Grassmannians and Gauss Maps in Piecewise-Linear and Piecewise-Differential Topology

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1366 Norman Levitt

Grassmannians and Gauss Maps in Piecewise-linear Topology

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Author Norman Levitt Department of Mathematics Rutgers, The State University New Brunswick, NJ 08903, USA

Mathematics Subject Classification (1980): 57 Q35, 57 Q50, 57 Q91,57 R 20 ISBN 3-540-50756-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50756-6 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specificatly the rights of translation, reprinting, re-use of i~lustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus 8eltz, Hemsbach/Bergstr. 2146/3140-543210

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CONTENTS

CHAPTER

0

Introduction

CHAPTER

1

Local

CHAPTER

2

Formal

CHAPTER

3

Some Variations

CHAPTER

4

The

CHAPTER

5

. . . . . . . . . . . . . . . . . . .

Formulae Links

and the

Immersion

Immersions Actions

for C h a r a c t e r i s t i c

Theorem

6

Immersions

CHAPTER

7

The

CHAPTER

8

Some Applications

CHAPTER

9

Equivariant

CHAPTER

i0

into

Grassmannian

Glossary

Triangulated

of

43

. . . . . . . . . . .

60

Respect

of

/~n,k

Manifolds

...... Immersions

to S m o o t h i n g

Theory

. . . . . . .

Differentiable Immersions

Immersions

Definitions

101 . 116 161 . 181

188

and

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

V

87

into

. . . . . . . . . . . . . . .

Important

70

to O r t h o g o n a l

Smooth

Manifolds

ii

....

for P i e c e w i s e

Piecewise

Constructions

REFERENCES

with

.....

. . . . . . . . . . . . . . . . . .

Differentiable

Riemannian

APPENDIX:

~,k

~,k

for S u b c o m p l e x e s

Equivariant

CHAPTER

Piecewise

PL G r a s s m a n n i a n

of the

on Rn+k

classes

1

198

202

0.1 O.

Introduction

This monograph b r i n g s t o g e t h e r a number of r e s u l t s centered on an a t t e m p t to i m p o r t i n t o the study of

PL m a n i f o l d s some geometric

ideas which take t h e i r i n s p i r a t i o n from the o r i g i n s of d i f f e r e n t i a l t o p o l o g y and d i f f e r e n t i a l

geometry, ideas from which many i m p o r t a n t

aspects of f i b e r - b u n d l e theory have developed.

The reader i s

presumed to be f a m i l i a r w i t h the c e n t r a l r o l e t h a t the theory of f i b e r bundles has played in the study of d i f f e r e n t i a b l e manifolds f o r the past f o u r decades.

The c e n t r a l theme here has been t h a t a wide

class of geometric problems can be r e f o r m u l a t e d as b u n d l e - t h e o r e t i c problems.

Typical r e s u l t s f l o w i n g from t h i s approach have been the

C a i r n s - H i r s c h Smoothing Theorem; The Hirsch Immersion Theorem, together with i t s

g e n e r a l i z a t i o n , the G r o m o v - P h i l l i p s Theorem, and

much of the i m p o r t a n t work in f o l i a t i o n

theory.

The g r e a t advantage

of a r e d u c t i o n to bundle theory as has been g e n e r a l l y been t h o u g h t , is

t h a t the geometric problem has become a homotopy - t h e o r e t i c

problem from whence, with a l i t t l e

luck,

it

can be made i n t o an

a l g e b r a i c problem. It

i s also presupposed t h a t the reader i s conversant with the

g e n e r a l i z a t i o n s of c l a s s i c a l v e c t o r bundle t h e o r y , g e n e r a l i z a t i o n s which a p p r o p r i a t e much of the machinery developed f o r d i f f e r e n t i a b l e t o p o l o g y f o r use in the study of PL m a n i f o l d s , t o p o l o g i c a l m a n i f o l d s , homology m a n i f o l d s , P o i n c a r e - d u a l i t y spaces and so f o r t h .

In

p a r t i c u l a r , the n o t i o n s of PL bundle, PL b l o c k - b u n d l e , t o p o l o g i c a l bundle, s p h e r i c a l f i b r a t i o n

( t o g e t h e r with t h e i r s t a b l e v e r s i o n s ) are

assumed to be f a m i l i a r t e r r i t o r y .

So, t o o , the c l a s s i f y i n g spaces

(and canonical bundles) associated with these n o t i o n s : BO(k)

f o r v e c t o r bundles,

BPL(k) f o r

BPL(k) f o r k-dimensional PL-bundles,

PL k - b l o c k - b u n d l e s , BG(k) f o r

(k-l)

- spherical f i b r a t i o n s ,

and so f o r t h . I now wish to observe t h a t these g e n e r a l i z a t i o n s and the theorems t h a t have e x p l o i t e d them have had a c e r t a i n f l a v o r ,

0.2 d i s p l a y i n g , so to speak, an i n c l i n a t i o n to move i n t o the homotopy theory as q u i c k l y as p o s s i b l e from the p o i n t of view of u n d e r l y i n g c o n s t r u c t i o n s as well as t h a t of u l t i m a t e r e s u l t s . h i s t o r i c a l overview might make t h i s The notion of bundle and i t s

A brief

clearer.

applicability

to t o p o l o g i c a l

questions goes back, of course, to Gauss, whose g r e a t work on c u r v a t u r e and i t s Gauss map in i t s

r e l a t i o n to the topology of surfaces e x p l o i t s the o r i g i n a l and most l i t e r a l

sense.

This of course i s

the map which, f o r any o r i e n t e d surface immersed in 3-space, takes each p o i n t to the c o r r e c t l y - o r i e n t e d u n i t normal v e c t o r to the surface a t t h a t p o i n t , the t a r g e t space of the map being thought of as the standard u n i t 2-sphere. In t h i s c e n t u r y , the f o u n d a t i o n a l work of Steenrod, Whitney, e t . al.

led to the formal d e f i n i t i o n of f i b e r bundles, with vector

bundles along with p r i n c i p a l Lie group bundles serving as the prime example.

The discovery of the r o l e of

the Grassmann m a n i f o l d as the

" c l a s s i f y i n g space" f o r v e c t o r bundles preserved much of the o r i g i n a l i n s i g h t of Gauss' c o n s t r u c t i o n . learn, it

helps one's i n t u i t i o n

As beginners in the s u b j e c t soon to p i c t u r e vector bundles as tangent

bundles to m a n i f o l d s , p a r t i c u l a r l y manifolds embedded or immersed in Euclidean space.

In t h a t case, one e a s i l y goes on to p i c t u r e the

c l a s s i f y i n g map, ( f r e q u e n t l y and q u i t e a p p r o p r i a t e l y c a l l e d the "Gauss map") as t h a t map which takes each p o i n t in the given n - m a n i f o l d to the p o i n t in the a p p r o p r i a t e Grassmannian corresponding to the unique n-dimensional l i n e a r subspace ( o f

the given Euclidean

space) p a r a l l e l to the tangent space a t the p o i n t . In the i n t e r v e n i n g decades, g e n e r a l i z a t i o n s of the notion of v e c t o r bundle have p r o l i f e r a t e d , and the notion of " u n i v e r s a l c l a s s i f y i n g space" ha3 become a f a m i l i a r one f o r many c o n t r a v a r i a n t homotopy f u n c t i o n s beyono v e c t o r bundles and p r i n c i p a l bundles. chief tool

The

here i s E. Brown's R e p r e s e n t a b i l i t y Theorem [ B r o ] and some

0.3 of

its

g e n e r a l i z a t i o n s , which guarantee t h a t a homotopy f u n c t o r i s

"representible" (i.e,

has a c l a s s i f y i n g space w i t h i n the category of

CW complexes) under very u n r e s t r i c t i v e c o n d i t i o n s .

In p a r t i c u l a r ,

Brown's Theorem is u s u a l l y c i t e d as the j u s t i f i c a t i o n

for asserting

the e x i s t e n c e of BPL, 8 Top, BG e t . a l . Despite the beauty and usefulness of the R e p r e s e n t a b i l i t y Theorem, however, I wish to a s s e r t t h a t there i s something p r o b l e m a t i c a l about i t s geometric problems.

use in connection with i n t r i n s i c a l l y

F i r s t of a l l ,

one sees t h a t the c l a s s i f i n g space

BF obtained f o r a given f u n c t o r F is object; it

t r u l y a "homotopy t h e o r e t i c "

has no " n a t u r a l " geometric s t r u c t u r e and, indeed, is a

geometric o b j e c t only in the most shadowy and a b s t r a c t sense. The same may be said of the map X + BF c l a s s i f y i n g an element of F(X). This is no map a t a l l of maps.

strictly

speaking, but r a t h e r a homotopy class

In some sense, to the degree t h a t we r e l y on the

R e p r e s e n t a b i l i t y Theorem, we "know" BF or [X,B F] p r e c i s e l y as well as we know F or F(X).

The r o l l

of BF as a space or an element of [X, BF]

as a map i s l a r g e l y m a t a p h o r i c a l .

N o t e how f a r t h i s

is in s p i r i t

from the o r i g i n a l Gauss c o n s t r u c t i o n , in which a s p e c i f i c geometric o b j e c t (an embedded m a n i f o l d ) was seen to a c q u i r e an e q u a l l y s p e c i f i c map i n t o a concrete geometric o b j e c t (the standard sphere), a map whose l o c a l p r o p e r t i e s , moreover, were of intense geometric i n t e r e s t . Gauss, a f t e r a l l ,

was not i n t e r e s t e d in the a b s t r a c t c l a s s i f i c a t i o n

of normal bundles of surface but r a t h e r in understanding the l o c a l geometry of c u r v a t u r e in i t s

r e l a t i o n to g l o b a l i n v a r i a n t s .

The present work is a f i r s t this

spirit

a t t e m p t a t r e c o v e r i n g something of

f o r the study of c o m b i n a t o r i a l m a n i f o l d s .

manifolds, a f t e r a l l ,

are by d e f i n i t i o n ,

Combinatorical

o b j e c t s which support

s p e c i f i c geometric s t r u c t u r e s , namely t r i a n g u l a t i o n s (more specifically,

m e t r i c t r i a n g u l a t i o n s where each simplex has a m e t r i c

c o n s i s t e n t with i t s

convex l i n e a r s t r u c t u r e ) .

There i s a rough but

0.4 useful a n a l o g y :

t r i a n g u l a t e d m a n i f o l d s are to c o m b i n a t o r i a l

m a n i f o l d s as Riemannian m a n i f o l d s are to d i f f e r e n t i a b l e

manifolds.

That comparison suggests, among o t h e r i m p l i c a t i o n s , t h a t the l o c a l p r o p e r t i e s of a t r i a n g u l a t i o n ought to bear some r e l a t i o n g l o b a l i n v a r i a n t s of

the m a n i f o l d .

The problem, of course, point.

to the

The view taken in

is

to g i v e t h i s

these notes i s

t r i a n g u l a t e d manifold gives r i s e

i n s i g h t some c o n c r e t e

t h a t the l o c a l

geometry of a

to a map (and the emphasis here i s

on map r a t h e r than homotopy class of maps) i n t o a u n i v e r s a l example which, local

so to speak, is geometrices.

