the appropriate bases, has a matrix

which is the same as Bj^ except for i^'^ row, which is now replaced th th by the sum of the i row + (i:6)time.s the j row, corresponding to the relation X We pull

to a handle presentation

J of

ψ Κ

'

rf ' and ^J^ , the matrices of the boundary maps are the same if we choose

2 ι

the corresponding bases. And tation

can be extended to handle presen-

of (M,X) by adding the

( ^k+1 )-handles as they are.

By doing a finite number of such changes, we have 8.9.2,

(Basis Lemma), j y is a handle presentation satisfying 8.9.ij

Bj^ is the matrix of the k^'^ boundary of map of^j^^

with

with respect to bases corresponding to handles. Given any p, % P, IC matrix Ε which is the product of elementary matrices, then 3 handle presentation

iC a

of (M,X) satisfying 8.9.1

(1) the number of i-handles in 3rC is the same as the number of i-handles

> for all i> and

(2) the matrix of the

boundary map of ^

with

appropriate bases corresponding to handles is Ε · Bj^. Ql As an application of the "Basis Lemma", we will prove a proposition, usually known as the "Existence Theorem for h-cobordisms". Let Μ be a PL (n-1 )-manifold; Μ

compact, with or without boiindary.

The problem is to produce a PL n-manifold

W

containing Μ in its

boundary such that (W,M) is a h-cobordism with prescribed torsion» 8.9.3. Proposition, If the dimension of Μ is greater than 4, then given any

^ ^ Wh( HVCM)),

there exists a h-cobordism (W^M)

.with Τ (W,M) = Tq. Proofs Let A - (a^^) be a matrix

(m >k, ^k+1) A^ +

f is an equivalence

onto itself.

(k+1)

If the attaching tube of % ^ " 1

mapping X does not intersect

any other k-handle except ^ , f can be assiimed to be identity /(k) on all , We assume that this is the case. Now ^(k) ^ /(k) ^(k+1) ,····,ρ are all the k-handles and TL^ »····, ^^ are all the

(k+1 )-handles of-3^-

abuse of notation^

_

, is a basis of

Thus (by. (B^,

^)

is a basis of ΓΓ (B , Β ). Let k+1 k+1 k ^

k+1

denote the

boundary map o f ~

Consider the following commutative diagram rf

(A , A k+1 k+r k

(A ) k k

^ ^^^^ ' 1

^ 1

(A = B ,Ar^B,A ^B k+1 k+1 k^ k k-^- k

J \ (T (A , A ) " k k k-1

In this diagram, the vertical maps are induced by inclusion, the horizontal: maps are canonical maps, and Now

and

/rp

Ν

i

Hence, for i>,2,

^

) - o>

= ^

,j

k+1

/ Pk

r#(k)

ς :

'

i=2 Thus, if the matrix of ^ ^^^ is

j^j),

is

then the matrix of ^ ^^^

j >^2, This we have as long as the attaching tube

/(k+l) [1 TL 1i keeps away from the transverse tubes of the handles '(k) i>,2. (It is easy to see ci oC, ^ = 1 1,2 '> pk

that

this case). It does not matter even if the attaching tubes of other (k+1)-handles intersect the transverse tube of 8.10.2. If f : A • k

| \

Ό.

A is an equivalence isotoDxc to the identity k

£ leaving X fixed, then in corresponding Since the

and

, the attaching spheres of the

(k+1)-handles represent the same elements in ^^ , (A ). ^ Κ boundary maps are factored through (V (A ), the κ k

corresponding matrices are the same after the choice of obvious bases in ^ a n d ^ ^ and hence in ^ and ^ ^ 1 ^. Q Proof of Theorem A. Step 1; Let^^= A^) be a handle presentation of satisfying 8.9.1. ^^e are given that we know that the sequence

(M,X) is ^-connected, th«

j.^

250

'V,' ν

ΓΓι^ν ν.' ^

···

°

is exact. Suppose that we have already eliminated upto handles of index" (i-l), that is inj^ > (*), TT^^, the matrix

= ^q " " " fTi

^P^+^^Pj^^

^^

i®

^ i+1 ^^^ bases corresponding to handles.

