Stratified Polyhedra (Lecture Notes in Mathematics, 252) 3540057269, 9783540057260

Citation preview

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

252 David A. Stone Massachusetts Institute of Technology, Cambridge, MNUSA and State University of New York at Stony Brook, Stony Brook, NY/USA

Stratified Polyhedra

Springer-Verlag Berlin . Heidelberg . NewYork 1972

AMS Subject Classifications (1970): Primary: 57B05, 57C40, 57C50 Secondary: 55F65, 57F20

ISBN 3-540-05726-9 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05726-9 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin· Heidelberg 1972. Library of Congress Catalog Card Number 77-187427. Printed in Germany.

Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

Introduction The present paper is a revised version of a much longer work (which had fewer results) entitled "Block Bundle Sheaves". This is an obsolete phrase for "stratified polyhedra"; and since my work has been referred to under the old title, I mention it as a sort of subtitle of the present work.

During the process of

revision, I profited greatly from lecturing in a graduate seminar at M.I.T., since it was necessary to describe technical definitions and procedures intuitively.

I talked about my difficulties to

many friends, especially Ralph Reid, my sister Ellen and Dennis Sullivan.

Above all, I am indebted to Colin Rourke, who suggested

the present method of defining stratifications, and who urged me to change terminology to conform better with Thorn's theory [28] of "ensembles stratifies".

I am also glad to express my gratitude

to William Browder. The two facts about block bundles, as defined by Rourke and Sanderson, which make them so important are:

t , submanifold of Q, then M has a normal block bundle in

1.

if M is a p.

Q,

which is in some sense unique;

2.

there is a relative transversality theorem: if M.

Q.

then there is an arbitrarily small isotopy f

t

block transverse to M; bdy M in bdy Q,

N are subrnanifol.ds of

of Q such that f N is 1

moreover if bdy N is already block transverse to

then we may require f to be the identity on bdy Q. t

That M has a block bundle neighbourhood in Q was shown by a simple construction: roughly speaking, take a simplicial triangulation B of Q in

IV which M is covered by a full subcomplex A. subdivision of B.

For each simplex SEA,

* tk(s. B') s*A = b * l.k(s. At) s*B =

let

(the simplicial join);

'"

where s E B' is the barycentre of s ,

f s*B

Let B' be a first derived

Then s*B is a block over s*A.

and

: s E AJ forms the required block bundle. This construction can equally well be performed when M and Q are

no longer manifolds but general polyhedra.

It seemed reasonably to hope that

by following the proofs of Rourke and Sanderson one could prove 1.

some sort of uniqueness result, and

2.

a relative block transversality theorem. It is the main purpose of the present paper to carry out this program.

In Chapter 1 we give an abstract codification of the kind of structure offered by our construction.

There are three stages of the analysis.

First

we group together all points of M that "look alike" locally in the pair (Q. M).

This expresses M as a disjoint union of open sets

m'. 1

which is a manifold and in fact a union of open simplexes of A.

fm.1 J

is called a "variety" of M in Q.

is a subcomplex A, of A. 1

neighborhoods of A. in Q. 1

each of The family

It follows that each closure cl

m.1

Our construction thus provides us with regular in M and in each A. (by Cohen's definition of

J

a regular neighbourhood [7]). These regular neighbourhoods fit together tidily enough, and we describe the situation in terms of a "normal regular neighbourhood system" for the variety {mil of M in Q. Q is a manifold, is a submanifold of Q.

Next we note that

mp

in case

and that our construction puts on

(very roughly) the regular neighbourhood of A. in Q the structure of a nor1

v mal block bundle - - this is Rourke and Sanderson's construction. that the regular neighbourhoods of A. in the general Q,

This suggests

M and the various

1

A. can also be given block bundle structures, if we allow block bundles whose

J

fibre is no longer a disk but any cone.

Our final step is to define such block

bundles and describe how they are to fit together. tion of a "stratification" of M in Q.

This completes our defini-

It should be observed that our construe-

tion for M in Q includes a construction for M in itself. stratification to think of

J

of M in Q includes a stratification as a "generalized bundle" over

Thus our final

of M.

It is fruitful

this is why I use the notation

us with Greek letters for the "bundle" space and Roman letters for the "base" space. In Chapter 2 we simply verify that the result of our construction satisfies the axiomatic systems of Chapter 1. Theorems about stratifications are proved by dealing inductively with one block bundle at a time.

(It is a meta-theorem that any geometrical proof

in Rourke and Sanderson 's [19, I and II] applies to a block bundle with arbitrary fibre whose base is a manifold. the Appendix; see also p. vii.)

This idea is developed in

In fact, over each

7ll i'

its block

bundles in Q, M and so on, fit like sub-bundles one of another, and can be treated simultaneously. over some

m.1

Difficulties arise when passing from block bundles

to block bundles over some other "I.. J

In proving the unique-

ness theorems of Chapter 3 for stratifications of fixed M in Q,

these

difficulties are overcome by a massive use of induction on "complexity" (that is, the number of terms in the variety).

I should add that the uniqueness of

normal regular neighbourhood systems of M in Q is a straightforward consequence of the usual uniqueness theorem for regular neighbourhoods. also prove that if K is a fixed stratification of the variety {"I.} of M. -

1

We

then

VI of M in Q which is over

there is an essentially unique stratification

This theorem reduces the description of abstract regular neighbourhoods of M to the description of abstract stratifications

over K - a generalization

of the description of bundles over a given base space. We have now generalized to stratifications the first main result about block bundles.

In Chapter 4 we consider transversality, proving this

theorem: Given polyhedra X,

Y in a manifold M.

Then there is an arbi-

trarily small isotopy of M which carries Y "block transverse" to a stratification of X in M.

Further, if a subpolyhedron Y'

of Y (which satisfies a

certain natural condition) is already block transverse to the stratification, then we may keep Y' fixed during the isotopy.

This theorem is new, I

believe, even in case X, Yare submanifolds of M and Y'

of Y.

The

difficulties I mentioned before prevent me from proving (or disproving) that block transversality is symmetric. As an application of the relative transversality theorem, we prove in Chapter 5 an analogue of Thorn's theorem [27] which exhibited an isomorphism between the group of cobordism classes of n-dimensional differentiable manifolds, and the n-th homotopy group of the spectrum of Thorn complexes of classifying bundles for the orthogonal groups.

One would like to define poly-

hedra X and Y to be "cobordant" if there is a polyhedron W whose boundary is the disjoint union of X and Y.

However, every polyhedron

X without boundary would then bound the cone cX; is trivial.

so this cobordism theory

We define non-trivial theories thus: Let 3' be a family of poly-

hedra (satisfying certain conditions).

A polyhedron X is an ":, -polyhedron"

if (roughly) for every point x E X,

its link tk(x, X)E J'.

the cobordism theory in which X,

Y and Ware all required to be

;J-polyhedra.

An ":;-classifying" stratification

'!../J!

Now "J'-theory" is

is a stratification of an

VII ;,-polyhedron in a manifold which satisfies an appropriate universal property for "morphisms" of stratified ;,-polyhedra K - 'Q.

By such morphisms into

U are classified all stratifications over K in manifolds.

Hence, in the light

of Chapter 3, we can classify all abstract regular neighbourhoods (Q, M). where M is a fixed polyhedron and Q a (variable) manifold. stratification of M in Q,

then the "Thorn space" of

is a

If

is defined by

identifying to a point the complement of a regular neighbourhood of M in Q. Then our analogue of Thorn's theorem is: For each family;"

there is an

isomorphism 'X : ;, -theory - the homotopy groups of the spectrum of Thorn complexes of ;'-classifying stratifications. As another consequence of the relative transversality theorem we prove:

Let u/K be a disk block bundle (in the sense of Rourke and

Sanderson) with K any polyhedron.

Let M be an abstract regular

neighbourhood of K which is a manifold.

By extending !-1 over M we ob-

tain an abstract regular neighbourhood !-1*M of !-1 which is a manifold. Then the function !-1 *: (Manifold regular neighbourhoods of K) - (Manifold regular neighbourhoods of u) is a bijection. In Chapter 6 this question is raised: given Xc Y,

when does X

have a (disk) -block bundle neighbourhood in Y? A necessary condition is that X be "locally fiat" in Y;

that is, for every point x E X,

there are

neighbourhoods U of x in X and V of x in Y such that U c V. the pair (V, U) is p , L, a disk,

isomorphic to the pair (UxD, UxO),

0 an internal point of D.

and

where D is

However, this condition is not sufficient.

A primary type of obstruction is defined and, in principle, classified. obstructions are not sufficient; the usual difficulties arise. elucidate the problem though I cannot solve it. in proving block transversality symmetric.

These

I have tried to

It is related to the difficulty

VIII In Chapter 7 I have listed various problems concerning stratified polyhedra that interest me; some spring directly from the material of the previous chapters; some are not yet related to this material and I think they should be. In the Appendix is outlined the theory of block bundle flags,

I have

used an even more general setting, in order to deal simultaneously with other structures needed in this paper,

The method of developing the

theory, including the statements of results and their proofs, are largely a straightforward generalization of the corresponding parts of Rourke and Sanderson's [19, I and II],

Accordingly, I have only given those details of

proof which might not be immediately apparent, As this paper was being prepared for the press, I proved that block transversality between a polyhedron and a submanifold of a manifold is symmetric.

I have inserted a proof as Chapter 8,

Contents

Chapter 1 § 1 § 2 § 3 § 4

Chapter 2

Definitions. • • •

1

General Polyhedra. • • • • • Disjunctions and Varieties • • Regular Neighbourhood Systems. Block Bundles and Blockings.

1 12 21

Existence.

43

29

Chapter 3 Uniqueness • •

56

Chapter 4

Transversality •

79

Chapter 5

Classifying Stratifications and Cobordism Theories •

98

Chapter 6

Obstructions to Block Bundles. • •

• 133

Chapter 7

Open Questions • • • • • •

• 152

Chapter 8

Symmetry of Transversality •

Appendix Bibliography and Related Reading. •

159 • • 179 • 191

Chapter 1.

Definitions

General Polyhedra This section contains our polyhedral and combinatorial foundations. We follow Cohen [7, section 1] with a few additions from Zeeman's more general treatment [32, chapters 1 and 2]. A simplicial complex B will always be assumed locally finite and contained in some Euclidean space R countable.

IB I

q.

Hence B is finite-dimensional and

denotes the underlying topological space of B.

The notation

A < B means: A is a subcomplex of B. We shall denote simplexes of a simplicial complex B by Roman letters r-, s, t; and if the vertices of s are v 0' s

= .

If s, t E B,

=

••. ,

va'

then we write:

we define

t h e simplex of B whose vertices are those of s together with those of t, if it exists; otherwise

f

A barycentre of s is any point

of the interior of s.

If a barycentre is

given for every simplex of B,

then we can form a corresponding first-

derived subdivision B' of B.

If A < B is a subcomp1ex,

the induced subdivision of A.

s*A

I

[

The dual cone to s in A is defined as

fI,

/I,

s




h : sttx; Y , ... , Y ,X , ... , X,) q 0 n 1

with

x

D x ctG, F)

v X c,

for some v E

Here (X , ... ,X ) is the filtration associated to (:x:.}. n

0

1

depends only on the equivalence class of {X.},

This definition

since one only needs the

1

equivalence class of the filtration (G, F).

n.

The filtration of X associated to

a variety of X in (Y) will be called a variety filtration of X in (Y). The most important varieties for our purposes occur this way: Given polyhedra X

Y and a point x in X,

dimension of x in (Y, X),

denoted d(x; Y, X)

are compact polyhedra G:: F, h : st(x; Y, X) with x

-;>

v

X

D

i

c,

dtx; Y, X) "i if it is

we define the intrinsic

an i-disk

d(x; Y, X) > i if there

by:

ni ,

and a p. t. isomorphism

x c(G, F) for some v E

>i

but

n

i

t. HI.

. The intrinsic variety of X in Y is

defined by: X." (x EX: dtx; Y, X) " i}. 1

It will be proved in Lemma 1. 10

that the intrinsic variety is indeed a variety.

For now we note that if X is

of dimension n (and all our polyhedra are finite-dimensional), then X is the disjoint union of X , ••• , X .

o

x in X such that st(x; Y, X)

n

z

p , L,

One can define 1. directly as the set of

n\ c(G, F)

1

as above, where the pair

(G, F) is neither the suspension of, nor the cone on, another pair. pair (G, F) will be called the basic link of x in (Y, X), b. tk(x; Y, X).