In view of

c o n s t r u c t i o n we c a l l This usage i s as we d e f i n e i t , is,

c o n s t r u c t e d from a l l tradition

p o s s i b l e p r o t o t y p e s of

and of

the n a t u r a l i t y

by the f a c t t h a t the Gauss map,

c a r r i e s the a p p r o p r i a t e bundle i n f o r m a t i o n . n a t u r a l l y covered by a bundle map ( i n

a p p r o p r i a t e c a t e g o r y ) of

That

the

the t a n g e n t bundle of the m a n i f o l d to some

c a n o n i c a l bundle over the u n i v e r s a l space (which i s be t h o u g h t of as a kind of mode of

the

t h i s map a Gauss map.

further justified

the Gauss map i s

of

"Grassmannian").

thinking yet further,

thus n a t u r a l l y to

Carrying t h i s

analogical

we might c o n s i d e r a t r i a n g u l a t e d

m a n i f o l d embedded in Euclidean space so t h a t the embedding i s a c o n v e x - l i n e a r map on each s i m p l e x . submanifold of Euclidean space.

The analogy here i s

to smooth a

One ought to suspect t h a t ,

j u s t as

t h e r e i s a n a t u r a l Grassmannian which r e c e i v e s the Gauss map of

the

embedded smooth m a n i f o l d , t h e r e might be a n a t u r a l space which r e c e i v e s the e q u a l l y n a t u r a l Gauss map of This s u s p i c i o n i s

quite justified.

the embedded m a n i f o l d .

Again, p r o t o t y p e s of l o c a l

geometries (where now the embedding in Euclidean space i s

to be taken

i n t o a c c o u n t ) can be assembled to form the a p p r o p r i a t e PL Grassmannian

which in

t u r n supports an a p p r o p r i a t e c a n o n i c a l bundle.

Once embarked upon t h i s mode of t h i n k i n g , we f i n d o u r s e l v e s n a t u r a l l y drawn i n t o g e n e r a l i z a t i o n s and e x t e n s i o n s of the main idea

0.5 of

c o n s t r u c t i n g Grassmannians

and Gauss maps to handle d i f f e r e n t

kinds of u n d e r l y i n g g e o m e t r i c s i t u a t i o n s . way of

To name but one example by

suggesting the f l a v o r of our approach, we might c o n s i d e r

whether a c o m b i n a t o r i c a l m a n i f o l d

M

admits a "bundle of

Grassmannians" so t h a t given an immersion Gauss map from

V

V

M, t h e r e w i l l

be a

to t h a t " b u n d l e " c o v e r i n g the immersion.

Leaving aside f o r

the moment an e x a c t enumeration of

g e o m e t r i c a l c o n s i d e r a t i o n s which g i v e r i s e

those

to "Grassmannians" and

"Gauss maps", we come to the f u r t h e r problem of

justifying

such

c o n s t r i c t i o n s beyond the l i m i t e d appeal of a b s t r a c t i n g e n u i t y . First

of a l l ,

we s h a l l

exploit

the n o t i o n t h a t a Gauss map ( i n

c o n t r a d i s t i n c t i o n to a h o m o t o p y - t h e o r e t i c c l a s s i f y i n g map i n t o a h o m o t o p y - t h e o r e t i c c l a s s i f y i n g space) i s

both c o n c r e t e and l o c a l l y

d e t e r m i n e d . This can be used to c o n v e r t g l o b a l i n f o r m a t i o n i n t o l o c a l i n f o r m a t i o n , a t l e a s t in p r i n c i p l e . here i s

The a n a lo g y to be borne in mind

to the Chern-Weil theorem [ M i - S t ] on c h a r a c t e r i s t i c classes of

Riemannian m a n i f o l d s .

Just as a u n i v e r s a l d i f f e r e n t i a l

classical

p u l l s back ( g i v e n a c l a s s i c a l

Grassmannian

form in

the

Gauss map) to a

de Rham c o - c y c l e r e p r e s e n t i n g a c h a r a c t e r i s t i c c l a s s , a " u n i v e r s a l c o - c y c l e " in one of our "PL" function.

(In

address t h i s

Grassmannians

performs a s i m i l a r

the subsequent c h a p t e r - b y - c h a p t e r o u t l i n e ,

we s h a l l

p o i n t more s p e c i f i c a l l y . )

Beyond t h i s ,

we are i n t e r e s t e d in

the r e l a t i o n between

" g e o m e t r i c a l s t r u c t u r e " on m a n i f o l d s and Gauss m a p s . s t r u c t u r e , in our sense t y p i c a l l y

means immersion of

Geometrical the m a n i f o l d s

i n t o a given ambient space, p o s s i b l y w i t h a d d i t i o n a l c o n d i t i o n s as to the " l o c a l

geometry" of

the immersion.

In the smooth case, such

g e o m e t r i c q u e s t i o n s u s u a l l y are phrased in data,

terms of

infinitesimal

so t h a t a "geometi'y" fo," the m a n i f o l d may be most u s e f u l l y

t h o u g h t of as a cross s e c t i o n o~ some bundle cf map germs s a t i s f y i n g , say, some f u r t h e r c o n d i t i o n d e f i n e d in terms of a j e t

bundle to which

0.6 the o r i g i n a l example is is

germ-bundle maps v i a d i f f e r e n t i a l s .

an immersion, which is

of maximal rank e v e r y - where.

theorem of H i r s c h ,

of course a smooth map whose l - j e t The t h e m a t i c r e s u l t

Gromov and P h i l l i p s ,

[P]

s e c t i o n of

sufficient

the germ bundle i t s e l f , section.

example, H i r s c h ' s o r i g i n a l

result

m a n i f o l d s is of

the

bundle w i t h the

evidence f o r

the e x i s t e n c e of a

whose d i f f e r e n t i a l

p r o p e r t i e s as the o r i g i n a l

here is

which assures us in a

l a r g e number of cases t h a t a s e c t i o n of the j e t a p p r o p r i a t e p r o p e r t i e s is

The s i m p l e s t

has the same

Again t a k i n g the s i m p l e s t

[Hi]

tells

homotopic to an immersion i f

it

us t h a t a map between can be covered by a map

t a n g e n t bundles of maximal rank everywhere ( w i t h some a d d i t i o n a l

assumptions necessary in codimension 0). Of course i t

is

well

known t h a t the Hirsch Theorem admits a

g e n e r a l i z a t i o n i n t o the PL c a t e g o r y , w i t h c o n d i t i o n s being phrased in terms of

PL t a n g e n t bundles.

Yet i f

we wish to study immersions

satisfying certain further restrictions, view of

PL geometry, the general ideas of

seem i n a d e q u a t e .

n a t u r a l from the p o i n t of the G r o m o v - P h i l l i p s Theorem

There are no d i f f e r e n t i a l s ,

jet

bundles e t c .

in the

PL c a t e g o r y . However, we s h a l l

see t h a t c e r t a i n kinds of geometries on

m a n i f o l d s - c e r t a i n kinds of immersions meeting l o c a l -

do correspond in n a t u r a l w a y s to the Grassmannians

c o n s t r u c t and, m o r e p a r t i c u l a r l y , immersion whose l o c a l map whose image l i e s Thus,

in

different

the s p i r i t

to subspaces

properties satisfy

we s h a l l

thereof.

some r e s t r i c t i o n

in an a p p r o p r i a t e subspace of of

specifications

That i s ,

an

has a Gauss

the Grassmannian.

the G r o m o v - P h i l l i p s theorem but w i t h much

c o n s t r u c t i o n s in

hand, we may ask the converse q u e s t i o n :

Given an a b s t r a c t map of a m a n i f o l d to the i n d i c a t e d subspace of

the

Grassmannian, covered by a map ~rom the t a n g e n t bundle to the cannonical bundle, can we then o b t a i n an immersion w i t h the a p p r o p r o p r i a t e geometry?

We s h a l l

prove theorems of t h i s

kind u s u a l l y

0.7 with the p r o v i s o t h a t the m a n i f o l d in question be open. We s h a l l also address f u r t h e r questions in a r e l a t e d vein having to do w i t h smoothing theory and with p i e c e w i s e - d i f f e r e n t i a b l e , r a t h e r than piecewise l i n e a r maps.

We s h a l l also consider versions of these

r e s u l t s in the c o n t e x t of a c t i o n s by f i n i t e

groups.

The reader may

f i n d the f o l l o w i n g o u t l i n e u s e f u l . Chapter 1.

Local formulas f o r c h a r a c t e r i s t i c classes.

The main t o p i c in t h i s

section is an e x p o s i t i o n of the a u t h o r ' s j o i n t

work with C. Rourke [Le-R] proving the e x i s t e n c e of l o c a l r a t i o n a l c h a r a c t e r i s t i c classes of PL m a n i f o l d s . here i s t h e m a t i c . is

formulas f o r

The methodology

A s e m i - s i m p l i c a l complex JQnJ is constructed which

the n a t u r a l t a r g e t of a Gauss map from t r i a n g u l a t e d n - m a n i f o l d s

w i t h a l o c a l o r d e r i n g of v e r t i c e s .

JQnJ n a t u r a l l y supports a

canonical n-block bundle which r e c e i v e s a n a t u r a l n- block bundle map from the tangent b l o c k - b u n d l e of such a m a n i f o l d , which map covers the Gauss m a p . The e x i s t e n c e of c h a r a c t e r i s t i c classes f o r the canonical b l o c k - b u n d l e e a s i l y leads to the e x i s t e n c e theorem.

The

chapter also contains a g e n e r a l i z a t i o n to homology m a n i f o l d s as well as a b r i e f discussion of various attempts to f i n d a concrete formula f o r the Pontrgagin classes and L - c l a s s e s . Chapter 2.

Formal l i n k s and the PL Grassmannian ~ / ~ k .

In t h i s chapter we c o n s t r u c t the "PL GrassmannianU~Jn,k, with i t s

canonical PL n-bundle

Yn,k.

together

This is the n a t u r a l

Grassmannian f o r s i m p l e x - w i s e l i n e a r immersions of t r i a n g u l a t e d n-manifolds into

Rn+k.

It

i s shown how a Gauss map a r i s e s n a t u r a l l y

and a u t o m a t i c a l l y f o r such immersions. Chapter 3.

Some v a r i a t i o n s ~

This chapter b r i e f l y

the ~n~k

constructfon.

e x p l o r e s the c o n s t r u c t i o n of spaces akin to / ~ n , k

and a p p r o p r i a t e to g~ometric s i t u a t i o n s other than s i m p l e x - ~ i s e l i n e a r immersions of t r i a n g u l a t e d m a n ; f o l d s .

I,i p a r t i c u l a r maps more

general than immersions and complexes more general than c o m b i n a t o r i c a l

0.8 manifolds correspond to c e r t a i n spaces defined s i m i l a r l y to / ~ n , k . Chapter 4. In t h i s /~,k.

The immersion theorem f o r

subcomplexes 2~/~{__11~_}~.

section we d e f i n e the n o t i o n of geometric subcomplex of

In s p i r i t ,

t h i s means a subcomplex which receives the Gauss map

of manifolds immersed in such a way t h a t a d d i t i o n a l geometric r e s t r i c t i o n s are observed. I f ~ manifolds

Mn

i s such a subcomplex, we consider

whose tangent bundles map to the r e s t r i c t i o n

the canonical bundle

Yn,k.

has image in

will

of

The main r e s u l t , g e n e r a l i z a t i o n s of

which occupy much of the remaining t e x t , be non-closed, then i t

to ~

immerse in

is

that i f

Rn+k

such a m a n i f o l d

so t h a t the Gauss map

~.

Chapter 5.