Then, there exists a (p +p)y.(p "^p) matrix E, which is i+1 i i+1 i the product of elementaiy matrices, such that Ε X

s

-χ

Ί Pi

0

If we attach p^ cancelling pairs of (i+l), (i+2)-handles away from the handles of index ^ (i+2) to the i^'^ level o f ^ , then in the resulting handle presentation of

·

,..., B^) of (M,X), the matrix \

appropriate bases is

Then, by the Basis Lemma, we can obtain a handle presentation of (M,X) satisfying 8.9.1, with exactly the same number of handles of each index >st as above, but (i+l; boundary matrix will now be

251

Ex

"l Pi 0 Pi.i'Pi

This means that starting from ^ ^ , presentation

»···> ^^^

(0

we can obtain a handle such

that

satisfies 8.9.1, >nd there are no handles of indices ^ i - 1

(2) the (i+O®^ bomdary map of

matrix

Now we can eliminate the i~handles one at a time as follows: Step 2! Consider ^ ^ ^ ^ and i^n-4. of δ

and

^ _ ^^

=

Hence by the Isotopy Lemma, there is an equivalence f such that

f takes an attaching sphere S^

to another ..i-sphere sj and

of ^ ^

^

S| intersects a transverse sphere of

κ(i)

at one point transversally. Moreover it can be assumed that f (i+o ρ (i+1) (attaching tube of ^ ^ ) does not intersect the transverse tubes of the other i-handles. f can be extended to an equivalence

f of

C. taking X onto itself and in the handles corresponding ^^^ and can be nearly cancelled. By 8. 5.7, there is an equivalenc f g (e;f) g of C^, so that in (IK ) = I X " , the handles corresponding and

can be cancelled. /(i-^Ox

(attaching tube of of the handles ^^^^ ,

Again, we can require that g*f

] sho\ild not intersect the transverse tubes Consider

This is a handle

{ CWAA ^ Crm be

252

•to V^e. iic-bfic Vo VW. lA-en-HV^ i(i) 0 (i-1) the handles Tt and .-X ''I ^(i+0 and fl,^ can be cancelled. By

presentation of (M,X), and in ^ say, corresponding to XioTlTthe

(i+1)®^ boundary map

Hence, we can go on repeating step 2 ^iPi-1) 0 to obtain a handle presentation of (M,X) vdthout handles upto index i. has the matrix

0'

Thus, inductively, the first part of Theorem A is proved. The second part is clear.

Q

Proof of Theorem B: By Theorem A, we can assume that there is a handle presentation

with handles of indices (n-3) and (n~4)

OT^y. ^ ^ obviously satisfies 8.9.1. Consider the map

δ Here Ah. . = -1

n-3 ' ' n-3 k . Let n-5

A^ n-3' n-U

rr^^

A be the matrix of η ^ with respect n-3

to bases corresponding to handles. A is a nonsingular matrix, say mKm

matrix.

Since

(M,X) = 0, A represents the 0-element in

Wh(iT ). Hence for some q^m

there exists an (m+q) >!^(ffl-Kj) matrix

Ε which is the product of elementaiy matrices, such that A 0 = I 0 I m+q qj Now we add q cancelling pairs of (n-3)- and (n-4)-handles to A^ ^ away from: the other handles, so that in the new handle presentation, say^]^,

of (M,X), the

the matrix ~A 0 η

(n-3)^'^ boundary map of ^ ^ has

253

Then by the Basis Lemma, we can obtain a new handle presentation of (M,X) with exactly (m+q) handles of indices and no others, and such that the matrix of the of

(n-3) and (n-4)

(n-3)^^ boundary map

with respect to bases corresponding to handles is Ij^+q·

by a repeated application of Step 2 in the proof of Theorem Ay all the handles can be eliminated .so that Μ V^·

^

8.11. Proof of the Isotopy Lemma. Vie begin with some elementary lemmas. 8.11.1. Lemma. Let

Q.C ^ ^ Q ^ be polyhedra, where is OKvdl ί Ρ >. GL ptS^Kiu

f 5"

and

jLk(r,(p)(—>(Lk(fr,%)^ The map \ an embedding; othervase if f r·^ = f

then

/^

(Lk(fe,A.)l is

, 'X , Τ η ί Lk(r'jp) and

f( Γ T^) = f(r· ^ 2 )

so that

^^ ,

contrary to the assumption- that. ^ is a principal simplex of ^

St(5',(p)) —> |st(f(r

Hence the map

embeddings is an embedding.

.

being the join of

Q

Proof of the Isotopy Lemma for k ^ n-5! The situation is: We have a handle presentation^^ of a relative n-manifold a special case, and

satisfies the hypothesis 8.9.1. We have

k-sphere 3 (what was called Σ representing

(M,X) which is

in 8.9.4 and 8.9.5) in

in

where

d (Α^^,Χ)

is a k-handle. We

deduced in 8.9.4 that in this case if k $ n-3 there is a homotopy ^ { 3)ς I —

t

h

a

t

h^ = embedding

S C0iA^v>)and h'^

(transverse tubes of all k-handles) is a k-cell C which is mapped by h^ isomorphically onto a core of

h,(s - c ) c ^ ( v x)n

, so that

X)·

In the iSotopy Lemma, we have further assumed that k ^ n-4. We first prove the simpler case when k dimension of ^ in δ (Aj^jX) "is ^4.