This

denoted

The associated filtration to {X.J is the intrinsic filtration 1-

of X in Y. By setting y " X we obtain definitions of the intrinsic variety and filtration of X,

and the intrinsic dimension and basic link of a point x in

1.14

14

X.

The intrinsic variety of X in Y refines the intrinsic variety of X. Akin [1], following Armstrong and Zeeman [4], has defined the

intrinsic dimension of x in (Y, X) slightly differently. if (to use our previous notation) h: st(x; Y, X) -> x

v

c,

X

where v is in the interior of

ni •

n i x c(G, F) with He defines the intrinsic

skeleta of X in Y by: Ii(y, X) = [x EX: d'{x; Y, X)

I].

to the disjunction of X by I! = (x EX: d' (x; Y, X) = i} 1

variety of X in Y;

His d'{x; Y, X) ::: i

We shall refer

as the skeletal

from it we derive the skeletal filtration of X in Y.

It will be proved in Lemma 1. 10 that the skeletal variety is indeed a variety.

By setting Y = X we obtain the skeletal variety of X,

etc.

The skeletal variety of X in Y refines the intrinsic variety of X in Y.

In fact, it will follow from Lemma 1. 10 that every variety of X in

Y refines the intrinsic one.

They are not always equal, as these examples

show: 1.

Let M be a manifold with non-empty boundary.

variety is (M - bdy M, bdy M}; 2.

Let (I.}. 1

J

but its intrinsic variety is just (M}.

be the intrinsic and skeletal varieties of X.

intrinsic variety of X x I is (I.

1

(Ix,J). = X. 1

3.

X

1

If G::: F

(O,l) U I. 1 1-

X

X

I);

Then the

but the skeletal variety is

(0,1),

are compact and (3) = (J"n'" .,J"k} is a variety of F in G,

then (c, (c.1.-c)) is a variety of cF in cG. 1

of F in G,

Then its skeletal

If (.1.} 1

is the intrinsic variety

then (c, (cJ". -c)} is the intrinsic variety if cF in cG - unless 1

(G, F) is a cone-pair or a suspension pair, in which case it is

1. 15

15

{c3: (c.7.-c) : i 1 k• F in G.

= k+1 •...• n}.

If {j! ••• .. :I"} is the skeletal variety of n 1,

then the skeletal variety of cF in cG is

(c U j! (cj!-c) uj!

t

1,

1,+1· .. ·'

«s:n-l -c)

U j!

n'

cj! -c}

- unless (G, F)

n

suspension pair, in which case it is (j!, cj! U j! 1"'" (cj' 1,

4. k

n-

1,+

Let (X , ...• X ) be a variety filtration of X in Y. n

0

= O••••• n,

•...• X

even if (X) is intrinsic.

o)

is associated to a variety of (X

k) 1

and not (8

1•

I-c) U j! .es: -c}. n

n

Then for each in Y.

However.

need not be; for example. the intrinsic filtra-

tion of the one-point union 8 V 8

5.

1,

is a

2

2, 1 is (8 V 8

8

1,

base point) is the intrinsic filtration of 8

base point), but (8

1)

1.

Whitney [29] has defined a "stratification" of the analytic variety V as

an expression of V as the disjoint union of a locally finite set of analytic manifolds. each of constant dimension. called the" strata", such that the frontier of each stratum is the union of a set of lower dimensional strata. I have used the term "variety" in our analogous p. t: context for two reasons: first, I wish a p. t. "stratification" to be analogous to Thom's definition [28] of an "ensemble

which has more structure than Whitney! s

stratification; second, an analytic variety has more structure than just its underlying topological space, so a "p. t: variety" may reasonably denote a polyhedron together with additional structure.

The connection between

Whitney stratifications and p. t: varieties is provided by Lojasiewicz (14].

One

can put on a stratified analytic variety a canonical p. t: structure in which the closure of every stratum is a subpolyhedron. analytic strata form a p. t. variety.

In this p. t: structure the

1. 16

16

Even when there is a canonical" simplest" analytic Whitney stratification,

6.

the resulting p. t. variety may not be intrinsic. Brieskorn varieties; say V,

of real dimension 10, in complex space C

(which we regard as real space R

+

+ .•• +

1 2),

+ z; = O. The only singular point of V is the origin, c.

with the unit sphere in R

12

simply a 9-sphere,

The intersect ion of (R 12, V

9

is a pair (Sl1, L: ),

an exotic homotopy sphere (see Milnor [16]).

L:

9

is

V is a p. t . 10 manifold, and its intrinsic variety is just

However, the L: 9 is knotted in Sl1 (see [16]) smoothly and piecewise

hand, this resource fails us if we include 10

10)

and it is known that L: 9 is

So in the p. t. category,

linearly; hence the intrinsic variety of V in R

v

6

is defined by:

So V has the analytic stratification (V- c, c).

V.

Let us consider the

in R

14

is just V,

12

is (V-c, c). On the other

c 6 ::: C 7; the intrinsic variety of

since local knotting cannot occur piecewise linearly

in this codimension; whereas the analytic stratification is still (V -c, c).

This

example explains why we work with varieties in general, and not just intrinsic (or skeletal) ones. 7.

A subsidiary reason for using general varieties is this: One of our main

questions is, given a polyhedron X,

how many regular neighbourhoods M

does it have which are manifolds (up to p, L isomorphism reI X)? We shall only be able to discuss the weaker question, given X and a variety of X, how many M are there such that the intrinsic variety of X in M is the given one? If dim M - dim X

3 (and under another simple hypothesis),

the intrinsic variety of X in M equals the intrinsic variety of X.

I shall

1. 17

17

often. I admit. be compelled to make these assumptions.

But at least the

reader may find it helpful to think of a general variety of a polyhedron X as its intrinsic variety plus other manifolds along which we allow local knotting when embedding X in manifolds. Theorem 1.6. Given (Y •...• Y). Xc Y q

(X •...• X ) of X in (Y). n

1.

0

0

0

and a variety filtration

Then:

Each:t. of the associated variety is an open manifold (see §1 for the 1

definition) ; 2.

If K is a cell-complex triangulation of Y in which each Y. or X.

J

covered by a subcomplex K. or L.. J 1

1

is

then bdy(Y. X) and

covered by subcomplexes bdy(K. L) and bdYKL of K. 3.

Let f be an isotopy of (X). t

Then f extends to an isotopy of (Y. X). t

4.

If in fact (X) is the intrinsic filtration of X in (Y) and K is a cell

complex triangulation of (Y) in which X is covered by a subcomplex. then each X. is covered by a subcomplex of K. 1

These results are due to Akin [1]. filtration of X in Y.

His work deals with the skeletal

but the proofs apply to the present situation.

virtue of 1. if [:t i) is a variety of X.

By

we shall henceforth speak of :t i as

a manifold oT the variety. Lemma 1. 7.

r

Let (F. F. ) ) and (F'. I F!) 1

1

be compact polyhedra with

families of subpolyhedra indexed by the same set .; E

o,

FA L'

n.

Assume that for some

Given a p. t: isomorphism f: Dnx c'{F",

n where D is an n-disk,

k D a k-disk.

Then n':::' k,

1

-- Dkx c [F', (F.) • 1

,k-n-l F f.. is either S or 1

1. 18

18

k-n-1 -1-1 D (here S = D = I

.g:

-> FIl1

1

and there is a p. t: isomorphism

* (F, (F.}). 1

The proof is by induction on n, If n

= 0,

we have

(F', fF!}) Z tk(c', c'(F', (F!}» 1 p. t. 1 k -;> tk(fc', D x c(F, fF.}» f

1

k

*

p,Z t; tk(fc', D x c) {F, I Fl'}) k k since fc' E D x cFA\ = D x c. Under this triple composite, 1

1

p, .

tk(fc', Dkxc)

* Fi:..1 = tk(fc ', Dkx c)

as fc' E int(Dkxc) or fc' E bdy(Dkxc).

k- 1 which is Sk-1 or D according Thus the lemma holds for n = O.

n For the general step, take a point x E bdy D;

k so f(xxc') E bdy D x c.

' then x x c' E bdy(Dn xcF/N' 1

n-1 Hence D x c'{F", fF!}>

r> p. i,

1

?d

p.. t

n tk(xxc', D xc'(F',{F!}» 1

k

tk(f(xx c'), D xc(F, (F k-1 D

i}»

x c(F, {F.}}. 1

Applying inductive hypothesis to this triple composite, we see that the lemma

,

holds in this case, and hence for all n, Observe that FI.' is a sphere or a 1

k

disk according as f(yxc')E int(Dkxc) or E bdy(D xc),

where y is any point

n• of int D Lemma 1.8. Given (Y , •.• , Y), Xc Y and a filtration (X , ••. , X ) of q 0 0 n 0 X.

Let D be a disk.

Then (X) is a variety filtration of X in (Y) if and

only if (DxX) = (DxX , •.. , DxX ) is a variety filtration of D x X in (DxY). n

0

Let (DxX) be a variety filtration of D x X in (DxY). r

D,

x E I.. 1

Then there exist compact polyhedra

(G , ..• , G , F , ... , F.+ q 0 n 1 1,

and a disk D" such that

Take points

1. 19

19

st«r, x); Dx Y , ... , Dx X.) ::::: Dl! x c(G, F). q 1 p. t:

On the other hand,

st«r, x), Dx(Y, X» "" st(r, D) X st(Xi (Y, X»i p.t.

and st(r, D) is a disk D', while

st(Xi (Y, X» is of the form c(G', .•• , G' , F' , ... , F! I' F!), q

coneon tk(x;(Y,X».

0

n

1+

being the

Thus we have D"xdG,F)::::: D'xdG',F'). p.t.

Lemma 1. 7, it follows that c(G', FI)::::: D*xc(G, F), p.t. That is,

1

Using

for some disk D*.

(X) is a variety filtration of X in (Y) "near x",

Since this

holds for all x E X, (X) is a variety filtration of X in (Y).

The converse

argument is easier, and we leave it to the reader. and X' eX c Y .

Lemma 1.9. Given (Y , ... , Y), q

0

a variety filtration of X in (Y), XI in (Y, X).

-

-0

Let (X , ... ,X ) be

n

0

and (W, V) a regular neighbourhhood of

Then (V) is a variety filtration of X' in (W).

If x is a point of V - fr V,

then st(x; W, V) is also a st(x; Y, X),

so a suitable product structure D X c(G, F) on st(x; Y, X) gives one on st(Xi W, V).

If x E fr V,

we use the fact that fr(W, V) is bicollared in

(Y, X) and Lemma 1.8 to show that fr(V) is a variety filtration of fr V in fr(W).

k Hence there is a structure st(x; fr(W, V»::::: D x c(G, F). p, t.

Now stfx; (W, V» "" st(x; fr(W, V» x I p.t. D p7t.

p7t.

k

X

c(G, F)

X I

k 1 D + X c(G, F),

and this is a suitable product structure.

This proves Lemma 1. 9.

Lemma 1. 10. 1. If (X , ... , X ) is the intrinsic filtration of X in Y, n

0

then the intrinsic filtration of D

k

X X

k k k in D x Y is (D xX , .•. , D xX ). n

0

1, 20

20

2. The intrinsic filtration of X in Y is a variety filtration of X in Y. 3.

The skeletal filtration of X in Y is a variety filtration of X in Y. We have the proof of 1, to the reader. Proof of 2.: We use induction on the dimension of X.

is trivial if dim X

The result

= O. For the general case, take a point x E X.; then 1

there are compact polyhedra G

i

F such that st(x; Y, X) z

D

p.t.

c(G, F).

X

To say x E X.-X. 1 is equivalent to saying that (G, F) is not a cone-pair 1

1-

or a suspension-pair, by Lemma 1, 7. hypothesis, the intrinsic filtration (F filtration.

By Remark 3, p. 1, 15,

filtration of cF in cG;

Now dim F n-

c(F

$ dim X.

So by inductive

1"" , F ) of F in G is a variety

n-

0

1"" , F ) is the intrinsic 0

the proof of Lemma 1, 9 shows that c(F

n-

is also a variety filtration of cF in cG.

By 1. and Lemma 1. 8,

ni

X cF in

X c(F) is the intrinsic filtration of

filtration.

ni

1"" , F ) 0

n i x cG, and a variety

But st(x; X , ••• , X ) agrees with the intrinsic filtration of n

0

st(x, X) in st(x, Y) at least in a neighbourhood of x.

This suffices to prove 2.