Immersions e q u i v a r i a n t with r e s p e c t to orthogonal a c t i o n s on Rn+k.

Here we g e n e r a l i z e the r e s u l t of the l a s t chapter to deal with t r i a n g u l a t e d manifolds on which a f i n i t e

group acts s i m p l i c i a l l y and

with orthogonal a c t i o n s by t h a t group on a u t o m a t i c a l l y acts on /~n,k as w e l l . )

Rn + k .

( T h e group then

The idea is

to o b t a i n

e q u i v a r i a n t immersions s u b j e c t to a d d i t i o n a l geometric c o n d i t i o n s corresponding to an i n v a r i a n t geometric subcomplex 7 .

The r e s u l t

holds f o r manifolds s a t i s f y i n g the so - c a l l e d Bierstone c o n d i t i o n . Chapter 6.

Immersions i n t o t r i a n g u l a t e d m a n i f o l d s .

This chapter contains the t h e s i s work of my student Regina Mladineo. As the t i t l e

suggests, we study immersion theory where the t a r g e t

space i s now a t r i a n g u l a t e d m a n i f o l d r a t h e r than Euclidean space.

We

s t a r t by c o n s t r u c t i n g , f o r a t r i a n g u l a t e d m a n i f o l d , an analog to the Grassmannian bundle associated to the tangent bundle of a smooth manifold.

If

Wn+k

is

t r i a n g u l a t e d we c o n s t r u c t ~ n , k ( ~ )

the n a t u r a l t a r g e t of a Gauss map from t r i a n g u l a t e d m a n i f o l d immersing in r e s p e c t to the t r i a n g u l a t i o n . images of s i m p l i c e s of

Mn, where

Wn+k

Here it

Mn

which is

is a

in geaeral o o s i t i o n with

is also assumed tKat inve~s~

W are subcomplexes of

M

and t h a t the map

0.9 is

simplex-wise c o n v e x - l i n e a r .

fiber

bundle over

In p o i n t of

-~n k(W)

is

not a

W but r a t h e r a s e m i s i m p l i c i a l complex assembled

from a c o l l e c t i o n of copies of

J~n-r,k

simplex of

r.

W of codimension

d e f i n e d and i t

fact,

is

w i t h one copy f o r each

Geometric subcomplexes are then

shown t h a t a r e s u l t analogous to t h a t of Chapter 4

can be o b t a i n e d .

If

a c t i o n s by a f i n i t e

W and

M

are f u r t h e r equipped w i t h s i m p l i c i a l

group then the analog to the r e s u l t of Chapter 5

can be o b t a i n e d as w e l l . Chapter 7.

The Grassmannian f o r

piecewise-smooth immersions.

Here we broaden our c o n s i d e r a t i o n s to study PL m a n i f o l d s equipped not w i t h a t r i a n g u l a t i o n but r a t h e r w i t h a s t r a t i f i c a t i o n "linkwise simplicial"

and where each stratum is

smoothness s t r u c t u r e so t h a t i n c l u s i o n s of are smooth.

If

provided with a

s t r a t a i n t o higher s t r a t a

we c o n s i d e r piecewise-smooth immersions of such

m a n i f o l d s M i n t o Euclidean space Rn+k, a p p r o p r i a t e n o t i o n of n,k

which i s

Grassmannian.

it

is

n a t u r a l to l o o k f o r an

This space, which we d e s i g n a t e

t u r n s out to be c l o s e l y r e l a t e d to the Gn, k of p r e v i o u s c

chapters.

In f a c t , ~?'n,k

realization

is

of a s i m p l i c i a l

,k r e t o p o l o g i z e d as a the geometric

space r a t h e r than a s i m p l i c i a l

theorem analogous to the main r e s u l t of Chapter 4 is Chapter 8.

Some a p p l i c a t i o n s t o

set.

A

obtained.

smoothing t h e o r y .

This c h a p t e r r e p r e s e n t s a d e t o u r from the main t h r u s t of the f o r e g o i n g Chapters 2-7 in

t h a t we are no l o n g e r concerned w i t h immersion t h e o r y

but w i t h smoothing t h e o r y . A°rd which i s , i

and of ~ ,

a locally

in

We begin w i t h the c o n s t r u c t i o n of a space

some sense a s i m p l e r v e r s i o n of

k as w e l l .

Aord i s

the JQnl of c h a p t e r

the n a t u r a l t a r g e t of Gauss map from

ordered t r i a n g u l a t e d m a n i f o l d Mn, y e t ,

N.B.,

it

is

not

c o n s t r u c t e d w i t h a view to s u p p o r t i n g a c a n o n i c a l PL bundle. has~ so to speak, one i - c e l l of

Si - I .

i-cell

Aerd

f o r each p o s s i b l e ordered t r i a n g u l a t i o n

We then go on to c o n s t r u c t a n o t h e r space ABr which has one f o r each "Brouwer s t r u c t u r e " on the cone on an o r d e r e d ,

0.10 t r i a n g u l a t e d Si - 1 ,

where a Brouwer s t r u c t u r e means a s i m p l e x - w i s e

l i n e a r embedding in Ri .

ABr i s

r e t o p o l o g i z e d to produce

,k)

then r e t o p o l o g i z e d ( a s ~ n , k was to y i e l d y e t another space

ACBr maps n a t u r a l l y i n t o Aord. Our theorem is if

and only i f

there i s a homotopy l i f t

ACBr.

t h a t Mn i s smoothable

in the diagram

ACBr Mn ÷ AOrd• What i s

i n t e r e s t i n g about t h i s

no a p r i o r i

r e s u l t is t h a t the p r o p e r t y sought has

connection with bundle t h e o r y .

Chapter 9.

E q u i v a r i a n t piecewise d i f f e r e n t i a b l e immersions•

We resume the main theme of these notes by c o n s i d e r i n g p i e c e w i s e smooth manifolds s u p p o r t i n g a compatible f i n i t e

group a c t i o n s and

e q u i v a r i a n t immersions i n t o a Euclidean space on which the group acts orthogonally.

We g e n e r a l i z e the r e s u l t of Chapter 7 j u s t as Chapter

5 g e n e r a l i z e d t h a t of Chapter 4. Chapter 10.

Piecewise d i f f e r e n t a i a b l e immersions i n t o Riemannian m a n i f o l d s .

We now consider piecewise-smooth immersions where the t a r g e t is a smooth m a n i f o l d equipped with a Riemannian m e t r i c •

For such spaces c

Wn+k we c o n s t r u c t an " a s s o c i a t e d Grassmannian b u n d l e ~ n , k ( W ) (now truly is

a bundle) whose f i b e r i s

the ~ cn,k of chapter 7 •

j ~ nc, k ( W )

the n a t u r a l t a r g e t of a Gauss map from Mn when Mn i s p i e c e w i s e -

smoothly immersed•

As Chapter 6 g e n e r a l i z e d the r e s u l t s of

Chapters 4 and 5, t h i s Chapter g e n e r a l i z e s Chapters 7 and 9. A b r i e f glossary of i m p o r t a n t d e f i n i t i o n s and c o n s t r u c t i o n s i s provided in the appendix•

10

1.1 1.

Local Formulae f o r

The p o i n t o f

view which l o o k s a t

m a n i f o l d as g l o b a l

summaries o f

I n s o f a r as S t i e f f e l - W h i t n e y be s a i d

the c h a r a c t e r i s t i c data is

the s u b j e c t .

c l a s s e s were d e v i s e d i n

bundles and smooth m a n i f o l d s , definition

local

Classes c l a s s e s of a

a r a t h e r o l d one.

classes are concerned, t h i s

t o have been born w i t h

Stieffel-Wbitney

Characteristic

it

approach may

In p a r t i c u l a r ,

though

connection with vector

became c l e a r e a r l y

on t h a t

the

readily

e x t e n d e d to c o m b i n a t o r i a l m a n i f o l d s . [In fact, i v i a the d e f i n i t i o n w = ( ~ ¢ ) - I S q ¢, ~ the Z/2Z Thom c l a s s of the i b u n d l e i n q u e s t i o n , i t is e a s i l y seen t h a t P o i n c a r e d u a l i t y spaces have w e l l - d e f i n e d interesting fold, that

or,

Stieffel-Whitney

a s p e c t of

the d e f i n i t i o n

w i a combinatorially

more c o r r e c t l y ,

the d e f i n i t i o n

c l a s s e s as w e l l ] .

is

local.

of

But the more

on a c o m b i n a t o r i a l manitriangulated

manifold,

is

We remind the r e a d e r how the f o r m u l a

works. n M

Let

be a c o m b i n a t o r i a l l y

r e s p e c t to t h i s centric

i-co-cycle n M

on

dual

triangulation T I "

subdivision

g i v i n g an ture

fixed

to

to

w , ]

formula is Let

(co-efficients T .

plices 1.1

of

the

n-i

by s p e c i f y i n g an

= ZT (in

n-i int

(n-i)

n-manifold.

we have the f i r s t

in

Alternatively,

extraordinarily y* n- i T'

T,

The f o r m u l a f o r

giving a representative for dual

triangulated

w i Z/2Z)

With

bary-

may be viewed as for

the c e l l

struc-

we may read the f o r m u l a as

homology c l a s s cycle in

w* n-i itself.

T

Poincare The

simple:

where t.l

if

Theorem (Whitney [Whn];

ranges o v e r a l l n-i ~ ~ 0). Then

the

s

see a l s o

[Chl],

[H-T]).

(n-i)-sim-

y

is

a

n-I

Z/2Z

c y c l e whose homology c l a s s ThUs,

Poincare

one may reed o f f d u a l s of

wishes t o t r a n s l a t e

is

directly,

wk n- i

H (M, n-i

on the c h a i n

the s t a n d a r d S t i e f f e l - W h i t n e y this

into

~i;

Z/2Z)~

level,

the

classes.

If

a corresponding statement about

11

one

1.2 co-cycle representatives Stieffel

(in

the dual

Whitney cohomology c l a s s e s

the t r i a n g u l a t i o n , The o r d e r i n g

at

least

cell

structure)

themselves, it

for

is

so t h a t each s i m p l e x i s

useful

to o r d e r

linearly

ordered.

c a n o n i c a l l y d e f i n e s a s u b d i v i s i o n map

l

the

X: T , + T,

and so we o b t a i n a c y c l e ~,¥ E C , ( T , T ~ ~M; Z / 2 Z ) . I f we l e t i n-i y be d e f i n e d ( w i t n r e s p e c t to the c e l l s t r u c t u r e T* P o i n c a r e dual i to T) by y ( ~ * ) = (number of ( n - i ) - s i m p l i c e s in ~ - I ~ c T ' ) (mod 2 ) . Then

1.2

Corollary.

Stieffel-Whitney Note t h a t

i

y

is

class

the v a l u e of

on the s t r u c t u r e the p a t t e r n f o r

of

y

i

on a dual

our g e n e r a l i z a t i o n ,

be a

(T*,

the o r d e r e d s i m p l i c i a l

ence t h e o r e m s , to a r b i t r a r y n-manifolds. n Let M

i C

a c o c y c l e in i w (M).

PL

at

i-cell complex

least

characteristic

Z/2Z)

r e p r e s e n t i n g the

o*

depends o n l y

st(o).

on the l e v e l

This

sets

of e x i s t -

co-homology c l a s s e s of

PL

manifold with a c o m b i n a t o r i a l t r i a n g u l a t i o n

T. 1.3

Definition.

the v e r t i c e s k st(a)) (o

of

A local T

such

orderin~ for t h a t each s t a r

a k - s i m p l e x of

T)

is

T

is

st(q

a p a r t i a l o r d e r i n g of k ,T) (abbreviated

thereby linearly

ordered.