n-5i that is when the co-

255

We can by general position suppose

(h) has dimension

^ 2 (k+1) - (n-l) = 2 k+3 - n. Now ^

is polyhedrally equivalent to D^

the transverse tube of any point JC k ii^t

corresponding to D

vdth

^

^o^

et X 0 Al is a transverse sphere; and any such

transverse sphere will intersect the core h^(G) transversally in exactly one point, since h^(C) corresponds to D

We try to apply Lemma 8.11.1 Q= "Shadow"

^ , for some

to this situation. Define

I. (h) = ^Projg r ( h ) } * I

Ρ = SΚ I Ρ— ^ S^ I —

X 3 Y y^ct becomes D

transverse tube of

^

The crucial hypothesis now is dim Q < k. Since, in general dim (projg ^ (h)) ^ dim Zl(h), we have dim Q $ dim X(h)+1 ^2k+4-n. To have this < k is exactly where we need k ^ '>-5. The conclusion then is: dC Κ ^ Δ X

b Δ »

There exists a transverse sphere Τ of

of the form

some dC tint D, so that h'^T) does not intersect

the shadow of the singularities X(,h) or, what amounts to the same, the "shadow" of h~\T), namely Ζ = _ [projg h'UT)]K I C does not intersect 51(h).

S X. I

Hence there is some regular neighbourhood

Ν of Ζ in 3 Λ Ι , with Ν Γ> 21(h) = 0. This implies, since Ko is an embedding, that h| S)ζΐ), that is at point (corresponding to «^^fi ) transversally. /

By being only a bit more careful, considering the transverse tubes of other k-handles, we can arrange for s' not to intersect the other k-handles at all (if k-handle other than ^ isotopy of

τ' is a transverse sphere^aome

, then h~^(T')nSM = 0, and there is an

X) carrying ^

X) - small reg\ilar neighbourhoods

of prescribed transverse sphere of the other k-handles to transverse tubes of the other k-handles). Remark; This already gives Theorem C for The case

k - n-k.

In case k = n-4, η

the above result is still true,

but this involves some delicate points. Since η w e

have

(for k = n-4) the crucial number

2k + 3 - n>0. We consider, as before h : SXI — >

^^

general position, so that dim X (h) < 2k + 3 - n. Remembering S = h(S)*,0), we further use general position so that

h~ Vs)f\ S X.(0,2J

is of dimension ^ k + (k+l) - (n-l) = 2k + 2 - n, and call

257

0(h) = closure Make

(0,3 ).

h simplicial, say vrLth reference to /{of

and refine/^ to /4 θ '

S

so that

0(h) is covered by a subpresentation

^y ^ subpresentation

is simplicial on

1,

> and the projection

S lU: — ^ S

'.

Now we have to pick our transverse sphere Τ =

xbA'^

In the transverse tube D^ X 'δ D""^ so that (1) h"\T) Ο

shadow X (h) is O-dimensional

(2) h-\T) a H i h ) - φ (3) h'^T) Π On S X O U

shadow

^ (2k + 2-n)-skeleton of Σ*^» 0

a neighbo\irhood of

[shadow h~ (T)"^

h

is a local embedding, using Lemma 8.11.3. Let Q = shadow h'^T).

The finite set of points

Q r\ X(h) does not intersect any point of h~^(T). Each point say X 6: Q rv^(h) belongs to a (2k + 3 - n)-simplex of ^ ^ say Since

has dimension

we can move

hood of X by a polyhedral equivalence move

SXI

f !S

in a tiny neighboup—^

3>-Ι

so as to

X around on C"^, that is, so that f(Q)nX(h) = Q n X ( h ) - I x ] +

,

where the choice of x' ranges over an infinite set. f will not move

h"^!) nor will it move

3^0 . There are only a finitely

many points to worry about, and so we can find a polyhedral equivalence f : SXvI— ^ of points points.

SXI,

leaving

h'^T) Vj SX>0 fixed, such that the set

f(Q) r\X(h) are mapped by h into pairwise distinct

258

At this moment, we see that on S)i.O VJ fCQ), h is an embedding.

Since h is a local embedding on some neighbourhood of

f(Q) (we restrict

f close to the identity so that

f(Q)C S X.I -

+ 2 - n)-skeleton of ^

, and an embedding on

f(Q); hence it is an embedding on some neighbourhood of f(Q). Q(h) is well out of the way, and so h actually embeds all of Sy^O U (a neighbourhood of f(Q)). We now proceed as before, usine f(Q) to move around along. This trick looks a bit different from piping, which is what we would have to do in the case η = 5, k = 1; this was the case when we had a null homotopic

vyf: 22/9/'67

1-sphere C

4-manifold unknotted.