Proof of 3.: If Fe G are compact polyhedra and (F' 1"'" F') are n-

intrinsic skeleta of F in G, skeleta of int

0

then as in 2. one shows that the intrinsic

ni x (cF-F) in int n i x (cG-G) are the int ni

inductive proof of 2. can now be applied.

X

J

J

The

1. 21

21

§3. Regular Neighbourhood Systems Given a filtered polyhedron (X) = (X ••..• X •...• X) and q n 0 X*

< bdy(X). A normal (q, n)-regular neighbourhood system (abbreviated

to: normaltq, n)-rns) for X

in (X) with respect to X* is a family of

n

subpolyhedra of X N = [N. . : 0 < i < n,

-

J.1

--

i _< j _
. then an isomorphism f: N

is a p. t. isomorphism f: (X) - (XI) such that: f : N. . FIll NI.. for all N. . E N; J. 1 p, t. J. 1 J. 1 f : (N' J • 1

FIll (N':) p.t . • 1

and a

for i = O..... n .

FIll

N'

1. 22

22

Remarks.

1.

Set N* -

= (N.J,1. n

x* : N. . E N_l; J,1

then N* is a normal

(q, n)-rns for (X*) in (X*). n 2.

The reader may be pleased, or exasperated, to know that in the main

applications of the machinery we are setting up, q will be either n or n+1.

I have left the general q in the notation partly to get two definitions

for the price of one; partly in hopes the general notion may prove useful to someone; partly from this consideration: Let us start with a filtration (X , ..• , X ) of X, n

and an n-z-ns N of (X) - thus q

0

trate on X in X,

n_ 1'

-

then

= (NJ,1 ..

namely: (n-l)N -

n-2,

either j (n, k)-rns

: 0 < i < n-j ] --

k or j = n 1 of (X

k)

= n+1

n_ 1) is

But if we concentrate on some

in X,

(k)N = (N .. : 0 < i < k , J,1 -1.

of (X

- thus the case q

then rather than a normal (k+1, k)-rns

Proposition 1. 11.

If we conceri-

includes a normal (n, n-l)-rns

inferrable from the case q = n. k

= n,

N' -

= (N.

. : 0 < i < k and J,1 --

it is more natural to take the normal

i _< j _< n} for (X in (X X k) k). n"'"

Given (X) and X* as above.

Then there exists a

normal (q, n)-rns N for (X ) in (X) with respect to X*. n

2. If N' is another such normal (q, n)-rns and

=

then there is an

isotopy f of (X) such that: t

f is the identity on t

x-

and on X

o'

Proof of 1.: Let B be a simplicial triangulation of X such that X* and each X. is covered by a full subcomplex B* and B. of B. J . J be a first derived subdivision of B.

Set

Let B'

23

1. 23

1.24

24

-Think of X as standing for a very long cone on a figure "8". To simplify the picture, I am only drawing the upper half near the cone-point.

A normal (3,2) - regular neighbourhood system for X in M.

diagram 3

1. 25

25

N.. J,l

U{SEB':sEN(X.,W), 1 J

for 0:5. i .::: n,

but

1-

i:5. j .::: q,

Using induction on n and Proposition 1.4 to prove the inductive step, one shows that N = {N. . J is the required normal (q, n)-rns. J,l Proof of 2.: uses induction on n and Theorems 1. 1 and 1.3 to prove the inductive step. Corollary. X,

Given equivalent filtrations (X, ... , X ) and (X', ... , X') of q

such that X

n

and given X* < bdy(X) and < bdy(X').

= X' .

m'

a normal(q, n)-rns for X

n

in (X) with respect to X*.

a normal (r-, m)-rns for X' m

Nil di}., k, t n X'i-I = l'

0

Let N_ be

Then one obtains

in (X') with respect to X* by renumbering

the components of the N . . 's. J,l N'. . = U{N" : N" J, 1 k, t k, t

r

0

To be precise, set

is a component of N

an d s et N"j, i = U{N'k, t

k, t

E

N'k'

, t n X!1 f

n N"k, t } .

This follows from the construction used to prove the first part of the Proposition: one simply chooses B to triangulate (X); triangulates (X').

Thus one obtains a normal (q, n)-rns M and a normal

(r-, m)-rns M' related as stated.

isotopy f (X'),

so

Remark.

t

of (X) carrying

¥

By part 2 of the Proposition there is an to

o

but f

t

also respects the filtration

= f (¥') is a normal (r-, m)-rns related to N as stated. 1 The assertion of Proposition 1. II, 1 is a tautology, since the

definition of a (q, n)-ns is constructive: of X

then B also

in (X),

one takes a regular neighbourhood

throws it aside, takes a regular neighbourhood of what's

1. 26

26

left of X 1 in what's left of (X),

and so on.

The proof of the assertion is

more important; it says that rather than successively taking smaller and smaller second-derived neighbourhoods, one can construct a (q, n)-ns with just one second-derived triangulation. Let N. = U [ N. . : i = 0, ••• , mintn, j)}. J, J, 1 polyhedron (N) (N

q,

= (N

q,

, ..• , N

0,

The proof of Proposition 1. 11 shows that

).

, ... , N +1 ) is a regular neighbourhood of X

n,

to be its frontier, filtered by (fr N N.

J,

n

fr N

=

= {N.J,1.

-

: 0

q,

, ... , fr N

in (X).

n

Define fr N

1); note that

n+ ,

for j:s. n.

Given numbers n N

Then we have the filtered

a family of polyhedra

< i < n, i < j < q}, and a polyhedron fr

-

-

--

Then N is a

N.

regular (q, n)-neighbourhood system (regular(q, nl-ns) if there is a filtered polyhedron (X , ••• , X ) such that N is a normal (q. n)-ns for q

X

n

0

in (X) with frontier fr N.

-

N is a weak (q, n)-neighbourhood system -

--

(weak (q. n)-ns) if there exist a filtered polyhedron (X , •.• , X ) and a q

normal (q, n)-ns N' for X -

all j, i;

n

in (X) such that N . .

and fr N is defined as fr N'

J. 1

n

N'

q,o

•

N

-

=

0

J,

.

1

n

N' , q. 0

for

is a (q, n)-neighbourhood

system «q, nl-ris) if it is either a regular (q, nr-ns or a weak (q, nr-ns , In case q

= n,

we write: N is an n-ns ,

Remarks 1.

We could avoid introducing the auxiliary (X) by giving definitions in

terms of weak regular neighbourhoods. 2.

In accordance with our notion of the equivalence of filtrations, there

1. 27

27

is no need for our indices to start from 0; what is actually used is a filtered polyhedron (X

, ..• ,X ) of length q and the subfiltration of q+a a

We stick to the convention that X. and N . . are 1 J,1

length n.

i I}; J,I -

then M' is a restriction

of N'; 3. M'. J,

n

N" = M" q,o j,o

for j

= 1. ... , q.

This is stronger than simply requiring for Z

n

in (Z).

We write: M < N. -

regular and weak (q, n)-ns's. is a restriction of Lemma 1. 12.

Similarly we define restrictions of

If Z < bdy(N

= [P.

,[N. . J), J,I

q,

in this case we say:

Given (q, n) -ns' s

and

and an isomorphism f:

< P -

!'i;

-

to be a normal (q, n)-rns

. : 0 < i < n, i < j < q} by: P. . J, 1 - - J, 1

then it follows that

< bdy restrictions


... ::> F. l' n 1+ p:

/I

and a p. t. isomorphism

a disk D.

...• Bi--:> Dx c (Fn+1 •...• /I

Now st(s; (BI» = oS * s*(B' +1' .... B1 (the simplicial join). and n •..• , 1)

=

S*«B'»

1\

s * t k(s; (B'» is a cone with subcones.

to s x S*«B'»

p.t.

we find a disk D' and a p, t , isomor-

D X c(F.

phism S*(B' 1••..• n+

1 p.t.

DI X

c(F

n+

1•...• F. i ' 1+

every s*B. has a block structure with base s*B.. 3 1 /I

that if t E bdy Xi'

Applying Lemma 1. 7

then t*bdy B

j

This shows that Similarly one shows

has a block structure with base

t*bdy B. • 1

Now let h: s*B .... s*B. J 1

X

cF. be a structure. J

corresponds s*B.) 3

s*B.

1

X

cF.) 3

.. tk(b. s*B.} X cF. U s*B. X F .. 1

3

1

3

On the other hand. as simplicial complexes.

Then h

2.4

46

A tk(s, s*B.) J

A

= st(tk(s, s*B.), t I

A k(s, s*B.» J

U tk(s*B.,s*B.). I J

1\

Pseudo-radial projection from s gives a simplicial isomorphism 1\

f: t kta, s*BJ -. [t k(s, BJ]' (the first derived), under which J

1\

J

t k(s, s*BJ [tk(s, BJ]' ; I

I

A A st(tk(s, s*B.}, tk(s, s*BJ) I J st(tk(s, B.), [tk(s, B,)]'); I J tk(s*B.,s*B,) I J

(tk(s,B.), [tk(s,B,)]'). I J

(See diagram 5, p. 2.5) By definition of a regular neighbourhood, and by virtue of the isomorphism f,

t

s*B.),

s*B.» is a regular neighbourhood of t J.

I

s*B.). J

s*B.} in I

By uniqueness of regular neighbourhoods, we may assume

that h corresponds 1\ 1\ A st(tk(s, s*B.), t k'(s, s*BJ) t kfs, s*BJ x cF ., I J I J

and hence t k(s*B., s*BJ I J

(-:>

A

s*B. x F.. Now if s E int X. , I J I

"

which is a manifold without boundary, then tk(s, s*B.) is a p. L sphere and equals bdy s*B..

Hence U{r*B. : s

I

I

structure on bdy s*B.; I

I

< rEB.} is a cell complex + I

and under h,

u[r*B. : s < rEB.} os*B., J + I J t k(s*B., s*BJ (s*B J • I J J This checks the block structure on s*B. if J

int X.. I

A similar calculation checks the block structure on A

when t E bdy Xi'

A

However, if s E bdy Xi'.

B., J we have the

47

dia gr am ..2 .

2.6

48

"

difficulty that ,t k(s, s*B.) is a p. t. disk, not a sphere, and bdy s*B.

1

" s*Bi) u = t kts,

1

s*bdy B.. We make a further calculation; 1

there is a block structure h: s*bdy B. - s *bdy B. x cF.. J 1 J

There is also

a p, t. isomorphism f: s*(B., B.) - s*bdy(B., B.) x I. J 1 J 1 Combining with h gives hit : s*B. -;> s*bdy B. x I x cF., J 1 J

extending

h: s*bdy B. -;> s*bdy B. x 0 x cF .. J 1 J

"

tk(s, s*B.) J

Now h corresponds

(3 s*bdy B. x I 1

U s*bdy B. x 1) x cF. 1 J U s*bdyB. x I x F .• 1 J The previous argument, using pseudo-radial projection and the uniqueness of regular neighbourhoods, shows that we may choose hit still to extend h and also such that

"

"

h : st(tk(s, s*BJ, tk(s, s*B.» 1 J (3 s*bdy B. x I U s*bdy B. x 1) x cF. 1 1 J and t k(s*B., s*B.> s* bdy B. x I x F .. J

1

J

1

"

"

Hence s*B. has edge: s*bdy B. u st(tk(s, s*B,), tk(s, s*B.» J J 1 J and rim: t k(s*B., s*B.). 1 J

From this follows the desired block structure

on s*B .. J Let A, C be any of the B. or bdy B.. Then J

r*A n s*C one of the B

*(An C), j

or bdy B

r

J

for any r, s E B; .

Also If

A

i:"

E Xi'

1\ S

note that An C is again E

and

49

then

1\

E 'I

2.7

k

This observation makes it easy to verify axioms about the intersection of blocks. ebb. and each N= -

.. n

J.l

We leave the reader to check that each

.) a flag.

.. is a J.l

The proof of Proposition 1. 11,1 shows that

,1

is a normal (n+1,n)-rns for X in (Y.X).

Theaxioms

for a blocking are easily checked, except perhaps part of bs 3: that for Nil = {(s*B)' n N . . 1 is a normal (n+1, n) -rns J.l

every simplex s E B. n

for (s*B )' in (s*(B»·. n

The proof uses pseudo-radial projection from

"s again to give a simplicial isomorphism

f: t k(s*B., s*B) -+ [t kta, B))'. 1

Then tk(s. B ) has the normal (n+l. n)-rns n

the simplicial triangulation t kfs, B). Proposition 1. 11, 1;

N!. J,l

r E N(,tk(s. B.), [tk(s, B.»)'), 1 J r

n Itk(s. B i -1) I

f :

. So

.