Abstractly,

an n - s t a r o f c o d i m e n s i o n i, i < n, s h a l l mean a n-i i-i n-i complex o f the form A * ~ , where A i s the s t a n d a r d n-i i-1 s i m p l e x and ~ denotes a c o m b i n a t o r i a l l y t r i a n g u l a t e d (i-1)sphere

(= 0

if

i

= 0).

An o r d e r e d c o d i m e n s i o n - i n - s t a r

is

such an

object

w i t h a l i n e a r o r d e r i n g of i t s v e r t i c e s , and an o r i e n t e d s t a r i-1 means one where ~ has been g i v e n an o r i e n t a t i o n ~. Isomorphism of

o r d e r e d ~ t a r s means a s i m p l i c i a l

o r d e r i n g and t h e f a c t o r s

A, Z

of

isomorphism p r e s e r v i n g both the the j o i n .

12

1.3 1.4

Definition.

A l o c a l - o r d e r e d formula f o r an i - d i m e n s i o n a l

co-chain with c o e f f i c i e n t s in (isomorphism classes o f ) t a k i n g values in ¢ ( A n - i , ~i-1 , , ~ )

G. =

is a function

~

defined on

o r i e n t e d , ordered, codimension-i n - s t a r s

I t is An-i 0(

-

G

further stipulated that ~i-1 , - ~ ) .

*

A l o c a l formula is merely a l o c a l - o r d e r e d formula such t h a t An-i i-1 i-1 ¢( * Z , (j) depends only on the s i m p l i c i a l s t r u c t u r e of n-i i-4 and not on the o r d e r i n g of A * Z n

If

M

is a manifold with a l o c a l l y ordered t r i a n g u l a t i o n T, n-i i t is c l e a r t h a t f o r any ( n - i ) - s i m p l e x o , the s t a r st(o,M) may n-i be regarded as an ordered, c o - d i m e n s i o n - i n - s t a r o * Ik(o,M). Thus, given a l o c a l ordered formula

m f o r an i - d i m e n s i o n a l i ~(T) E C ( T * ; G ) , where T*

G-cochain, we o b t a i n a co-chain n the c a l l s t r u c t u r e on M Poincare' dual to @M ~ ~ ,

T*

is a c e l l

M,

but t h i s

by n o t i n g t h a t f o r each dual i - c e l l to

M

an o r i e n t a t i o n

should m

o , o + ¢(0* I k ( o ) , co-chain theory

of

We note t h a t f o r

s t r u c t u r e on a deformation r e t r a c t of

r a t h e r than the whole of

interior

T.

@M ~ )

is a minor p o i n t .

o*

C*(T*;G).

an

an o r i e n t a t i o n

Ik(o,M) = I k ( o ) .

~) ~ G

(o

denotes

M

We see t h i s

n - i - s i m p l e x of o

T,

may be regarded as

T h u s the assignment

defines an i - c o - c h a i n in the ( o r i e n t e d ) We denote t h i s

class

@(T).

We consider an i - d i m e n s i o n a l c h a r a c t e r i s t i c class c for i n-dimensional PL m a n i f o l d s , i . e . an element c ~ H (BPL(n); G). 1.5 D e f i n i t i o n . c

if

The l o c a l

and only i f ,

t r i a n g u l a t i o n s T, [¢(T)]

( o r d e r e d ) formula

¢

is said to r e p r e s e n t n f o r a l l c o m b i n a t o r i a l manifolds M , and a l l i (T) i s a c o - c y c l e with the co-chain

: c{M) E Hi(M,G).

Our main r e s u l t i ~ : 1.6

Theorem. [Le-R] Given any c h a r a c t e r i s t i c class

there e x i s t s a l o c a l - o r d e r e d formula

13

¢

c E H (BPL(n),G),

representing

c.

1.4 A special rational 1.7

case of

numbers.

Let

G

a local

Proof:

interest

occurs when

We then have the f o l l o w i n g

Corollary.

Then t h e r e is

particular

By 1 . 6 ,

be a d i v i s i b l e

let

is

the

relevant corollary. i c ~ H (BPL(n);G).

group,

formula r e p r e s e n t i n g

G

c.

be a l o c a l - o r d e r e d f o r m u l a r e p r e s e ~ t i n g 1 c. D e f i n e the l o c a l ( u n o r d e r e d ) f o r m u l a # on an ( u n o r d e r e d ) n-i i-i co-dimension i n-star ~ , ~ by

n-i

¢

i -i

n-i

~T R where

q

is

over a l l

possible linear

manifold

M

we have of

the number o f

with

an o r i e n t a t i o n

on

1.8

o

characteristic

There i s class

P o n t r j e g i n class

for

p. I

T,

T,

#(T)=

Clearly,

~*,

c o r r e s p o n d s to

Corollary.

of

and

these v e r t i c e s .

triangulation

local-orderings c(M).

o

n-i i -i A * Z

of

a co-cycle representing

also representing

(where

i

o r d e r i n g s of

(finite)

~1(T,x)

distinct

vertices

i -1

i

Thus,

~ ~I(T,~ )

given a simplex

we have

~(T)

~*,o)

ranges

Then,

and l o c a l

c(M).

A

for

ordering if

a ~,

m = number

is a c o - c y c l e n-k ~ C i n t M and

: ~(~*lk(~),,~)

W). a local PL

f o r m u l a r e p r e s e n t i n g any r a t i o n a l

manifolds;

and the r a t i o n a l

n particular L-class

L

i

the r a t i o n a l are so

represented.

1.9

Corollary.

(n-i)-simplex simplicial

Suppose ~ C int

n M

is

M, ~ k ( o )

self-homeomorphism.

characteristic

classes of Let

~

M

triangulated

by

T

so t h a t

for

any

a d m i t s an o r i e n t a t i o n - r e v e r s i n g Then a l l

i-dimensional rational

must v a n i s h .

Proof: n-i i-i A * ~

be a l o = a l

formula

be a c o d i m e n s i o n - i n - s t a r

orientation

reversing simplicial

with

with

co-efficients in Q yi-1 a d m i t t i n g an

self-homeomorphism.

14

and

Then g i v e n an

1.5 i-1 n-i 4-i ~ on , it follows that A * Z , ~ is i s o n-i i-i n-i i-i n-i i-1 m o r p h i c to a * ~ , -~. So ~(A * ~ ,~) = ~(~ * Z ,-(J). n-i i-I n-i i-i n-i i-I But ¢(A * ~ ,~) = - ~(A * ~ ,-~). So ~(a * ~ ,~o) = O. orientation

Thus, since tic

¢

¢(T)

m O,

with

the g i v e n h y p o t h e s i s on

all

such

c l a s s e s v a n i s h on

B e f o r e moving to the a c t u a l d i s c u s s i o n is

in

conjecturing

that

to,

in

1.1 -

say,

order. local

p r o o f of

1.6,

( o r d e r e d ) f o r m u l a e must e x i s t

result

Riemannian s t r u c t u r e s

c o n n e c t i o n s ) these c l a s s e s

forms.

is

for

the example

F u r t h e r m o r e , the t e m p t a t i o n t o g e n e r a l i z e

P o n t r j a g i n c l a s s e s or L - c l a s s e s i s

manifolds provided with

r e p r e s e n t e d in

some p h i l o s o p h i c a l

O b v i o u s l y , the p r i m a r y i n s p i r a t i o n

1.2 above.

rational

characteris-

M.

s t r e n g t h e n e d by the d i f f e r e n t i a l - g e o m e t r i c

tial

Therefore,

may be chosen to r e p r e s e n t any g i v e n r a t i o n a l

class,

cited

T.

(with

real

further

that

for

smooth

(or merely a f f i n e

co-efficients)

are c a n o n i c a l l y

de Rham cohomology by " l o c a l l y

That i s ,

determined" differenn g i v e n a Riemannian m a n i f o l d M , the r e a l

P o n t r j a g i n class p.(M) i 1 P (M) ~ ~ (M), dP = O. i i f o r any open s e t U of

is

r e p r e s e n t e d by the Chern-Weil

form

P i s " l o c a l " i n t h e sense t h a t i M, P (U) = P (M)IU. For d e t a i l s the r e a d e r i i may c o n s u l t the book of M i l n o r and S t a s h e f f [ M - S ] . It

will

Moreover

be a c o n t i n u i n g theme of

ment of a s p e c i f i c

triangulation

to a

a n a l o g o u s to c h o o s i n g a s p e c i f i c fold.

That i s ,

specific, that

rather

global

geometry.

rigid

PL

monograph t h a t manifold is

Riemannian m e t r i c

endowing a " t o p o l o g i c a l " geometry.

the a s s i g n -

in many ways

f o r a smooth mani-

object with a

The t h e m a t i c p r i n c i p l e

then emerges

i n f o r m a t i o n a b o u t the m a n i f o l d s h o u l d then be seen as a

summary, so t o

and t ~

one i s

this

speak,

of l o c a l

Both the Chern-Weil

contributions forms f o r

Whitney c y c l e

formula for

be seen as i l l u s t r a t i v e

examples.

characteristic

(d~al)

real

d e t e r m i n e d by l o c a l characteristic

Stieffel

classes

S h i t n e y c l a s s e s may

The c o n j e c t u r e t h a t a r b i t r a r y

c l a s s e s a r e r e p r e s e n t e d by l o c a l

15

formulae t h e r e f o r e

PL

1.6 becomes q u i t e n a t u r a l .

In p a r t i c u l a r , one expects t h a t the l o c a l

i n f i n i t e s i m a l data on a Riemannian m a n i f o l d g i v i n g r i s e to the ChernWeil

forms ought to be replaced by " s i n g u l a r " d a t a , i . e .

c o n t r i b u t i o n s f o r each b i t

of r e l e v a n t l o c a l geometry.

discrete For an

i - d i m e n s i o n a l class the " r e l e v a n t " b i t s should be the l o c a l geometry in the neighborhood of each n - i

s i m p l e x , in other words, the

s i m p l i c i a l s t r u c t u r e of the l i n k s of such s i m p l i c e s .

The example of

the S t i e f f e l - W h i t n e y classes suggests, a t l e a s t , t h a t o r d e r i n g data should f i g u r e as well f o r such a l o c a l

f o r m u l a , a t l e a s t in the

absence of an averaging procedure l i k e t h a t in Cor. Historically,

the f i r s t

example of such a l o c a l

1.7. formula comes

from the papers of G a b r i e l o v , Gelfand and Lossik [GGL] on the d e t e r m i n a t i o n of a cocycle r e p r e s e n t i n g

p

of a smoothly t r i i angulated smooth m a n i f o l d which turns out to depend merely on the l o c a l c o m b i n a t o r i a l s t r u c t u r e of the t r i a n g u l a t i o n .

The procedure i s

complicated and somewhat obscure, although c l a r i f i e d

somewhat by the

papers of MacPherson [Mac] and D. Stone [ S t I ,

We s h a l l

describe t h i s

St2].

not

c o n s t r u c t i o n here, a l t h o u g h , a t the end of t h i s

s e c t i o n , we s h a l l make some remarks on G a b r i e l o v ' s a t t e m p t to extend these methods to higher P o n t r j a g i n classes. however, t h a t ,

although

p

1

It

is noteworthy,

is an i n t e g r a l class on

PL

manifolds,

the method of [GGL] do not seem to r e s u l t in a l o c a l - o r d e r e d formula f o r an i n t e g r a l r e p r e s e n t i n g c o c y c l e . * Cheeger [Ch2] has a t t a c k e d , with some success, the problem of finding local briefly

formulae f o r the real

L - c l a s s e s , and we s h a l l also

describe the general idea of his approach a f t e r proving 1.6.

We must take note, a t t h i s

point,

e x i s t e n c e theorem, as the proof w i l l to describe an e x p l i c i t f o r a given c l a s s .

t h a t Theorem 1.6 i s purely an

make c l e a r .

c o n s t r u c t i o n of the l o c a l

No a t t e m p t i s made ( o r d e r e d ) formula

Nevertheless, the e x i s t e n c e proof i s s u r p r i s i n g l y

quick and e l e g a n t , and demonstrates the power of the v i e w p o i n t taken ~See [ M i ] ' f o r a computational example. d i f f e r e n t approach.

16

See [Le 2] f o r a somewhat

1.7 in t h i s monograph as a whole: on a m a n i f o l d , i t

Given some notion of e x p l i c i t

geometry

becomes p o s s i b l e to replace the idea of " c l a s s i f y -

ing space" f o r the a p p r o p r i a t e kind of bundle by a "Grassmannian." That is

" c l a s s i f y i n g spaces" are,

traditionally,

o b j e c t s in the

homotopy category whereas a "Grassmannian" means a s p e c i f i c space with i t s

own e x p l i c i t

geometry.

At the same t i m e , the " c l a s s i f y i n g

map" f o r the tangent bundle of a m a n i f o l d ( d e f i n e d up to homotopy) is r e f i n e d , in the presence of geometry, i n t o a "Gauss map" i . e .

a

s p e c i f i c , canonical map i n t o the Grassmannian which somehow keeps t r a c k of the l o c a l The proof of

geometry of the m a n i f o l d . 1.6,

which we now g i v e , r e c a p i t u l a t e s t h a t to be

found in the paper of the author an C. Rourke [ L - R ] . First,

some t e r m i n o l o g y .

An s - b a l l IKi

is a l i n e a r l y - o r d e r e d s i m p l i c i a l complex

K

such that

i s a Euclidean b a l l . An s - c e l l

complex i s a p a r t i a l l y

ordered s i m p l i c i a l complex

t o g e t h e r with a f a m i l y of subcomplexes (1)

Each

L i

is

totally

{L } such that i ordered and, as w e l l , an s - b a l l

K,

(of

some dimension).

IKI, { I L l }

(2)

is a c e l l

1

Thus, an s - c e l l

complex is b a s i c a l l y a c e l l

t r i a n g u l a t e d so t h a t each c e l l angulation p a r t i a l l y

complex (as in

[R-S,

p. 3].

complex, f u r t h e r

i s a subcomplex, and with the t r i -

ordered so t h a t the subcomplex f o r any c e l l

is

l i n e a r l y ordered. An isomorphism plicial

h: K ÷ K between s - c e l l complexes is a simi 2 isomorphism p r e s e r v i n g the o r d e r i n g on each c e l l .

As an example, consider a c o m b i n a t o r i a l l y t r i a n g u l a t e d manifold with a l o c a l o r d e r i n g .

K

is

the t r i a n g u l a t i o n , the Poincare'

,~ual c e l l

structure

that

has a d e r i v e d l o c a l o r d e r i n g ( v i z . ,

K'

o r d e r i n g on i t s

K*

If

thep becomes an s - c e l l

complex, in the sense

the l e x i c o g r a p h i c

v e r t i c e s a r i s i n g from the o r d e r i n g on

17

K); moreover,

1.8 each dual

cell

the p a r t i a l

of

K*

is

a subcomplex o f

K',

o r d e r of

and i s

k)

(i)

linearly

c o n s i s t s of

o r d e r e d by

t h e r e f o r e an s - b a l l .

o v e r an s - c e l l c o m p l e x

An s - b l o c k bundle dimension

K'

K

(of

fiber

the f o l l o w i n g :

A partially-ordered

complex

Q

with

K C Q,

(preserving

ordering)

(2)

L i n e a r l y - o r d e r e d subcomplexes

{R } , 1 an s - b a l l such

L

C K) with L. ~__ R and i I i forms a k - b l o c k bundle o v e r An isomorphism of

which i s linear

R i IKl,{IL

}I. 1 s - b l o c k bundles i s

a simplicial

IQI,{IR.I} i

isomorphism

o r d e r on each b l o c k .

i n g of

PL

of

Recall over

that:

each c e l l

a l s o a b l o c k - b u n d l e isomorphism and which p r e s e r v e s the

n n+k M C_ W

To t a k e an e x a m p l e , l e t

tion

(one f o r

W

the M

of

manifolds. such

that

Suppose n

M

construction

is

of

the embeddinp:

r e p r e s e n t s the dual

cell

a k - b l o c k - b u n d l e over

If

is

a simplex of

in

K*,

M = IKI

become s - b a l l s

example, i t

(so-called

PL

bundle

to d i s t i n g u i s h

TM)

K,

specific

way ( i . e .

the c e l l

way, f o r

any p a i r

O,T

ordered).

with

Moreover,

of KxK

blocks

complex w i t h

cells

K, then ~* ~ ,

K.

bundle and E : over and,

vw(M} ~ , o a

is

o*.

o f an s - b l o c k - b u n d l e o v e r

K*

As

since

(when r e g a r d e d as subcomplexes of

K it,

of

M,

P')

(of

the t a n g e n t b l o c k - b u n d l e

as a f o r m a l i t y ,

a c q u i r e s the s t r u c t u r e

g i v e n the o r d e r i n g on

triangula-

becomes p o s s i b l e t o see how, g i v e n a

ordered t r i a n g u l a t i o n

~M

normal b l o c k

respectively,

= IK*I

an s - c e l l

v (M) a c q u i r e s the s t r u c t u r e W f i b e r dimension k}.

P*

embedd-

by the subcomplex

the

the c e l l s

locally

triangulated

~

flat)

a locally-ordered

of

K

From t h i s

is

JR-S]

we have seen, ~*

is

P

be a ( l o c a l l y

o f an s - b l o c k b u n d l e .

we may t r i a n g u l a t e O×T

IKI×IKI

becomes t r i a n g u l a t e d

simplices, is

from the t a n g e n t

since

locally-ordered

18

O,T

= MxM

For in a

in a standard

are l i n e a r l y

by t h e induced

1.9 l e x i c o g r a p h i c order on i t s v e r t i c e s . F u r t h e r , the diagonal map A A M + M×M i s a s i m p l i c i a l map K + K×K, and thus v (AM) M×M acquires an s-block bundle s t r u c t u r e as above. But, of course v (AM) i s , by d e f i n i t i o n , TM. M×M Our purpose now is to c o n s t r u c t a canonical s - b l o c k bundle over a u n i v e r s a l space f i b e r dimension To t h i s

end,

which w i l l

n

c l a s s i f y s - b l o c k bundles of

n.

some f u r t h e r t e r m i n o l o g y :

An s - C e l l cell

Q n

y

is an s - c e l l

of which a l l

complex with a s i n g l e t o p - d i m e n s i o n a l

others are faces.

The category .> n

(Cairns

[CI],

P

of

v

general].

depends on the chosen embedding).

N m C(K) K

such t h a t

in

U

{ba-v},

(Note

Proposition

t h a t an

x

in

M M

at such

meant an embedding l i n e a r

~-n

1.16

P

[Wh]

and c o n s i d e r a " g e n e r a l

the s e t of N K

planes

R ,

Recall

s a i d to be t r a n s v e r s e to

any

position"

the moment, we

is

vertices Let

N K

t r a n s v e r s e to

Whitehead [Whd]).

K

K, denote at

b A.

For a s u i t a b l e

j,

x Rj .

Now c o n s i d e r a l o c a l l y

ordered B r o u w e r - t r i a n g u l a t e d manifold Mn £ and a general p o s i t i o n embedding M C__ R , meaning t h a t the embedding is

a g e n e r a l p o s i t i o n embedding on the s t a r

N(M)

1.17

= ( x , P ) E MxG IP ~-n~n Proposition.

N(M)

t r a n s v e r s e to

is

at

implies that

N st

34

we note t h a t o

is

Let

x}.

homotopy e q u i v a l e n t to

Without g i v i n g a d e t a i l e d p r o o f 1.16 which d i r e c t l y

M

of e v e r y s i m p l e x .

B(M).

this

f o l l o w s from

homotopy e q u i v a l e n t to

1.25 the c o n f i g u r a t i o n ticular B(M)

C(st

c h o i c e o f embedding is

clearly

classes begins, g:

space

intrinsic

o).

M C R~ to

M.

however, w i t h

N(M) ÷ G

via

Of course

(x,P)

for

N(M)

a particular

Our a n a l y s i s of

N(M).

depends on a p a r ~,

whereas

characteristic

Note the o b v i o u s p r o j e c t i o n

map

+ P.

~-n,n

Consider a r a t i o n a l normal r a t h e r

characteristic

than t a n g e n t i a l )

class

~

(for

convenience,

t h o u g h t of as an e l e m e n t of H*(N(M),Q).

H*(G

; Q). This p u l l s back under g to g*~ = A(M) ~-n,n Since ~ i s a r a t i o n a l c l a s s , i t is d e f i n e d on t h e PL

manifold

M

and we c l a i m 1.18

Proposition. Briefly,

(~-n)-vector the

(k-n)

the

PL

~(M)

this

follows A

bundle

plane

= ~*~(M).

v

P.

from the f a c t

over

Moreover,

normal bundle of

In v i e w of

1.18,

class,

which t a k e s i n t o E u c l i d e a n space.

M.

of

viz,

whose f i b e r

at

a canonical

( x , P ) E N(M)

is

v ~ ~* v(M) where v(M) denotes PL Since ~'(M) = e ( ~ ) 1.18 f o l l o w s .

finding

for

the moment, a s l i g h t l y

a local

the problem o f

a c c o u n t an embedding o f That i s ,

there is

A

we may c o n s i d e r ,

e a s i e r problem than t h a t characteristic

N(M)

that

formula for

finding

a given

a local

the m a n i f o l d in

we may c o n s i d e r l o c a l

formula some

formulae

¢

which

a s s i g n a number t o each t r i p l e

K = a

n-i

* 5

~- i

0 = orientation e:

K C_R ,

( w i t h embeddings deemed e q u i v a l e n t E u c l i d e a n t r a n s f o r m a t i o n s on Given a m a n i f o l d in

Mn

general p o s i t i o n i ¢(T,f) ~ C (T*,Q). tic

class

~,

it

by

r e s p e c t to

To f i n d

Z

a general-position they d i f f e r

embedding

by an a c t i o n o f

the

R .)

triangulated (with

if

on

T),

and an embedding

F:M n C R

we t h e r e b y o b t a i n

r e p r e s e n t i n g the c h a r a c t e r i s -

such a

would c e r t a i n l y

T

suffice

35

via

1.18 t o have a

1.26 "locally-defined"

section

M .

N(M).