N' in tk(s. (B» -

dual to

was constructed in the proof of

U(r E [t k(s, B)]': but

But f gives an isomorphism of ns' s,

is a normal (n+ 1, n) - rns, as required.

This completes the proof of Theorem 2.1. Proof of Addendum: We are now given (Y*,X*) < bdy(Y, X). restriction filtration.

Set Y" = cl[bdy(Y.X)-Y*).

and let (Y", X") be the

Let 'B be a simplicial triangulation of Y in

which y* and each X. is covered by a full subcomplex, 1

'B* and 'B.• 1

of 'B.

Since, by Theorem 1. 6,2, bdy(Y, X) is covered by a sub complex

of 'B,

it follows that Y"

is covered by a subcomplex B" < B.

collar neighbourhood y* x I of (Y*, X*) in (Y, X).

Then

Take a

2.8

50

bdy(Y, X)

n bdy(Y*x I, X*x I) = y*

Hence Y

n

bdy B* = B*

Y" :: bdy(Y*, X*),

n

B" of B.

°

0 U bdy(Y*, X*)

X

X

I.

and is thus covered by the subcomplex

Similarly,

°

bdy (Y. X) n bdy (Y*xI. X*xI)

= y*

°

X 0 U bdy (Y*, X*) X I.

Hence, though bdy(Y*, X*) is not necessarily collared in (y .., X..)

=Z

(for example, take X

X

X* = Z X 0,

I,

where Z is the region of

the plane bounded by a figure "8"), we do have that bdyo(y*, X*) is "locally collared" in A S E bdyy*X*.

definition of bdyo. (y.., X.. »

X.. ) in the sense that: if s E B

n

with

A A then st(s; bdy(Y*. X*» is collared in st(s; YO'. X''»,

it is also collared in

u st&

Y*, X*».

By

And

(Y*, X*» = stfs; bdy(Y, X»

A

is collared in st(s; (Y, X»,

A

since s E X*

bdyyX,

•

Usmg these collars,

the methods of proving Theorem 2.1 apply without further innovation to The stratification ...'!1 that we obtain is defined by:

prove the Addendum, Tl •• J.l

= {s*B.

A

: s E l:.} IJ J 1

A J

s E l:;!,} 1

E bdyyOX n r} u (s*bdy B'!': 1 J J The block structures on these blocks are: U {S*B ': :

s*B. has base: s*B.; J 1 and edge: U (r*B. : s u

of e :

I).

k, +

*

So " (cr,

is a stratification,

-

The general step in our induction on k is complete, and

Proposition 3.6 follows. Proposition 3. 7(n).

Given an (n+l, nr-bs

Then there is a subdivision

!l./!:

and a subdivision L of K.

of

The proof also applies to a (q, n)-bs constructs a (q, n)-bs

!l.

subdividing

We use induction on n. Appendix, Theorem I, 1. 5.

S

)/ L

0,0

n-bs is

whose n-bs is

one

!:.

Now assume Proposition 3. 7(n-1):

the Appendix, Theorem I, 1. 5. ,0

whose

Proposition 3.7(0) is a special case of the

There is a flag subdivision (iJ., L

completion of (iJ., L

S

Let

,0

)/L

0,0

of

K

,0

)/K

0,0

be a weak (n+l, nr-bs

.

I claim that P" can be chosen to subdivide " (L). 0-

For by

,by

3.18

73

Lemma 3.2, there are a subdivision

(IL ,0 I) of (L

pI

-

of I\*(P II ) and an isotopy at of 0-

such that alP' subdivides r-. *(L) and a -

,0

) for all t ,

0-

By inductive hypothesis, Proposition 3. 7(n-1), there

is a subdivision isotopy of

of

(I r; ,0 I),

any (n+1, nr-bs}.

1\6 (!:!.").

since (IK

,0

I)

We cannot directly extend at to an need not be a variety filtration

However, for each cell cr E K

*(e .

restriction

is a flag isomorphism

t

each P_"1.

J.

1

q + 1.

, =

J,

1

1

for

An application of Cohen's Stellar

Neighbourhood Theorem (quoted as our Theorem 1.3) shows that is a normal (s+1, s)-rns for (y ) (F,. • • (Y')• n

s

Since (y ) s

n

IF,. I in

n F,. • is collared in cl[(y')- I I] - this is a

to a normal (s+1, s)- rns N' of (y ) in M.

-

M;

= fV J,l ..J

IF,. I with respect to

typical use of our strengthened notion of block transversality -

(q-rI; q)-rns

V -

of (Y) in M.

s

For let

Now N'

-

y.

extends

extends to a normal

be any normal rns of (Y) in

then there is an isotopy of (M, Y) carrying the normal (s+1, s)-rns

90

J,

0

I

:

0

< i < s,

-

i

-

..

*I )/" H* . Subdivide 11 11 J'+-

r; so that every block of (" 11 *)

and the rim of every such block, are covered by subcomplexes

Then *);

=

1)/"H. We may assume (lin) extends the flag . ,

, •.. ,

"T)*, ••• , "

and (lin). of K.

s

(liT)

r;

K*I)/'H*. q+ s J+ regular neighbourhood of Nf.I n X in J to a normal disk block bundle

T)

IH

J

it follows that

of NI.\ in M; J

I'HI is a

Since I

T)

q+l

I 'H

extends

this is the required

block bundle. We now choose a flag structure (v) = (v q+l' ... , (N +1 ' •••• q

on

By the uniqueness theorem for normal flags. we shall be

able to isotop M till the sub- flag (v r' v , ... ,Vf.l 1) I H equals (r) q+ s J+ effect of this step, then, will be to improve from having X

i

I H.

The

(Nt). 11) to J

having X (approximately) block transverse to the whole system (Nq+ l' ... , (v ) is chosen to extend ('" *) I 'H* and such that (v s,·· "'1rl)/H = (n s'· ··,r)1+1)/H.

This can be done, since any choice of

(v ) can be isotoped till it meets these conditions (see the Appendix, Theorem

93

I. 4. A).

There is now an isotopy c

t

c 1 : ('V q+1''V ...• s' of the Appendix.

4.15

of M reI bdy M U N ••.•• T)t+l)/H.

Henceforth I shall suppress c

s

such that

by the same theorem

and assume that

t

(I I, X)

is already "transverse" to the flag ('V )/H. (I leave the reader to define "transversality to a flag" . ) The final step starts with choosing a disk block bundle plementary to (\I) of

'H.

I :: Ir;" I.

such that and

Then (N

Then (N

q+ 1

) :t

namely: K":: K and its rim are covered by sub-

comes from a subdivision of

/IS.

"I; /"K of «A"» complementary to (\I)

'H.

extends to a blocking

I claim we can choose

*/"K* be a subdivision of I; and which extends

to "K be a subdivision of S

some subdivision of

By subdividing

which extends «A *» and

as a sub-family of block bundles.

Let

"H>.

1

"/K"

'H defines a block decompo-

'H

Then s"s and

of «sX» such that hI: s"

)} I n bdy M.

s'

= [(v s • • • • • VA. )/"H; T'1-1

s

t

K 2(dim -

q,

I + 1);

where t is

u+t

thus C = B X A triangulates D

.

Let l:! be obtained from 11 X C X I by subdividing 11 x C X 0 to

l:!" = (11 X B)" X A

X

0 and 11

tively that for each j

X

C X I to (11

X

B)'

X

A x 1.

Assume Indue-

< i-I we have found a subdivision iJ. of

-

-

II (1-1 • .,1-1 ) J, J -

(which is a weak (n-j+1, n-j)-stratification, a restriction jy of subdivision jh:

rv -+ -

II

. . X I, ,. X I),

J, J

..:i.

:il:!.

and a

such that, whenever k < j:

j _1-1 restricts to a subdivision of II(lI *(kIJ ) . ., kIJ ) k - J, J i_v restricts to a subdivision of II (lIk*(kv) . .,kIJ), - J, J -

and jh extends kh there;

i _v meets 1-1" and 1.1' in subdivisions of II (1-1'.' .,1-1") and II (u! .,1.1 ,), J, J J. J Thus l:! has the subdivision }!/\ an (n+1, n)-stratification (such as !:!./\,

OA

l' l'

,

= U{II*(). . .,). ). . :

J, J

1,1

i -< i

- 1};

< bdy A. .' Similarly we define 0 v . .

= !:!. + X

I,

+ , .. + (i-I)}!. If A is etc.), define

thus 0 A. . is a cell complex 1,1

U{J\*(v . . ,1-1") •. : j J, J 1,1

Then the jh fit together to define a subdivision

3 such that for every (G, F) E .&-, G is a sphere. Then

for each r,

there is an r-universal J! -stratification.

Take a standard ordering of J!,

J! (k)

= {(Gj,

F

j)

: j

= 1,

... , k}.

and set

We use induction on k:

Assume that for

each integer s there is an s-universal .t (k) -stratification such that

11. (k) I

cS-block bundle

1.(k) 1'Q.(k)

(which is a manifold) is connected and has trivial tangent 'I

11. (k) I.

Our task is to construct, for each r ,

an r-

universal .. (k+1)-stratification satisfying the analogous conditions. . k+1 k+1 wr-ite (G, F) for (G ,F ), filtration of F

and let (Fn' ••• , F 1) be the intrinsic

in G.

11m Note that (G , F ) must be (S dim

= -1).

We

Hence a

over a manifold.

for some m::: 2 (since

(I)-stratification is just a cSm-block bundle

Now ry IrU exists by Rourke and Sanderson's [19, I,

110

§ 2], and Ir y I is connected. including how to arrange that general case,

5.13

The argument below gives some details; 'T

I ty I is trivial.

We proceed to the

(k+l):

First we construct a

«o. F n' ... , F I} -flag

... , 'I) I H',

where H' is a simplicial complex, which is r-universal in this sense: for every simplicial complex P

.: rip I. IHI] -+

of dimension < r,

the function

the set of isomorphism classes of c(G, F n' ... , F I)-block

bundles over P,

defined by

(h) = h*(

u '),

done using Rourke and Sanderson's [22, II].

is a bijection.

This is

Let Pt-(G, F) be the A-

group which has for a-simplexes all p. t . automorphisms

t

f:

X cG -+ Aa x cG (Aa is the a-simplex) such that:

a f is the identity on A x c; f

-1

f-

a

a

(A XcF) = A x cF

1(AaXG)

=AaxG;

1 Ab " v f - (Au b v" cG) = u c G f or every f ace Ab < Aa .

Since F is of codimension

at least 3 in G, the intrinsic filtration of F in G is (Fn' ... , F 1)' by Lemma 1. 10,

f

-1

a

a

(A x F.) = A x F., 1

a Pt- (G, F)-block bundle

e

1

for i

I, ... , n -1.

over a simplicial complex P,

Hence

It follows that

in the sense

of Rourke and Sanderson, can be given canonically the structure of a c(G, F n' .•• , F I)-flag

re. \)n' ..• , \)1)IP;

and conversely, given such a flag

(e, v >/p with P a simplicial complex, then Pt "" (G, F)-block bundle structure.

e/p

has canonically a

The existence of an r-universal

c(G, F n' ... , F 1) -flag now follows from [22, II, Corollary 2. 6] applied to

pC

(G, F).

111

We can replace u ") / H" such that

"I

Then

I'J

IH" " -

IH" I

is a connected manifold without boundary. Let

'J/

I

II

I

be its stable normal

Let H be a simplicial triangulation of

'J'

IH" I

Iv 1>.

polyspace of and now

by another r-universal flag

is a connected manifold.

cS-block bundle.

5.14

(regarded as a polyhedron, and not as a sub-

Then we can choose an r-universal flag

IH I is a manifold without boundary, and

manifold with boundary

I I

whose tangent bundle,

'I"

is a connected

I I,

is trivial.

Let 21/!:: be an (n+1,n)-bs completion of (l.l n'" .,l.l1)/H in (see Lemma 2.2). to form 21.

Thus 1\ *(21) consists of the blocks added to

o

Since (G, F) is not a cone- or suspension-pair,

1\ * (n) is a j;(k)-stratification.

o .:...t

> dim II\*(L)I

-

0-

= dim u" , n

stratification such that ,.

Pick an integer s> r + 1 and -

and let V(k)/U(k) be an s-universal --

Iv_ (k) I

is trivial.

Let h: A*(L) 0-

\-+

U'(k) -

classify 1\*("'). Pick a structure (Du,E, (U(k)xE)', f\* (L)',h ') for h o.:...t 0 such that dim I.,! (k)

I + u?:.

2 dim

+ 2.

Set

(U(k)xE) I,

and let

1. -tv: be a subdivision of 1. (k) x E; then 1. * is again an s-universal (k)-stratification with trivial tangent bundle. a subdivision h*: 1. * Since

(!::>

I

-+

21' We write

is boundary-less,

restriction bdy identified to bdy

1. *

f\ 0




0

-

F::I

(this uses the fact that p" is now an inclusion of one flag in another).

'1." = eo(ry"); then '1." is an s-universal "(k)-stratification. By definition, the restriction of p" to p*: I\. *(K") 1-+ U" is a classifying 0-

-

ry") Set

5.19

116

map for

0-

By subdividing e-

finely enough, we may assume there is a

0-

restriction

I, ... ,

suchthat

regular neighbourhood of I fI.

-« ") 1

J"

say

{/3

1.

X

I/31

some Io ],

I:

with

defined by the properties: e-(J") < -

n c (:!..If). Then

q'" : e-(:!..If) .... u"

II

classify it.

I"

1:!..". 0-

I,

-

and p*,

concordance.

I;

here we

I:!.." to

II

Alternatively,

-

c0(K"); c0(K") -

-

(JIf); -

St(k)- stratification; let

Since

can be amalgamated over

-

q* classify its restriction to the two ends, we have Let t: A *(K") 0-

X

IH U" be such a

We have the subdivision u: J" .... fI. (K") -

0-

U".

-

-

0-

x I 1 reI

X

1.

Note that

By Theorem 3. 1 there

0-

I

-

Let the subdivision

-+ fI. *(K") X I be part of a structure for t ,

is an isotopy w of t

c

" J" U

e,(J") is

We have the restriction q*: 1:!.." .... "Q."

tou extends p* U q* : fI.*(K") U 1J" -

J"}.

Q'

(:!..") is a

that p* and q* are concordant.

"t : K

X

By amalgamation, we may regard q* as a strati-

fied map q*: fI. *(KIf) .... U". X

and

*

0-

-

0-

classifying

0,

X

0-

we

0-

Define the (n-l)-stratification e,(JIf) to be

E e, (K") :

1:!..1f ,,:!.."

0-

is a subdivision of fI.

0 -

fI.*(F If) 0.2

Since

fI. *(K") corresponds to fI. *( S")

-

isa

in

0-

(lc0.2 (F")I, lc (K") I, ... , lc (K")l I). o-n, 0-, may also arrange that

I)

IJ'{,

x [0, I} I such that

I

and W10u(:!..") have a common subdivision,

*K.

Then the subdivision

10Itt : ("t) -1 (*'!9 -+ u(.:!.") can be used in a structure for an improved t such that t-u is a stratified map and still extends p* U q*. defined to be pA on A(K"

K")

0" 0' -

c (:!..If)

is such that v*(ry") "

a stratified map.

By Lemma 5.1,

Note that v:

'

t on J" -

Then A

•

and q

on

.... r"Q." is by definition

v is concordant to a stratified map

5.20

117

VI :

I=

such that, if p :

projection, then pov 1 : (povI)*(ry)

=

is a stratified map.

t+

Thus cp

Proof that

is the Clearly

is onto.

is 1-1: Given stratified maps f

cp

such that dim

x IB I ...

r and f

6(ry) = fi(ry).

f : o' l

Then, using a similar

argument to the one just given, one finds a stratified map F: such that on K

X

-

concordant to f

1

0, X

F is concordant to f

X

0,

rU

and on K x I, F is

0 -

1.

This suffices (in view of Lemma 5.4) to prove

is 1-1.

that cp

This completes the inductive step.

Hence Proposition 5.5 is

proved. Corollary. Proposition 5.5 still holds if J1 is finite-dimensional (and not just finite). This time let J1 (k)

= ( (G, F) E

J1 : dim G

is a family of link pairs. By induction on k, J1(k).

Say

(k)

1

1

k} ;

then each J1 (k)

assume the corollary for

..

= (G , F ), ... , (d, F J), ••• }. For each

j we construct an r-universal flag () j,lJ.j)/H with fibre (G

j,

F

j),

as in the proof of Proposition 5.5. Observe that we can make

the Hi all have the same dimension; then the j

Then the disjoint union n/!::

stratification; the point being that a polyhedron.

In

I will all have the

Let 11 j /!::j be a completion of

same dimension (since dim G = k+1). for each j.

I

I

U(.nj/!::j} isa

is finite-dimensional and hence

Now continue with the proof of Proposition 5.5, noting

5.21

118

that the integer z will do for all the .!l j at once. Our requirement on n-stratifications K that (IK j, n,

Remarks 1.

... , IK I)

be the intrinsic filtration of

using more general definitions.

IK I

can be set aside by

One would define a family

var,

each

of whose elements is a pair (S, F) together with a variety filtration (F , ...• F ) of F in 8. q

0

such a

I leave the reader to state the axioms that

var should satisfy.

family a var Uvar), stratifications.

and

One would also define the associated var-(n+1, n)-stratifications and a var-n-

There would then be a classifying theorem for

stratifications over a given a var-stratification. the proof is that

q

a

2.

x S

.

10

The only difference in

u ) would not be constructed using P t - (S, F) but

its /::,- subgroup P;;(8, F , ••. , F ), -+ /::,

var-

0

which consists of those f: I::. a x 8-+

-1 a a Pt(S,F) suchthat f (I::. xF.) = I::. x F., 1

1

for

.

1

=

0, ... ,q.

A more promising generalization is simply to restrict the I::.-group

Pt-(S, F),

by choosing

to be universal for some I::.-subgroup of

P t - (8, F).

For example, one could take only those f in P t - (8, F)

which preserve orientation of /::, a x S. which are the identity on /::,a x F;

Or one could use only those f

thus one would Classify ,I-(n+1, nl-

stratifications over a very special kind of a-n-stratification. 3.

The restriction in the hypotheses of our theorem that for all

(G, F) E

G be a sphere, is necessary for the proof.

point where we have to "fatten up" the flag that

,u)

u Y (k)

u ):

does indeed define a variety.

Consider the

this is necessary so If

-; •

is not a

119

5.22

e

manifold, then I do not know that the cS-block bundle

exists; this is

related to (what I call) the main unsolved problem of stratified polyhedra (see Chapter 7). if

The same problem arises in the proof of Lemma 5.6:

I" n+ 10 is not a , 1

manifold, I cannot prove . ./

and a,-l(E) isotopic.

Even if these steps could be managed, (as they can - see Chapter 8) if existed and

e

1-1' lay in

- how is one to extend

1) 1 unless one knows that

this except by requiring

e

e

e

over all of

is trivial? And I do not know how to ensure

1 1 and 11. (k) 1 to be manifolds so that one can

use their tangent bundles. (X , ... , X ) and (Y , ... , Y ) variety non 0

Let X, Y be polyhedra, filtrations of X and Y.

A p , L, map f: X ... Y is stratifiable with respect

to these filtrations if there are n-stratifications K, L of (X) and (Y) such that f:

Is-a stratified map.

f : (X , ... , X ) 1-+ (Y , .•. , Y), non 0

We write:

or if they are intrinsic filtrations:

f:X \o+Y. Stratifiable maps f ' f

O 1

: (X) i"+ (Y) are concordant if there is a

stratifiable map F: (X) x Il-+ (Yx I) such that Fand F equals f

o

x 0 on X

X

0

and f

1

X

1(Yx{0,1}}

1 on X

X

=

X x

to, 1} ,

1.

One can prove the analogues of Lemmas 5.1, 5.2 and 5.3, using Theorem 3.1 instead of Lemma 3.8.

One can also prove this analogue of

Lemma 5.4: Given polyhedra Y c; Z and X,

a variety filtration

(X , ... , X ) of X and a variety filtration (Y , ... , Y ) of Y in Z. n o n 0 Then any stratifiable map f: (X)

Ho

(Y) induces a pair f*Z:2 X such that

(X) is a variety filtration of X in f*Z.

The equivalence class (in the

120

5.23

sense of Chapter 3) of (f*Z, X) depends only on the equivalence class of (Z, Y) and the concordance class of f. filtration of Y in Z,

Moreover if (Y) is the intrinsic

then (X) is the intrinsic filtration of X in f*Z.

Hence there is a category B whose objects are polyhedra each with a variety filtration and whose morphisms are concordance classes of stratifiable maps. Putting together Theorem 3. 9 and the Corollary to Proposition 5.5 gives: Theorem 5.7.

Let.il be a finite-dimensional, boundary-less family of

link pairs of codimension at least 3, such that for every (G, F)E.iI, is a sphere.

Then for each r there is a

,j, -

pair

G

(rE, rV) which is

r-universal in the sense that ql

:

Mor[X, rV] --::> the set of equivalence classes of

is a bijection whenever dim X

,j, -pairs

(M, X)

r-,

Here Mor[X, rV] is taken using the intrinsic filtrations of X and rV.

We also note that rE and every M are manifolds.

The most important case of this theorem is when .iI is of the form I q(:J),

where :J is a finite-dimensional, boundary-less family of links

of constant local dimension, and q:: 3.

In particular, if X is a polyhedron

of constant local dimension rand bdy X is collared in X,

then

I q(:J (X» satisfies the hypotheses of Theorem 5.7; and if (rE, rV) is an

r-universal pair, then Mor[X, rV] classifies equivalence classes of pairs (M, X) such that M is a manifold whose components are all of dimension

5.24

121

q + r,

bdy M n X

= bdy X

and (bdy M, bdy X)

< bdy(M, X) .

If :J is not of constant local dimension, we can take

for each q 3

3,

defined to consist of all pairs (8, F)

dim 8 - dim F

Remarks 1.

=

)

such that

q.

The theorem can be generalized to classify, for each X

and variety filtration (X) of X,

equivalence classes of pairs (M, X)

over (X) in which M is a manifold. 2) for local knotting of X in M.

This would allow (in codimenston

It seems to me that not every variety

of X can occur as the intrinsic variety of X in some manifold; also that given some variety {I1 of X which does so occur, then the possible types of local knot along the different I. must satisfy some non-trivial 1

relations. If I am correct, then this generalization is of limited value. 2.

Using Remark 2 after the Corollary to Proposition 5.5, one can classify

equivalence classes of pairs (M, X) over the intrinsic filtration of X such that M is an orientable manifold. 3. Another use of that remark is in dealing with "orientable" polyhedra. polyhedron X is orientable if: its n-th integral homology group, H (X) = Z:

n'

map H (X)

n

for every x E X,

H (sttx, X), t ktx, X» n

= Z;

and the natural

H (sttx, X), tk(x, X» is an isomorphism for all x in X.

n

A choice of generator of H (X) is an orientation of X. n

n-stratification

of X,

for each block bundle K the block bundle K

In terms of an

these conditions can be expressed: ./K. .,

n,l

1,1

its fibre F is orientable;

. has as structural t:. -group the D. -subgroup

n,l

A

122

5.25

SP t - (F) of P t - (F) consisting of those f:

a X F ...

f::"

f::"

a X F such that

the induced homology isomorphism takes either orientation of F into itself (and not its negative). Thus for fixed orientable polyhedron X,

one can classify the equivalence

classes of pairs (M, X) over the intrinsic filtration of X in which M is an orientable manifold. 3.

By using a slightly different definition of a "family of link pairs"

(namely, replacing link pairs 1 by "link pairs 1: (G, F) is not a suspension-pair") and a slightly different notion of "boundary-less" (namely, allowing a single point to be "boundary-less"). and by using the skeletal filtration of X (instead of its intrinsic filtration), one could similarly classify equivalence classes of pairs (M, X) such that M is a manifold, Xl:;; int M.

X is of local codimension at least 3 in M,

and

However, for the applications to cobordism theory which we

shall now give, the use of intrinsic filtrations, and not skeletal filtrations, is appropriate. Let :J be a finite-dimensional, boundary-less family of links of constant local dimension. bdy X =

= bdy Y.

Let X, Y be compact :J polyhedra with

Then X and Yare

- cobordant if there exists a

compact :J -polyhedron W such that bdy W is the disjoint union X U Y. If also V is an :; -cobordism between Y and some Z,

then we

can form T from the disjoint union W U V by identifying the two copies of Y.

Then T is an :J -cobordism between X and Z.

It follows that

123

5.26

a: -cobordism is an equivalence relation. Note that if X is a: -cobordant to Y,

then dim X

= dim

Y.