But remember [ C l ] ,

have any s e c t i o n w h a t e v e r ( w h e t h e r " l o c a l l y that

the m a n i f o l d

that a "local all

PL

M

is

smoothable.

formula for

[Wh]

d e f i n e d " or n o t )

Thus i t

a section" exist,

is

for

clearly that

a transfer

homomorphism

t:

rather

factors

the i d e n t i t y .

are in

Q

or

subdivision

would i m p l y t h a t

if

T

p r o j"* -~ H*(N(M))

on

a ,

is

the diagram i n

t* ~

chains;

(a subcomplex o f

H(M)

T )

is

p a r t of

in

efficients,

is

*

g m : m(M) E define, K~

for R~

d*

as f o l l o w s :

ai

(N(M))

a star

represents n-i i K = a * ~ ,

seeing t h i s ,

at

least

for

formula R~ real

is co-

a Gauss map

is

the dual

i-cell

o,

barycentric

in

the f i r s t ¢

~(M)

(in

de Rham c o h o m o l o g y ) . We i orientation o on ~ , embedding

the number

its

represents

triangulated t#m(M)

we would have a l o c a l

Since t h e r e i s

a ~ g e b r a i c sum o f

that

~ C_ R~ and n o t

sense where an embedding i n

An easy way of

¢ (K,o,e) Here

flst

÷ G , and s i n c e the i - d i m e n s i o n a l r e a l c h a r a c t e r i s t i c B-njn ~ i s c a n o n i c a l l y r e p r e s e n t e d by a d i f f e r e n t i a l i-form ~,

class

e:

C.(N(M))

determined, -i + C.(proj (~*)) with

tIC.(o*)

map t o e x i s t ,

the more r e s t r i c t e d

the d a t a ) .

barycentric

M.

Were such a t r a n s f e r least

coefficients

locally

the homomorphism depending o n l y on the embedding on the r e m a i n d e r of

cohomology

subdivision while

the t r a n s f e r I

plan might

a chain-level

t o be the c h a i n s on the f i r s t

or on some f i n e r

need m e r e l y be s i n g u l a r .

so t h a t

i.e.

Here we u n d e r s t a n d the f o l l o w i n g :

R; C.(M) T'

than a s e c t i o n ,

C.(M) + C.(N(M))

H*(M)

g:N(M)

implies

m a n i f o l d s are smoothable,

be t o ask f o r

(at

to

hopeless

However, a l e s s e x a c t i n g , and t h e r e f o r e more f e a s i b l e

i,e,

that

to

n-i A

in

K, r e g a r d e d as the

simplices appropriately

~,

m a n i f o l d M,

representing

= ftd.m(e(K)).

s u b d i v i s i o n of

since f o r

K.

It

an embedding f :

the c o - c h a i n

t*~(M)

ordered c o n s i s t e n t with

= m(M). 36

¢(M,f)

is

is

now o b v i o u s

Mn ÷ R~ of

the

by d e f i n i t i o n

1.27 A slightly

more cumbersome way of p r o v i n g the same t h i n g ,

(specifically

for

more i n s i g h t

into

the

k th

P o n t r j a g i n class

pk ),

one which o f f e r s

G a b r i e l o v ' s approach is as f o l l o w s :

Let

Q

be an

arbitrarily-chosen

n - p l a n e in

V C G be d e f i n e d Q ~-n,n by V = {P E G Idim P N Q ) 2k} A l t e r n a t i v e l y V may be Q ~-n,n Q t h o u g h t as the set of n - p l a n e s RE G such t h a t o r t h o g o n a l n,~-n p r o j e c t i o n of R to Q (equivalently, Q to R) has n u l l i t y ) 2k. It

V is a s u b m a n i f o l d of G with Q ~-n,n n o n - s i n g u l a r p a r t is a s u b m a n i f o l d of codimension

is

well

known t h a t

singularities. 4k is

R~ and l e t

Its

whose normal bundle is

naturally

p r e c i s e l y the d e f i n i n g f o r m u l a f o r

class

oriented. the

I n t e r s e c t i o n with

kth i n t e g r a l

V

Q

Pontrjagin

p

in the sense t h a t , given a smooth m a n i f o l d Nnc R and k n a t r i a n g u l a t i o n of N such t h a t the Gauss map g: Nn + G is ~-n,n i n general p o s i t i o n w i t h r e s p e c t to V , then Pk i s r e p r e s e n t e d

Q

by the c o c y c l e which assigns to each o r i e n t e d 4 k - s i m p l e x integer

g(o)

the

• V .

Q

To extend t h i s trary

~

complex

X

t r a n s v e r s e to

slightly,

is

VQ

if

classified

the

~-n

bundle

by a s p e c i f i c

on the 4 k - s i m p l i c e s of

map

X,

~

over an a r b i -

u:

X ÷ G , u ~-n,n pk(~ ~) is rep-

then

r e s e n t e d by the c o - c h a i n a s s i g n i n g to each o r i e n t e d 4 k - s i m p l e x the i n t e g e r real)

transfer

the c o - c h a i n ber

u(~)

t(c)

N(M) + G

Q

where p

t

is a ( r a t i o n a l

is

interpretable

k th locally

as a l o c a l

Of course, the a , - b i t r a r i n e s s of eliminate it standard

we s h a l l

have,

(for

or

generic

measure on

or r e a l )

hum-

co-cycle repre-

determined, it formula f o r Q

is

G

p (M), and, k is c l e a r t h a t p

Pk"

somewhat u n p l e a s a n t ; we

choices of

n,~-n

37

Q)

P o n t r i j a g i n class

s i m p l y by a v e r a g i n g over a l l

0(~)-invariant

the ( r a t i o n a l

c ¢ C (M) the ( r a t i o n a l or r e a l ) k is now the obvious n a t u r a l map

or r e a l )

on the assumption t h a t in f a c t ,

g

Clearly

s e n t i n g the ( r a t i o n a l

is,

the presence of

C.(M) + C.(N(M))

a s s i g n i n g to V

.

So, in

Q

t:

p -1

• g

• V .

Thus,

Q,

using the

the d e f i n i t i o n

1.28 of

the c o - c h a i n

p

equally a local

on

C,(M)

formula,

(2)

is

r e p l a c e d by a new d e f i n i t i o n ,

viz

p(c) : E

(t(c).g-~VQ)

QEG n,~-n where

E

now denotes e x p e c t e d v a l u e o v e r a l l

We thus

see how the c o n s t r u c t i o n of a t r a n s f e r

local

formulae for

folds

of

that

R

characteristic

We s h a l l

essentially

formula in

c l a s s e s of

show now how to

the same t r a n s f e r

our o r i g i n a l

First

choices f o r

of a l l ,

it

t

Q.

l e a d s to

n - d i m e n s i o n a l submani-

sharpen t h i s

construction

f o r m u l a t i o n so

generates a l o c a l

sense. is

quite

obvious t h a t

what must be e l i m i n a t e d

from t h e f o r m u l a above i s any s p e c i f i c dependence on t h e embedding o f n M in R and on 4. F i r s t of a l l , we n o t e t h a t , g i v e n a l i n e a r n o r d e r on the v e r t i c e s of the t r i a n g u l a t i o n o f M (say t h e r e a r e vertices

in a l l ) ,

there is

extending linearly

on s i m p l i c e s

t h e i th v e r t e x of

M

Noting t h a t ent of local

with

this

ordering,

and

vertices,

global

course,

data,

albeit

simplex

c o n s i d e r the subspace of

R

think

of

in

this

them) o f

~he s t a n d a r d

( i • e.

star

o.

as the s t a n d a r d

t r a n s v e r s e to

L(o)

is

t)

R~. M

(i.e.

Let to

v

is i R~.

independ-

be the s e t of ~)

at R j

b

such ~ in

× s t ( G ) C N ( s t o) C N(M).

38

is ~

that

of rather

Given a

spanned by the v e r t i c e s simplicity

RJ-space embedded i n

L(~) st

dependence.

For the sake of

the o r t h G g o n a l complement o f

Note t h a t

p

of a v e r y weak s o r t , this

of

for

the c o n s t r u c t i o n o f

We show how to e l i m i n a t e

j

(2)

by

the r e m a i n i n g o b s t a c l e t o an e x p l i c i t

than l o c a l .

(say,

R~

v

our f o r m u l a a dependence on t h e number

which i s

o,

in

+ b where i i s t a n d a r d b a s i s v e c t o r of

b

we see t h a t

t h e r e remains in

the a s s i g n m e n t

the i th i embedding, the f o r m u l a

f o r m u l a (modulo, of

Mn

a s t a n d a r d embedding o f

the s t a n d a r d way

~-n

planes

that

R L is

we may

P

PcG

n , j - of n a summand

P).

1.29 1.19.

Proposition.

For

y a 4 k - c h a i n of

E y.g-IV = E QCG Q QEG n,~-n n,j-n

We o m i t our

the

proof,

considerations,

lated

manifold

sition

indicating

is

routine.

We now r e p l a c e

space.

a more-or-less

w i t h one r - h a n d l e

Recall

that

standard

N(M),

given

h

t h e s i m p l i c e s and broken l i n e s

o.

in

a triangu-

handle-body

f o r each r - s i m p l e x o the s i t u a t i o n f o r a 2 - m a n i f o l d ,

below i l l u s t r a t e s lines

Q

which

is

x st(o)

y.g-IV

by a s m a l l e r

there

L(o)

decompo-

The diagram

with

the s o l i d

the c o r r e s p o n d i n g

handles

Let

~(M)

be g i v e n by

L ( o ) C_ L(T)

which t e l l s

forming this

union.

abstract M ~--R

U hq x L ( s t

Note t h a t

d e t e r m i n e d p u r e l y by l o c a l space ~ ( n )

"pullback"

the " m o c k - b u n d l e "

whenever

M

is

over

M

~(M)

e(n)

~(M).

or,

Then,

equivalently, since

as an

in a n a l o g y to

over

We f u r t h e r

~(M)

locally

we r e a l i z e

~(M)

assembled from p i e c e s

so t h a t

B(n)

NOW ;bp~ose we NaJ a ( r a t i o n a l C.(~(n))

is

In f a c t ,

ordered.

c a n o n i c a l homOtODy e q u i v a l e n c e

~ < z,

M, n o t on the s t a n d a r d embedding

data.

may form the

if

are t o be made in

the c o n s t r u c t i o n o f

nor on the o r d e r i n g o f

M ÷ e(n)

Note t h a t

us what i d e n t i f i c a t i o n s

space depends o n l y on

of

o).

~(M)

39

B(M); or r e a l )

B(M) e(N)

is via

the gauss map there is

~(M)

transfer

t:

determined t r a n s f e r s as a subspace of

we

the

note t h a t thus

B(n)

a

N(M). C.(e

) n t:C.(M)

N(M)

via a

+

1.30 s t a n d a r d embedding, we o b t a i n , c h a i n r e p r e s e n t i n g the follows of

k th

from 1.19 t h a t

a cell

dual

to

of

if

st p

s u b d i v i s i o n of

an ( n - k ) - s i m p l e x ,

(2)

this ~

To e x p l i c a t e ,

co-

However, i t

c o - c h a i n on an o r i e n t a t i o n depends on n o t h i n g b u t the

o.

is

an ( o r i e n t e d )

the o r i g i n a l

Pk (p)

simplex of

triangulation,

the f i r s t p ~ o ,

o

=

Z

v ) Q

E { t { p F) h )

to compute each summand on t h e r i g h t h a n d s i d e ,

c o r r e s p o n d i n g to a simplex

T

with

o < 3,

p l a c e d by the o b v i o u s c h a i n on a f i n e are a l l

a real

then we may compute

{3)

p ~ h

a formula for

Pontrjagin class.

simplex

n-k

combinatorial structure

barycentric

real

the v a l u e of

some

More e x p l i c i t l y ,

as in

subcomplexes.

we t h i n k

of

s u b d i v i s i o n of

M

We moreover i d e n t i f y

L

T

p

as r e -

where

p,

w i t h a subT

space o f

G ( j ( T ) = # v e r t i c e s of st 3), and t h e r e b y j(3)-h,k average over Q~ G with V t h o u g h t o f as a s u b v a r i e t y o f n,j(T)-n Q Gj(t)_n,n. Prop. 1.19 i s used to show t h a t the r e s u l t i s the same as if

the a v e r a g e in each i n s t a n c e were to be t a k e n o v e r a l l

But clearly, ture of

~k ~.

pk(p)

depends

Hence,

only on

taking

p (M). k Of c o u r s e , the f o r m u l a

pk(o*)

Q~ G

p

and the combinatorial

=

Z

pk(p)

n,~,n" struc-

we get a local

formula for

that

even a r a t i o n a l

co-efficients

(3)

transfer

is

still

yields

not q u i t e

only a local

satisfactory formula with

s i n c e we are a v e r a g i n g o v e r a n o n - f i n i t e

However, we a s s e r t

(without proof),

the f o l l o w i n g

in real

measure space.

slightly

stronger

fact.

j:20

Pr0ppsitip9.

In f o r m u l a (3)

E(t(p ~ h ) • V ) 3 t h o s e spanned byQ ing co-chain s t i l l

above we i n t e r p r e t

t h e term

as an a v e r a q e o v e r t h e b a s i c n - p l a n e s n

standard basis

v e c t o r s ) in

Rj(3 )

r e p r e s e n t s the P o n t r j a g i n c l a s s

40

Pk"

Q

(i.e.,

The r e s u l t -

1.31 Here, a c l e a r consequence is

t h a t a r a t i o n a l t r a n s f e r leads to a

l o c a l formula with r a t i o n a l c o - e f f i c i e n t s . A final

refinement in our a n a l y s i s is a computation of

t ( p ~ hT) • VQ v

where

... i(1) t h i n k of

v

manifold

p ~ hT

Q

is

the n-plane spanned,

of the standard i(n) t a c t i n g on Co ~ hT )

difficulty

to

Rj ( ~ )

were merely a map of the

t ( p ~ h T)

co-efficients).

L(T) x St T + C(St T),

it

( t h e r e i s no e s s e n t i a l new

i n t r o d u c e d when t r e a t i n g

with r a t i o n a l or r e a l

For the sake of s i m p l i c i t y ,

as i f

L(T ) × st T,

say, by v e r t i c e s

as a s i n g u l a r chain

Then there i s a p r o j e c t i o n

the space of c o n f i g u r a t i o n s of

st T,

fact,

of course, t h i s

p r o j e c t i o n has c o n t r a c t i b l e f i b e r ) .

tion

t(x) E V Q

x E p ~ hT

ing way:

~t(x)

s t ( z ) ~ Rn;

for

The a s s e r -

may be c h a r a c t e r i z e d in the f o l l o w -

i s a c o n f i g u r a t i o n of

the vectors

(in

St(z),

i.e.

an embedding

v

... v spanning Q are v e r t i c e s i(1) i(n) of st(z) as w e l l . Thus {,t(x)v } i s a set of n veci(j) j=l,2..n t o r s in n-space: We claim t ( x ) E V p r e c i s e l y when the vectors

Q

{~t(x)v

i(j

)}

h a v e rank

With t h i s

n-2k.

r e - c o m p u t a t i o n of

( t p ~ hT)- VQ

o b t a i n e d G a b r i e l o v ' s formula [Gab, Prop. 5 . 1 ] ,

we have e s s e n t i a l l y or a t l e a s t the

( u n s t a t e d ) C o r o l l a r y of t h a t formula which r e s u l t s from averaging over choices of " h y p e r s i m p l i c i a l f i l a m e n t . " It

would be very g r a t i f y i n g to cap the present a n a l y s i s by con-

s t r u c t i n g the l o c a l l y - d e f i n e d t r a n s f e r o p e r a t i o n has been assumed throughout the a n a l y s i s above.

t

whose e x i s t e n c e

Unfortunately this

c o n s t r u c t i o n seems r e a l l y to be the h e a r t of the m a t t e r . skirts

the d i f f i c u l t y

question a l l and i s

by assuming ad hoc t h a t the t r i a n g u l a t i o n s in

have the p r o p e r t y t h a t f o r a l l

r a t i o n a l l y 4k-codim ~

rather trivial up to dimension

connected.

c o n s t r u c t i o n of a t r a n s f e r 4k,

Gabrielov

v,

C(st ~) i s

connected

This p r e c i s e l y a l l o w s tile C,(M) ÷ C,~(N), a t l e a s t

by wishing away the o b s t r u c t i o n s to such a

t r a n s f e r m a p . However, e v e n with these s p e c i a l assumptions, there i s

41

1.32 no a p p a r e n t c a n o n i c i t y to the c o n s t r u c t i o n , and c e r t a i n l y geometric c o n t e n t .

What does seem c l e a r i s

be understood about the t o p o l o g y of restriction tion

maps C(st ~) ÷ C(st T)

o < T.

C(st o)

and the

S u c h an i n v e s t i g a -

seems long overdue, c o n s i d e r i n g the e a r l y appearance of

spaces and maps in (Recall

foundational studies

t h a t C a i r n ' s p r o o f of

reduced to showing merely t h a t Thus,

t h a t something more must

the spaces for

no c l e a r

much of

the f o r e g o i n g is

in

these

geometric t o p o l o g y .

the s m o o t h a b i l i t y of 4 - m a n i f o l d s C(K)

is

connected f o r

K=AI*~ 2.

designed as m o t i v a t i o n f o r

an i n v e s t i g a t i o n .

42

just

such

2.1 2.

Formal l i n k s and the PL Grassmannian

We begin our discussion O f ~ n , k formal l i n k of dimension

by i n t r o d u c i n g the notion of

(n,k;j).

If

n

and

merely r e f e r to the dimension of the l i n k as Let

U n+k R

space

be a

/~n,k

k

are understood, we

j.

j + k - d i m e n s i o n a l subspace of the standard Euclidean

S denotes the u n i t sphere in U (centered a t the U o r i g i n ) and D the u n i t d i s c . If zJ-Ic__, s is a t o p o l o g i c a l U U j-1 ( j - 1 ) - s p h e r e , an a d m i s s i b l e t r i a n g u l a t i o n i s a t r i a n g u l a t i o n of (as a c o m b i n a t o r i a l m a n i f o l d ) such t h a t : (a)

r-simplex n+k dimensional subspace of R (b) n+k R ,

For e a c h

If

c(o)

then (c)

~

is

is

o

of

containing

the convex h u l l

c(~)

In p a r t i c u l a r ,

this

o

Definition.

(UL,ZL) and

where

~LC_ SUL

o.

of the v e r t i c e s of

ZL : O,

k-plane

U L Let L

a v e r t e x of

is a

L

S . U of dimension

is a pair n+k ( j + k ) - d i m e n s i o n a l subvector space of R

and thus a formal l i n k n+k in R

L

0

ZL •

We s h a l l

v

We l e t

Let

are d e f i n e d .

In t h i s

merely corresponds to a

be a formal l i n k of dimension

as f o l l o w s :

U.

(n,k;j)

i s an a d m i s s i b l y - t r i a n g u l a t e d ( j - 1 ) - s p h e r e .

(n,k;j-1) in

in

S of c(a). U corresponds to the n a t u r a l con-

Note t h a t formal l i n k s of dimension case

Z

set of assumptions i m p l i e s t h a t no two

A formal l i n k U

(r+1)-

under the a f o r e s a i d r a d i a l p r o j e c t i o n .

p o i n t s of a simplex are a n t i p o d a l in 2.1

there i s a unique

the r a d i a l p r o j e c t i o n upon

The convex s t r u c t u r e of

vex s t r u c t u r e on

~j-1,

(n,k;j),

and l e t

v

be

d e f i n e a new formal l i n k p

L of dimension v denote the segment from the o r i g i n to

U be the ( j + k - 1 ) - p l a n e orthogonal to p in U. v Let U° be the a f f i n e ( j + k - 1 ) - p l a n e of U p a r a l l e l to U and v passing through the m i d p o i n t m of p. Let S' be a small ( j + k - 2 ) -

43

2.2

2.3

sphere of r a d i u s ~k(V,ZL),

let

x

in

T(o)

U

centered at

S' (~ P(o)

Let

P(~)

to p o i n t s of then

~

is

1 the s i m p l i c e s of

range over a l l

m.

If

~

i s a s i m p l e x of

denote the c o r r e s p o n d i n g s i m p l e x of one

st(v,o).

from the o r i g i n

d e f i n e d as

U ,

= ~*v

dimension g r e a t e r in in

i

be the union of a l l

T(~).

We c l a i m t h a t i f

homeomorphic to

~k(V,~L),

then

o.

If

~o I

rays o

1 we l e t

is

forms a sim-

O

plicial

complex

). Now L by the homeomorphism u + 1 / x ( u - m ) . Let Z be v under t h i s map. Then Zv is seen to be an ad-

Z'

S' ,

in

~' ~v

with

i s o m o r p h i c to

~k(v,Z

JV

map

S'

to

S U ~' the image of V

missibly-triangulated as

(U ,~ V

( j - 2 ) - s p h e r e in

S U v

).

We may thus d e f i n e

L v

V

This c o n s t r u c t i o n may be e x t e n d e d . j - d i m e n s i o n a l formal

link,

and

o

If

L = (UL,ZL)

an a r b i t r a r y

is

r - s i m p l e x of

a ZL,

let

v ...v be i t s v e r t i c e s , o r d e r e d in some f a s h i o n . Let L be o r o the (j-1)-dimensional link L C l e a r l y t h e r e are v e r t i c e s 1 1 Vo v1...vr of ZL c o r r e s p o n d i n g to v1...Vr. T h e n set LI = (Lo) 1' vo v1 2 2 thereby o b t a i n i n g v e r t i c e s v ...v of Z c o r r e s p o n d i n g to i 1 2 r LI v2...v . C o n t i n u i n g i n t h i s f a s h i o n we o b t a i n L = (L) , r i+1 i i+1 vi+1 for

i

< r,

and the process t e r m i n a t e s w i t h

link

of dimension

2.2

Lemma.

L r

L , r

which i s a formal

j-r-1. is

independent of the o r d e r i n g of

We merely sketch the p r o o f . and l e t

m

Let

X

denote the unique

let

U

be the

Let

b

the v e r t i c e s of

denote the b a r y c e n t e r of

be the m i d p o i n t of the ray from the o r i g i n

be the a f f i n e

r + 1 - p l a n e of

(j+k-r-1)-plane (j+k-r-1)-plane

of

o.

U in which L o r t h o g o n a l to

U L p a r a l l e l to

U

to ~

b

in

lies,

U L" and

X Let U' o" and passing through

O

m,

and, as b e f o r e ,

tered at p(~)

= ~*o

m.

let

S'

be a small

Given a s i m p l e x st(~,~

L

)

and l e t

T

of

P(T)

44

( j + k - r - 2 ) - s p h e r e in

ck(~,~L)

U' cen-

we set

be the union of a l l

rays to

2.3 p(~).