The

set of equivalence classes of compact n-dimensional a: -polyhedra is n denoted 7l (a: ) .

7l

Under the operation of forming the disjoint union,

n

(:J )

becomes a Z2-module; the identity element being the class of X U X (which bounds X

X 1),

for any boundary-less, n-dimensional a:-polyhedron

x. Let ry q/r!!q be an r-universal S q(;1 )-stratification. Thorn space MrS q(a:), union

Iryq I U

(fr ry q)

(fr ry q) X 1 to a point, a polyhedron), but in which

Define the

or Mrq for short, to be formed from the disjoint

I by identifying fr ry q to (fr

X

"".

r: q)

X

0 and

Mrq is a CW - complex (and not, in general,

Ir!!ql has closed polyhedral neighbourhoods in Mr

q

Ir!!ql has the stratification ry q/r!!q with respect to its

intrinsic filtration. Theorem 5.8.

There is a natural isomorphism 71 n(a: )

F:j

7r

+ (Mr g q(:J»

n q

as abelian groups, whenever q::::' n + 2 and r > n + 1 (here tt the t

th

t

denotes

homotopy group).

:J -polyhedron X,

Given a compact, boundary-less) there is an embedding i: Xc:..... Sn+ n+q . S Since q::::' n + 2.

Let

q,

/J!i. be a stratification of i(X) in S

with respect to its intrinsic filtration. Take a structure (D

which is unique up to isotopy of

Let g: J!i.

u,B,(r!!qxB)I,J!i.I,g")

and let y" /!I" be a subdivision of ry q

X

for g. B.

l-+

r!!q classify

n+q .

Set !I" = (r!IqxB) 11 ,

Then gil -1 extends to a

124

I, fr

p, t . isomorphism g*: !:

(11." I, fr y").

point in Mr

q

Since fr y" U

x D

q

*:'1 n(:;)

-+

U

ry q x D

I -+ (fr ry q) x I x D

U

7r

K"

Il

q

can be contracted to a

+ (Mr

!:

Mr

q

q)

x D\l in

Let

•

by mapping the class of X to the class of p e f,

Conversely, given a map f: Sn+q -+ Mr homotopicto f such that f',-I(lry !:

U

x D

q

be the projection; then we define a function

n q

. n+q manifold P

I, fr 1."

g* extends to a map f: Sn+q -+ Mr

,

x D -+ Mr

(I.x

) -+

U

which Sn+q p : Mr

= fr

5.27

q

lU

q,

(frryq) x

we can find an f"

[O,tn

n+q. -1 q S • with bdy P = (" «fr ry )

isap.t.

1

U

on P.

We think of Sn+q as naturally embedded in a disk D

large.

Define a map f*: Sn+q -+ Mrq x DU by f*(x) = (fll (x), x);

f*: Pc... (Iryq, U (fr ryq) x

[O,tn

l.

ry q x B,

,

U X D is ap.t. embedding.

Theorem 4.1 there is an isotopy of f* r-el bdy P to some

I'(P)

p. t .

and ("

X 2)

I'

U

then By

such that U

where B is a cell complex triangulation of D

•

Since block transversality implies polyhedral transversality, it follows that X,

defined to be

less :; -polyhedron.

I' -1( Ir!:!.qx B I)

is an n-dimensional, boundary-

We define a function 'X. :

7r

n+q

(Mr

q)

-+ 'In(:;) by

mapping the class of f into the class of X. The proof that

* and

'X. are well-defined, homomorphisms and

inverse is the same as Thorn's original proof [27, section IV).

One

needs the full force of our Addendum 2 to Theorem 4. 1 to prove that 'X. is well-defined. Examples.

1.

Fix n,

and take J

to consist of all boundary-less

125

5.28

polyhedra of constant local dimension .:::. n.

Then 71 q(;J) :: 0 for q':::' n •

q What is 7l (;J) for q 2.

n + 1?

Let ;J be a finite-dimensional, boundary-less family of links, such

that every F Theorem 5.

in ;J is orientable (in the sense of Remark 3 following

7) •

Then one can define" oriented" ;J -theory.

On the other

hand, one can take an 8 q(;J) - stratification which is r-universal for \

orientable 3' -polyhedra.

Let its Thorn space be MSr3 (;J).

can prove the orientable version of our theorem: n + 2 and r

whenever q 3.

Cp

n n(;J)

Then one

1'::11f

n+q

(MSrgq(;J»

n + 1.

Let ;J consist of all joins of complex projective space 2

* Cp2 * ... * CP2

(including CP2 and

Consider an orientable

;J -theory in which all the block bundles in Proposition 5.5 are required to be trivial.

The analogous construction in the differentiable category gives

simply integral homology theory, as Sullivan has pointed out.

What is this

p , t . theory? 4.

Let k be a fixed integer.

Let ;J ::

, the set with k points}.

Again require block bundles always to be trivial.

Then we have a p. t .

version of what Sullivan has called Zk -theory. Here is a definition (unpublished) suggested by Akin: Polyhedra X and Yare cobordant if there exists a polyhedron W such that the disjoint union X lJ Y < bdy W,

and every component of bdy W meets Xu Y,

and

such that for every point w E W there exist points x E X, Y E Y and a p. t: embedding h: I

W such that:

126

5.29

wE im h; h(O) = x and h h(l)

=

y and h

-1

-1

(X) = 0; (Y)

= 1;

the intrinsic dimension d(h(tL W) is constant as t varies in [0, 1]. Note that the intrinsic 1-variety of W from X to Y. : tk(x, X)

consists of arcs running

Hence on O-varieties, W induces a bijection X

= F}

Theorem 5.9.

{y E Y

o

: tk(y, Y)

F

= F} = Y 0

,

F o

= [x

EX: 0

for each compact F.

Let X and Y be compact, boundary-less polyhedra of

constant local dimension.

Then there exists a compact cobordism

between them if and only if ;;(X) = ;'(Y), and (the finite sets)

x Fo

X and Yare ;;(X)-cobordant,

and Y F have the same number of points, for all F. 0

Let W be a compact cobordism between X and Y.

X

o

and Y

0

are finite since X and Yare compact, and we have just seen that W induces a bijection X

F F y , o 0

for all F. Since bdyX = bdy Y

are unions of components of bdy W; point x" EX. x".

hence bdy W

X U Y.

Then there is an embedded arc h[O, 1]

with h(l) E Y and with h-

1(X)

= O.

l:

st(h(l),Y)

And since X < bdy W, p. i,

p. i,

st(x",X); so b.tk(x",X)

can be chosen arbitrarily in X, :;(Y) c :;(X).

M

Let N be a

Since the intrinsic filtration of

h[O, 1] in W is just h[O, 1], we have that (N, h[O, 1]) some M.

Pick any

W which contains

Hence h(O) = x",

regular neighbourhood of h[O,1] in W.

X and Y

sttx ", X).

p , i,

p.t.

(M, h(O» x I for

Hence

b.tk(h(l),Y).

we have :;(X)

J(Y).

Since x"

By symmetry,

A similar argument shows that W is an ;;(X)-polyhedron.

5.30

127

Thus W is an ,(X)-cobordism between X and Y.

= ;J(Y)

Conversely, if ;J(X) X and Y,

and W is an :J 2 (note that m ponent lr

cannot be 0).

which meets X,

If m:::: 2, Ir

since FE:;: (X).

F

=1

must have a com-

For each component lr

F s

F pick points ;.F E int 1; F and x E int lr F s s s m-l F m-l Then t k(x , W) F::l S *F (the join), and tk(x , W) F::l S *F. s p. t, s p, t. F F If m 1, we must choose points x and x more carefully: lr con-

of lr F different from l;F,

s

s

sists of circles, which are disjoint from X U Y, XU Y in their end-points. List the circles as lri, ••. ,

lI:

and arcs, which meet

There is at least one arc l;F which meets X. and for each s

= 1, ... , r

pick points

F F F x E int lr , x E int Ir • Some arcs of Ir run from X to Y. Since s s s there is a bijection XF is indeed inverse to

completes the proof of Theorem 5.10.

e , This

Chapter 6. Obstructions to Block Bundles There is an urgent problem left from Chapter 5: when is a p , t. map f: X -+ Y stratifiable? This immediately reduces to the case that X is a sub-polyhedron of Y, replace Y by Y x D,

and dim Y - dim X is large (since we

D a large-dimensional disk).

condition is that X is locally flat in Y;

Then a necessary

that means, for every x E X

there is a p, t. isomorphism f: sttx, Y, X) -+ stfx, X) x (D, v) such that f'[x)

=x

X

v,

where D is a disk,

v E int D. We show that this condition

is not sufficient. We shall be at the same time discussing another question: given X

Y,

when does X have a cS-block bundle neighbourhood in Y? It is

again necessary that X be locally flat in Y, insufficient.

and again this condition is

It is tempting to conjecture that these questions are equiva-

lent; that is, the inclusion X'-+ Y is stratifiable if and only if X has a cS-block bundle neighbourhood in Y;

we allow cS-block bundles to have

different dimensions of fibres over different components of the base space throughout this chapter.

(This conjecture is proved in Chapter 8).

present chapter we shall see that if X is locally flat in Y,

In the

then the

"primary" obstructions to finding a cS-block bundle neighbourhood of X in Y,

and to exhibiting the inclusion X

Y as a stratified map, are the

same.

But there are definitely further obstructions to both problems. Thus the results of this chapter should be viewed, not as a descrip-

tion of stratifiable maps, but as a discussion of why block bundles in the

6.2

134

sense of Rourke and Sanderson are not sufficient to describe regular neighbourhoods even in the case of a locally flat subpolyhedron of a larger polyhedron.

We are thus generalizing Stone [24]; our methods

of proof generalize straight-forwardly those of Rourke and Sanderson's [21].

A p. 1,. map f: X

Y is locally flat if for some large-dimen-

sional Euclidean space lRu there are a locally flat sub-polyhedron X"

Y

lR

X

u

where p: Y x lRu If f: X

f

-1

(bdyY)

If h:

A homotopy f t:

X x I

t:

b

1

= fog;

o

-1

(bdy''Y) = bdyo X.

is also locally flat, then h X

= pog,

Y.

A map f: X

and b

and f, g, at' b

t

Hence

o

Y is locally flat if it is locally flat regarded Y is a locally flat homotopy

equivalence if there exist a map g: Y of Y such that a

such that f

Y is the projection.

Y is locally flat, then f

= bdyX.

as a map f

x''

and a p. t : isomorphism g: X

0

X and homotopies at of X,

are the identity of X and Y;

a

1

b

t

= gof,

are all locally flat.

For the rest of this chapter, when we write: X s; Y,

we shall

assume that X is locally flat in Y and that Y is a weak regular neighLet (X , ••• , X } be a variety of X equivalent to its n 0

bourhood of X.

intrinsic variety and such that every X. is connected. 1

m

, ••. ,

0

} be a similar variety of Y.

that each X. is a submanifold of some 1

u,J

has some 1.. as submanifold. 1

Let

Then our conventions imply

J

and conversely that each

Hence m = n,

and we may re-number

135

11.

the

J

so that :L. is a submanifold of 1

6.3

11.,

for i" 0, ... , n .

1

This we

shall henceforth assume done; the phrase "intrinsic variety" applied to X or Y will mean the appropriate one of these varieties, and "intrinsic filtration" will mean the appropriate associated filtration. Now let 'B be a cell complex triangulation of Y in which X is covered by a subcomplex 'A. 'B. < 'B covering Y., 1

By Theorem 1. 6 there are subcomplexes

for i" 0, •• ., n;

1

and then 'A." 'A 1

n

'B. covers 1

Let Band B' be first- and second-derived subdivisions of 'B,

X.. 1

and let A and A'

be the induced subdivisions of 'A.

Using these sub-

divisions, the construction of Theorem 2.1 gives n-stratifications dual to 'A,

X

and L of Y dual to 'B.

'IS. of

We shall refer to K and L

as simultaneously dual to (IB, 'A). The proof of Theorem 2.1 shows that for each i" 0, ... , n we have a decomposition bundle and

W"»,,

»/K

i,

i'

where It; is a cS-block

an (n-i)-flag, defined thus: Recall that

"

K. . "{s*A. and s*bdy A : sEA and s E 1.}. Then 1,1

1

1

1

or s*bdy A) "s*B. or s*bdy B..

1 1 1

1

block bundle for the submanifold

IK. .

1,1

1

1

(In fact,

of the manifold

ted according to Rourke and Sanderson's [19, I, §4].) (i (K

n,

., ... , K'+

1

.> /K.1, 1.;

so i"

1l, 1

i"1 .(s*A. or s*bdy A) ,J 1 1

0,

is just a normal I L .. 1 1,1

is just the flag

." K.+. . for j " 0, ... , n-i. J 1 J, 1

= s*(B.I+J.

or s*bdy B.

constr-ue-

Finally

.>.

1+J

In sum, we see that X has a normal (n+1, n)-ns N in Y such that for each i" 0, ••• , n,

(N

n+

1""" N . .) can be given the structure ,1

1,1

6,4

136

»,

of a decomposition «i'l1» = (i!;; (liD .1"", 7n-l

70

I)

variety of N

l',

n+ ,1

(N

where

."",N. . ), n,l 1,1

and li!;1

isthe i

th

intrinsic

By the uniqueness of normal rns' s and Theorem I,

4. A of the Appendix we see that the equivalence class of each «i'l1» (in the sense of the Appendix) depends only on X and Y, Lemma 6,1(n),

Given X

the intrinsic filtration of 1.

Y (with the conventions stated before), say Y

is Y "." Y • Assume that either: n 0

X has a cS-block bundle neighbourhood in Y,

or:

2.

n-stratification in which X is covered by a restriction. normal (n+1, n)-ns for X in Y, for i

0" , . ,n,

Lemma 6,1(n-l).

J,o

0

in (Y, X).

ture of a decomposition

By subdividing !;/K if necessary, we

is covered by a subcomplex of K.

0> I; then o

Now assume

Let (N' " .. , N' ) be a regular neighn,o 0,0

n

= Is

bourhood of X

giving rise to decompositions «i'l1»'

Lemma 6,1(0) is trivial.

in (X " , . , X ).

may assume each 0

be a

First consider the case that X has a cS-block bundle

neighbourhood S/K in Y.

o

Let

Then each «i'l1» must be a block decomposition.

We use induction on n,

bourhood of X

(Y) has an

0' ... , NO, 0) is a regular neigh-

We can give

«OT!»

Let

0' , .. , NO, 0) the struc-

which has S

K

0

(c)

X

D'.

6.10

Hence F: K

n,

0

(0') Co+

L· n,

0

(FA(cr)

is

a locally flat homotopy equivalence. For s E A o' If cr

is of the form s*A,

IBI

C (recall that

F>

Fs

we take dual cells to

F

the simplicial maps s

--p> fs

are 1-1.

and fs in Band

o s u) = [c] X lR to obtain (Fs)*B p1>$ (fs)*C , to

X D,

where D is a u-disk; similarly if 0' is of the form s*bdy A, o (Fs)*bdy B

1>$

p.t.

we use

(fs)*bdy C xD, This gives a p , t: isomorphism c such

that the diagram

IL n,o (F" (0' » I

c :

;>

IJn,o (t(o» I X D Iprojection

v

nl YxlR

IJrr. o (f'lO'» I

nl

projection

Y

;>

commutes. It follows from the natural way in which c is defined that c corresponds

ILn,o (FA(o

o)

L· (FA(cr» n,o

I IJ n,o (r"(cr» I X

J . (r"(o» n,o

X

D.

D;

and hence

Thus the projection p : Y

X

lR

Y

restricts to a locally flat homotopy equivalence p : L· (FA(cr» .... J . (t(o». n,o n,o

Combining with F shows that f

restricts to a locally flat homotopy equivalence

= po F

6.11

143

d: K'

n.o

(o ) -+ J'

n.o

(I'

IJn,o I

is an embedding.

Now set

6.12

144

x"

lc 0(K) I, y" -

II K'

n,o

.

lc 0(L) -

=

I,f" = f'

The hypotheses of Proposition 6. 3(n-l) are thus satisfied; so of f" reI bdy X" to an embedding.

there is a locally fiat homotopy Then hI'

defined to be ft on

quired embedding. sition 6. 3(n). Lemma 6.4.

IK

n,o

I and h"

1

lc 0 (K) I -

on

is the r e-

This completes the inductive step in proving Propo-

Hence Proposition 6.3 holds for all n. where S is a

Given a decomposition «n» =

cS-block bundle, and

a cF-block bundle.

I

)' IK with

bundle

Then bdy X" = (X"nbdy X) U

I«n» I such that the cbb's

form the block decomposition [s;

S in «n». Further

theidentityon lsi;

"l-

$

I

t

of l«n»1

suchthat f

and for every a E K,

S'17'

and nl,1

is called complementary to

another complement to

if

then there is an isotopy f

Then there is a cF-block

S in «n»,

$";

1:

ftl«n»(a) I =

f is t

I«n»(a) I

for

all t , The proof is essentially the same as that of Rourke and Sanderson's f19, II, Theorem 5.1], and uses the fact that since

s

I I is a manifold

(recall from Chapter 4 our convention according to which fold),

I«n»

is just v

I has the structure of a cF-block bundle

Is I -+ Inl

suchthat f(a)

vi Is I.

Then

IK I.

Let F;jK and niL be cF-block bundles. f:

IK I is a mant-

A p , t. map

is blockwise locally fiat if for every a E K there is a r E L In("') I and f:

and f(s' (a» \; n' (.,.).

In(.,.)

I

is locally fiat, with

6.13

145

If

IK I

locally flat.

IL I

and

are manifolds, it follows that f itself is

Similarly one defines blockwise locally flat homotopies

and homotopy equivalences between cF-block bundles. For any compact polyhedron F we can now define the G-(F) to have for k-simplexes any blockwise locally flat homotopy self- equivalence of the trivial cF-block bundle, such that f is the identity on

k 16

I = I I xc.

son's [22, I] for the definition of a to be the set of f in G-(F) notation of Chapter 5, G-(F) is a

f:

k

xcF/

-+

k

xcF /

k

,

(See Rourke and Sander-

The 6-set P ['(F) is defined

such that f is a p , t. isomorphism.

Pt-(F)

(In the

would have been called Pt-(F, F).) Then

-rnonoid, and Pt -(F) is a t:. -subgroup.

the

k

Let G/pr(F) be

G/Pt-(F) can be geometrically realized as a

CW-complex [22, IJ, which we again call G/pC(F). Let BPr(F) be the base space of a universal Pt-(F)-block bundle, constructed according to [22, II, Corollary 2.3].

Let BG-(F) be

the base space of a universal G-(F)-block bundle; according to [22, II, Proposition 3.10].

As in [22, II, (3.18)] one can show that there is (up to

homotopy type) a fibration 7r: BPr(F) -+ BG-(F) whose fibre is G/P['(F).

Let p: BP['(Sq) X BPC(F) -+ BPnF) be the projection, and

let A : E -+ BP[ '(sq)

X

BP

induced from 7r by 7rop.

r (F)

be the fibration with fibre G/p ['(F)

Thus A is covered by amap of fibrations

A- : E -+ BPt-(F). E can be represented as

tion s defined by stw, x)

= (w, x, xl ,

6.14

146

Theorem 6.5. (cS

q-

Assume q::: 2.

; cF-) decompositions.

Then E is a classifying space for

That is, for every cell complex K,

$)/K

Dec(K) be the set of isomorphism classes of decompositions with

.ez

a cSq-block bundle,

let

a cF-block bundle; then there is a

functorially defined bijection 8 : [IK I. E] Dec(K). The proof is essentially the same as the argument of Rourke and Sanderson's [21, section 3], so we omit some details of the present proof. Let «T1» = mentary to ing

/K be a decomposition. and let

in «T1».

and g. g' :

Then we have maps f: IK I -+ BP C(sq)

IK I -+

BP C (F) classifying

Let (V.[V(o)}) be a regular neighbourhood of

n

=

= W n T1 i ,

1(0),

such that V W(o)

and V

neighbourhood of both trivial over 0;

n

respectively.

in (T1;.1'["\:1(0)})

Then (W. [W(o)}) is a weak

and of

'(0) and of

and

classify-

Set W = cl[T1;. 1-V],

for every 0 E K.

regular neighbourhood of

be comple-

(For W(ri! is a weak regular

(0)

since (I;;

and

are

and one fits the Wtc) together by induction on a

standard ordering of K.

using Proposition 1.4 to ensure one always has

a weak regular neighbourhood.) By Lemma 6.2 the inclusions and

Ware blockwise locally flat homotopy equivalences.

we have a blockwise locally flat homotopy equivalence h: That is, 71"og = (71"Op)

W

and )' are isomorphic as 0

(fxg') (up to homotopy).

Hence

-+

•

and so

Thus associated to «T1» we have

a map y : IK I -+ E such that Aoy = fxg', A- OY = g (up to homotopy).

6.15

147

This process defines a function

ql :

Dec(K) .. r1KI, E].

Conversely, given a map y : IK I .. E, bundle

classified by Pl

classified by po\oy;

0;"'oy,

we have a cSq-block

and a cF-block bundle

where Pi and p are the projections of

BPr(Sq) x BPtlF) onto its first and second factors. cF-block bundle

i /K

classified by rro\ - oy;

locally flat homotopy equivalent to

and

We also have a is block-wise

'7 I].

Set «Tl'»

Then the

previous argument shows that we have a blockwise locally flat homotopy equivalence h:

... W'

'1')1: i '

!:

where W' is defined as before.

The

argument of [21] shows how to find a blockwise locally flat homotopy of

W': one uses induction on a standard

h to an embedding h": ordering of K,

and if h:

W'(oa) is already an embedding,

then by a double application of Proposition 6.3 one can homotop h on • (0) reI

r

(0 a)

to an embedding, first on bdy

By the standard procedure, ding h' :

1«'1')1»

I.

by the block bundles

a:

• (a),

and then on

f·

(e),

extends to a blockwise locally flat embed-

h"

Then we have a decomposition «'I'))/K defined

h l (7 )'

rj'l, i:

This process defines the map

[IK I, E] .. Dec(K). The proof that

a

and

ql

are well-defined and inverse is the same

as in [21]. Remark.

The statement of the theorem is imprecise because our polyhedra

have to be finite-dimensional.

(The same difficulty occurred in Chapter 5).

A precise statement would say that for each integer r > 0 there is an

6.16

148

"appr-oximatton" rE to E such that

ql

[IK I, E]

Dec(K) is a

bijection whenever dim K ::: r-, Corollary 1.

Let

cF-block bundle. classify

be a cSq-block bundle with q::::, 2, Let f: IKI

and

Corollary 2. block bundle,

Then

Let

«.ry)

q::::' 2,

BPf"(Sq),

g: IKI

and VK a

BP.c(F)

is classified by so(fxg) : IKI

=

;

and

E.

be a decomposition, with a cF-block bundle.

Then

a cS

«Tj»

q-

is a

block decomposition if and only if its classifying map y: IK I

E is

homotopic through sections of A to soAoy. The obstructions to doing n( so lie in the cohomology group s H IK I, 71" G/ pC(F» . n

Corollary 3. block bundle.

Let

as before, and let

By extending

'over

;

Then

position

=

and only if

I«Tj» I,

«Tj»

be a cS

q l-

one obtains a new decom-

is a block decomposition if

is.

Theorem 6.6. least 3.

«.ry) =

Given X

Y locally flat and of local co dimension at

For each F E :J (X),

let 1

F

= [x

EX: b. tk(x, X) = F}

Then for each F there is a sequence of obstructions an(F) E W(l F, 71"nG/p

for n

0, 1, 2, . ..

all previous a. (F) are 0) such that if either: 1. 1

bundle neighbourhood in Y;

(each defined only if X has a cS-block

or 2. there is an intrinsic n-stratification

of Y in which X is covered by a restriction, - then the a (F) must n

be 0 for all n and F. Moreover these obstructions are stable in the

6.17

149

sense that if Z has the structure of a cS-block bundle over Y,

then

the obst ructions for X in Yare the same as those for X in Z. This follows from Corollaries 2 and 3 just given and Lemma 6. l . In part icular, take F to be the disjoint union of Sr and a point.

= G/Pi'{Sr>.

Then G/P'c{F) is isomorphic to (the standard) G/Pt

r

r:::. 2, odd;

the homot opy groups if n

Z,

n

(G/P t ) are known to be: 0,

r

and Z2 if n

O{4);

E

1f

E

2(4)

If

if n is

(see Sullivan [25],

Kervaire and Milnor [11), or Rourke [18].) Hence whenever

q:::'

2, r> 2

and n is even, there are locally flat embeddings of Sn x cF in n

q+l

S x cF x D

which do not admit block bundle neighbourhoods.

By modifying this example we can show that the obstructions described in Theorem 6. 6 are not sufficient to ensure that X has a cSblock bundle in Y: Write Sn v+ be the centre of "cr)K x c(G, F) can be completed to a weak (n+l, n)-stratification by adding (>"cr)K

x£..

Thus we have two (n, n-ll-stratifications of

8.10

168

(K,O)' HAO')K in

1 h- « Aa)K x (i; , K

,0

)

f'-(AO')K,

namely: A*(A(O'),t) and

both of which define completions of the flag

HI.. O')K.

isotopy at of

The proof of Lemma 3.2 shows that there is an

I(i;, K ,0 )

and (AO')K

a

l

I reI II.. (a) I

: h -l«).,O')K x

morphism of the flag (i;, K

,0

such that (after subdividing

)

and at is an iso-

) !--(AO')K for all t ,

Now let (B, A) be a cell complex triangulation of the sequence (G, F) in which every block of plexes. on

0' X

"Ip "

x cB is a cS-block bundle structure j.i

Then (A cG in

and its rim are covered by subcom-

IA (a) I x cG. ',.)." is clearly complementary to the

stratification ()., O')K x c

«A O')K x ere,

F»

U «Aa)K x

1).

So '',.)./ 'p,

defined to be a h -1',.)."/ "i h -l p " is complementary to the stratification 1 A (A

(This is a trivial consequence of the definitions, since at

acts on h ­1 ',.)."

and on h ­1 «).,O')K x c$.).

Let ?}',.)./o'P be the restriction of 'u

=

n+1,0

to o'P

=

and let olJ./op be U {p',.)./pp : p . Let be a " , block decomposition blocked by "K 1)' and let "iJ./"P be a comple-

,

mentary blocking, We may assume that «"ri1)' and hence "1.1,

*

III (A(o-),I)n+1, 11 as cr

of L,

varies in L.

assume "u and r'u

agreein

respect each

By induction on a standard ordering

*

III (A(ocr),I)n+l,l'.

ByLemma

*

8.2, there is an admissable isotopy of IA (\(cr),I)n+l, 11 r-el III *(cr,1)n+1, I'

which carries "u

isomorphically to 1'1-1

in

'II *(A,(cr), 1;) 1 1 I. It follows -n+. ,

from the proof of that lemma that we may keep our isotopy fixed on

III*o,(ocr), -1;)n+l, 1 I.

By the standard procedure, this isotopy extends to an

admissable isotopy of

III*('A., -1;)n+ 1• 11

r el

1II*(L,I;) 1 11 = 11' P I, - n+ ,

induction there is an admissable isotopy of

IA *(L1)n+1,

which carries I ' u isomorphically to "u ,

By Lemma 8. 1,

complementary to (" S, "K, 1)'

11 reI

SO by

IA *(L,1)n+l,

t 'u

is

Details are left to the reader, who will

also see that the index 1 has played no part and is just a notational convenience for dealing with the general case, in which we consider the flag (A*(A,I) J, , 1

Hence u is complementary to the stratification II (A,1).

Next we have to arrange that 1-1 v*.

1-1

extends the given cS-block bundle

(n+1,O)v* is complementary to !IC(O, O)v*, there is a restriction 1-1 * /p* of u

this stratification.

Ip*I

such that f

I>;

and by construction of

which is also complementary to

t

We need an admissable isotopy f

1: 1-1*

F

in

The proof of Proposition 8.3

has a normal cS-block bundle neighbourhood v' /Q'

in M which is complementary to 11'

Then v / Q

\i'

Q'< IKn,

I>

is the

required cS-block bundle. Proposition 8. 4(n).

Given X, Y!; M such that M is a manifold,

submanifold of M.

Let (Y , ... ,Y ) be a variety filtration of Y in M,

and let

n

X a

0

be a normal cS-block bundle of X in M such that (Y) ":t'

Then there is a stratification nI!; of (Y) in M which is complementary to

s;

so in particular,

is < bdYM(Y)'

X

1.

11' Further, given that (y*) = (Y) n bdy M

and given a stratification

of (y*) in bdy M

8,17

175

which is complementary to S, We use induction on n,

then we may choose

to extend

Proposition 8,3(0) is just Rourke and

Sanderson's [19, II, Proposition 4.10],

Now assume Proposition 8, 4(n-l):

(Y) n X is a variety filtration of Y n X in X, normal (n+1, n)-ns for (Y) n X in X; the normal (n--L, n) -ns

{In>!;J,l. I n

by subcomplexes (j, i)K and (j, a normal (n+ I, n) -ns S

we may assume that

J, 1

u: K

of K,

Is I

in

I n Is I,

to a normal (n+l, n)-ns liN of (Y) in M,

In* I.

flag (I-l, An' ., " AI) /P,

Let

!:!:..'/£'

S L*

0, 0

J, 1

{I S I' (j,

Then

IsI

i)K\}

defines

which respects S'

Since (Y)

:I:'

S, S

and

extends

which we may assume

,

so there is a complementary

We may assume that (I-l, A) extends

where we write S L*

0, 0

for L*

0, 0

.

be an (n+l, n)-ns completion of (u,)..) which extends

i\(SL*

0,0 -

We have the normal (n,n-l)-ns's 1i\*(oP,I-l')! and -

(N. . n N' +1 O} of ()J' J,l n,

oP in u '

is an isotopy at of (1-l,)J'l'oP reI a

By

will be constructed as a blocking of

S I'(n+ I, O)K satisfies comp 1

, 0

extends

of (y*) n bdy X in bdy X.

X}

for (Y) n

which extends the (n+ I, n) -ns

(n*, L* )

be a

S we may assume that every N. . and N', . are covered

subdividing

extends

1!

Let

1: I

I

R
is a variety filtration of

on (I i; I) which is complementary

IK I is covered by a restriction

< !:..

One can check that the proof of Proposition 8.4 applies to this situation. f: IK I that fwith IK'I.

Or, as in the proof of Corollary 2 to Proposition 8.3, let M be an embedding into a large-dimensional manifold such

1bdyM

IK'I

=

1i;' I

M,

bdylKI.

Then S extends to a cS-blockbundle

and (IKJ ... , IK I> is a variety filtration of IK I in o

is a manifold, and

Ii; I .L i;'.

So the proof of Proposition

8. 4 shows there is an (n+ 1, n) - stratification which is complementary to S'.

!LI!:.

of

(Ii; I)

in

Ii; I I

Then L is the required n-stratifica-

tion. Putting together Propositions 8.3 and 8.4 gives: Theorem 8.5.

S'/K'

Given X, Y \;; M with M a manifold and X a

8.20

178

submanifold of M;

then X is block transverse to Y if and only if Y

is block transverse to X. From Corollary 2 to Proposition 8.3 and Corollary 2 to Proposition 8.4 we infer: Theorem 8.6.

A p. t. map f: X -+ Y is stratifiable if and only if there

are a large-dimensional disk D and a factorization f p : Y X D -+ Y is the projection,

= po g

such that

g: X -+ Y X D is an embedding, and

gX has a cS-block bundle neighbourhood in Y

X

D.

It is not hard to see (using the method of proof of Lemma 5.2)

that a p. t. embedding f: XC-+ Y is stratifiable if and only if the pair (Y,X) is equivalent to a pair

IK!) for some cS-block bundle

- in other words, one can cancel the disk D in the statement of the theorem.

A.l

179

Appendix

In order to develop the theory of flags and of decompositions, we introduce the notion of an "0 cone-block bundle", where 0 is a partially ordered set. Our treatment of 0 cbb's now follows Rourke and Sanderson's [19. I and II].

Thus our Result X, Y. Z is a reformula-

tion of [19. X, Result Y. Z]. It is usually left to the reader to reformulate Rourke and Sanderson's proof; we confine ourselves to details that did not arise in the case of block bundles. Let 0 be a finite partially-ordered set with a unique minimal element 0 and a unique maximal element a.

{s} over a cell complex K is a family (s

w

An

Q.

: w EO}

bundle of cone-block

bundles over K such that:

o cbb

1 if w < w' in 0,

Ocbb 2 "':>0

then

I w I l::: I w,I;

= K''

n cbb 3 for each o E K, there are a compact polyhedron with a family of subpolyhedra indexed by

n,

structure h: S (c) a

a

a

and a block

x cF (cr) which, for every w EO, restricts

to a block structure h: I; (c)

cr

w

We write:

say (F (e ). {F (cr)} ), a w

/K is an

x cF (o). w

(1 ebb,

We leave the reader to define the restriction

of an

(1

cbb

/K to a subcomplex L < K. If

(1;11 K

and

frl1 / L

are

n cbb' s,

then an

n cbb

isomorphism

A.2

180

f:

{,,} is a p , t . isomorphism f:

to a cone-block bundle isomorphism f: Given 0

I a I -+ 11

w

w

111

a

I

which restricts

for every w EO.

as before; then a family of compact polyhedra

(F , (F }) indexed by 0 is an 0 family if F = F a w o w ever w < w' : and F

o ebb

w

n F w, = some

F

w'

whenAn

whenever w, w' EO.

is a c{ F} -block bundle if for every cr E K,

(F (c), (F (. a w

Let

(1.

Then

W

Pick points x E int IL I and y E int p.

n.

p. t. isomorphism p: (V ,{V}) ... (XUy)

-> 0 P

for all w E

(ocr X cF U o X F ).

is a regular neighbourhood of

w

Q'

X

a

n IK I

).

Q'

= p,

Let By

I * (Fa ,(FW}»-;> extend to an

This, together with t,

gives an isomorphism (different from p) t

1

: (V ,{V}) -;> (00" X c(F ,{F }» U (o X (F ,{F

a

w

a

W

a

D).

W

The proof now follows that given by Rourke and Sanderson, except that one uses Cohen's Regular Neighbourhood Theorem (our Theorem 1, 1) instead of the regular neighbourhood theorem they quote. Proposition I, 1.4*. Let

Any

(1

ebb (sl!K with dimlKI':::n has an atlas.

{TI}!L be

(1

ebb's.

Then {TI}

is a subdivision of

if TI ! L is a subdivision of S ! K (as cone block bundles) for all

w

wE

w

(1 •

Theorem 1,1,5*.

1.

Let {sl!K be an

(1

let L be a cell complex subdivision of K. {Tl}!L of 2.

cbb with dim/KI':::n,

and

Then there is a subdivision

to.

Let {sIlK be an

(1

ebb with dimlKI.::: n.

Given K' < K and a

A.5

183

subdivision (Tl'}/L' of set of blocks.

=

U (Tl'} (as a

Then {Tl) is an n cbb and subdivides

Theorem I. 16.*. Assume

Set (Tl}/L

Let

be ncbb's with dirn l K]

and

IK I \l IL I (collapses geometrically). where L is a sub-

complex of K.

Given a cell complex isomorphism h: K

extension of h to an isomorphism of n cbb's h! :

F:$

K'

F:$

(S'HhL'.

Then h and h' extend to an n cbb isomorphism h" : (S) Corollary I. 1. 7*. IK I

< n.

Given an

o ebb

F:$

(S'}.

/ K such that dim IK I

{

and an

:s. nand

O. Then (S) is trivial. The proofs are the same as in [8. I. § 1.] In particular.

Theorem I, 1. 1(n) holds, and the inductive step is proved.

Theorem I.

1. 1 and the starred results now hold without restriction on n, Corollary I. 1. 8. L

Given n cbb's

/K x I and (Tl) /K x I, a subcomplex

< K, and an isomorphism h:

ULxI)

F:$

('I1)

Lx I) which

is the identity on K x 0 u L x I. Then h extends to an isomorphism h! :

(Tl)

which is the identity on K x 1.

Theorem I, 1.9. (S J/K.

Then

Let (Tl}/L and fn'}/L be subdivisions of an n ebb

Inl

(Tl').

Given an n ebb (S'}/K' and a cell complex K of which K' is a subdivision.

For each o E K.

let h:

tr-tvtaliz atton, given by Theorem I. 1. 1.

e:

'til

(c) so that h:

(0) -+ cr X

w

cF

w

-+ K' x c] F)

be a

For each wEn define a block

is a block structure.

Then

A.6

184

'"

(m : o E K,

w

w E o} is an 0 cbb over K, called the

amalgamation of {S'} over K.

Note that {S'} is a subdivision of

{s} . Given Ocbb's {sJ/K and {-I1}/L with IKI", ILl. and {'I'l}

11K' of {s},

are equivalent if there are subdivisions

{'I'l'}/L' of {'I'll,

and an isomorphism h: (S'}

identity on IK I. We write: {S} -