S' ~ P(~) = T is seen to be homeomorphic to T and ~_~ T 1 T 1 is a simpticial complex ~' isomorphic to ~ k [ ~ , ~ ). Once m o r e , t h e ~ k obvious t r a n s l a t i o n f o l l o w e d by d i l a t i o n i d e n t i f i e s S' with S , U and the i m a g e ~ of 5' i s an a d m i s s i b l y - t r i a n g u l a t e d ( j - r - 2 ) -° 0

sphere.

0

T h u s we o b t a i n a

(j-r-l)-dimensional

link

L

= (U ,~ ). (~

We claim t h a t the o r d e r i n g

L is the same as the l i n k o of the vertices of ~. Thus,

O

O

L d e f i n e d above, given r since L is obviously 0

independent of t h i s

o r d e r i n g , so is

L .

If

K = L

r

plex

e

of

~L

we s a y

K

Given a formal l i n k of

ZL.

v*

is

Let

v*

the f i r s t

Thus,

v

~L

is

L

~L

In t h i s

= '~"

as

subdivision

L

(written

j,

(j-1)-cell

in the f i r s t

K < L).

consider a vertex to

v

in

a simplicial of

complex

~.k(V,ZL).

homeomorphism

is a I - d i m e n s i o n a l l i n k ,

instance,

h(L,v): ~L

we i n t e r p r e t

V

to denote reduced cone on

X+;

~'L"

to

On t h e

thus

is a

the c~

cone

C~L

~ ÷

the

ZL.)

cone

other

v

(i.e.,

b a r y c e n t r i c s u b d i v i s i o n of

n a t u r a l l y isomorphic to the complex

we o b t a i n V a n a t u r a l

( I n case

to

of dimension

isomorphic

barycentric

the sphere

L

incident

denote the dual

the s t a r of

This is o b v i o u s l y

is

f o r some sima

on

hand,

~k(V,~L). V*

C.Z L .

v O-sphere and construction

cX

is a p o i n t and, o b v i o u s l y , V

h

L,v

identifies If

o

it

with

v* = { v } . )

is a simplex of

ZL

spanned by v e r t i c e s

o b t a i n a chain of i n c l u s i o n s (1)

CLr + ZL r-1

c~ L

r-1

Lr- 2

c~ L

÷

ZL

1

c~ L

45

v ...v o r

we

2.4 i-1 h(Li_1,Vi_l).

where each h o r i z o n t a l map is of the form c o r o l l a r y of 1.2 ,

As a

we assert t h a t the composite map

pends only on the simplex

C~L + Z deL r and not on the order of the v e r t i c e s .

o,

We leave t h i s to the reader.

Since

L

= L ,

in the n o t a t i o n of 1.2

r

we denote t h i s

homeomorphism by

h(L,a)

and note t h a t i t

takes

cL O

homeomorphically onto simplex of

Z, and L 1.2, we see t h a t L

o*, T

the dual c e l l

a face of

= (L) T

O

o,

of

a

in

Z

(2)

cZ

L

o

is a

where T

i

is

the simplex of

~L _

T I

T of o such t h a t 1 we get the diagram

o,

If

then by a simple extension of

corresponding to the face another face of

~.

T

C

C

Z

L

T*T

T

i

: O.

If

T

p

is

~

o

ZL i

ZL C p

cZ L p

and we claim t h a t t h i s diagram s t r i c t l y We may now form a j-cell

for each j-dimensional l i n k .

take the union of a l l h(L,o)

CW complex.

in

F i r s t we take one t o p o l o g i c a l Think of t h i s c e l l as

such, i d e n t i f y i n g

~L ~ C~L"

commutes.

C~L

with i t s

We denote t h i s comple~ by / ~ n , k "

CZL.

We

image under The n o t a t i o n

is meant to suggest an analogy with the c l a s s i c a l Grassmannian the space of l i n e a r e

L

n-planes in

to denote the c e l l

(n+k)-space.

G n,k' We use the n o t a t i o n

of y ~

which is the image of c~ L. n,k We now attempt to j u s t i f y t h i s n o t a t i o n a l analogy. Consider a

t r i a n g u l a t e d c o m b i n a t o r i a l manifold embedded, or merely immersed, in n+k n R , so t h a t every simplex ~ of M is l i n e a r l y embedded ( i . e . the image of

o

is the convex h u l l of the images of i t s

46

vertices).

2.5 In p a r t i c u l a r ,

under such an i m m e r s i o n , the s t a r of e v e r y s i m p l e x i s n+k n embedded i n R Let M denote L.) o* where t h e union i s taken o o n over those s i m p l i c e s o not c o n t a i n e d in aM , and , as u s u a l , n n n denotes the dual c e l l o f ~. If M has no b o u n d a r y , then M = M ; o n n n if @M ~ ~ , then M is a codimension-0 submanifold of int M o n n and d i f f e r s from M m e r e l y by a c o l l a r n e i g h b o r h o o d o f aM j n Given a s i m p l e x ~ of M , o aM we a s s i g n t o i t a c e r t a i n formal

yj

link,

L(~,M n)

be t h e a f f i n e

o the

n+k R

in

(n÷k-j)-dimensional affine

( n + k - j - 1 ) - s p h e r e in n ck(o,M ) let p(T) i

!

:

S

usual S',

containing

~,

b

U'

of

o.

Let

centered at

S

b . U

d e n o t e the s i m p l e x

Let

p(T),

of

~

Z

Z'

=

~

i

i

to

{T"

U

be be

and p a s s i n g

O

i

d e n o t e a small

Given a s i m p l e x T of n T*o C s t ( o , M ), and l e t

Map

S

i

onto

x ÷ i

(x-b

be the homeomorphic image o f

whose s i m p l i c e s are

Let

and l e t U yj ~, . Let

o r t h o g o n a l to

plane p a r a l l e l

O

is

as f o l l o w s :

L) T ~ S . T translation-followed-by-dilation ~

and l e t

tion

(n,k;n-j),

t h r o u g h the o r i g i n

t h r o u g h the b a r y c e n t e r

%

dimension

j-plane

(n+k-j)-plane

an

of

),

~'. i

= image T }

S by the UO ~ = radius

The t r i a n g u l a is

a d m i s s i b l e and

n

o b v i o u s l y isomorphic to

(n-j)-link

(U ,~

)

~k(o,M ).

which i s ,

We thus o b t a i n a f o r m a l

by d e f i n i t i o n ,

L(o,Mn).

This

a s s i g n m e n t g i v e s r i s e t o a n a t u r a l map, which we c a l l the n Gauss map, g: M + , g d e p e n d i n g , o f c o u r s e , on the t r i a n g u l a n o n~k t i o n of M On the c e l l l e v e l i t may s i m p l y be d e s c r i b e d as sending

~

the c e l l ency o f

o

.

this

of

plexes follows n

M

with

T

n

N to the c e l l e n of . The c o n s i s t o L(~,M ) ,k a s s i g n m e n t w i t h f a c e r e l a t i o n s in the r e s p e c t i v e comfrom the o b s e r v a t i o n t h a t

a face of

o

(i.e.

o

.

if

T,o

a face of

are s i m p l i c e s o f T*)

then

L(o,Mn) < L (T,M n )" For a more s p e c i f i c of

o*

There i s

description

as the cone on the f i r s t a natural

simplicial

of

the map

barycentric

isomorphism

47

g

pointwise,

think n ~k(o,M ).

s u b d i v i s i o n of n ~k(o,M ) + Z L ( ~ , M n ) ,

and

2.6 thus a c a n o n i c a l homeomorphism g ~

which e x t e n d s t o

C~L(~,Mn )" If

we compose t h i s

we d e s c r i b e dual

@o* ÷ +~L(o Mn) '

cell

gI~*.

complex

with

the map

CZL(o,Mn) +

eL(o,Mn)~___~n,k,..

A g a i n , we a s s e r t t h a t t h e f a c e i n c l u s i o n s on t h e n M are c o n s i s t e n t , under g w i t h the f a c e o

relations

on the c e l l

n,k The easy v e r i f i c a t i o n

point-by-point. observing that

complex

not only c e l l - b y - c e l l , of

this

fact

is

but also

a m a t t e r of

the diagram g L(a,M n )

(q g T

T

is

strictly

~

c o m m u t a t i v e , where

C~L(T,Mn)

h

is

the map

h(L(T,M),p)

and

p

n

is

the s i m p l e x o f =

Pl

*T.

This

Z c o r r e s p o n d i n g to p L(T,M) i o b s e r v a t i o n i s a m a t t e r of d i r e c t

The s i m p l i c i t y

and n a t u r a l i t y

of

in

~k(T,M

)

with

inspection.

the Gauss map are s e l f - e v i d e n t .

However, t o d e s e r v e d e s i g n a t i o n as a Gauss map, as the r e a d e r w i l l

no

d o u b t o b s e r v e , t h e r e should be an e q u a l l y n a t u r a l c o v e r i n g by a PL n b u n d l e map, j u s t as the Gauss map g: M ÷ G o f a smooth immern,k s i o n i s n a t u r a l l y c o v e r e d by a map from t h e t a n g e n t v e c t o r bundle n TM to the c a n o n i c a l n - p l a n e bundle over G . We a r e thus n,k obliged, first of a l l , t o show t h a t t h e r e e x i s t s a c a n o n i c a l PL n-bundle y over ~ , and then t o show t h a t t h e Gauss map n,k n,k n d n g: M + ~ i s n a t u r a l l y c o v e r e d by a PL n - b u n d l e of M . Hence"-n ,k f o r t h , we s h a l l use t h e term Gauss map so as i m p l i c i t l y t o subsume this

c o v e r i n g b u n d l e map. We need some f u r t h e r

X L

definitions.

d e n o t e the o r t h o g o n a l complement o f

48

Given a f o r m a l n+k U in R L

link Let

L,

let

Q C_U L L

2.7 denote the union of a l l points of

EL.

infinite

In p a r t i c u l a r ,

rays in

U L

QL = ~ Q~'

from the o r i g i n through

where

o

ranges

over

0

the simplices of

~ and L (If L is

Q

points of

O-dimensional,

~.

the o r i g i n . )

If

L

is the union of a l l QL

rays through

is understood to mean

is a

j - d i m e n s i o n a l l i n k , i t is c l e a r t h a t Q n+k L is a p i e c e w i s e - l i n e a r j - p l a n e in U R X i s , of course, L L ( n - j ) - d i m e n s i o n a l . Thus QL × XL" the vector sum of the sets QL n+k n+k and X in R , is a piecewise l i n e a r n plane in R We L denote t h i s space by VL. We now construct a c e r t a i n " t a u t o l o g i c a l " map "~n ,k

n+k +

R A

begin by f i r s t

supplementing the n a t u r a l c e l l

an a d d i t i o n a l decomposition.

We .

s t r u c t u r e on -~-, k

We are going to represent ~ n , k

by as the

union of c o n t r a c t i b l e subspaces

e , one f o r each formal l i n k L. L HowIn general, ~L w i l l n e i t h e r contain nor be contained in e . L ever, i f C is any c e l l u l a r subcomplex of , say C = ( J e , n,k i~ Li f o r some indexing set,J( , then ~ = L) ~ w i l l contain C as a i e ~ Li deformation r e t r a c t . To define

EL,

is a simplex of

we f i r s t ZL"

~L,o'

where

We may think of these as subspaces of

C~L,

w i l l r e s t r i c t to a homeomorphism on L T a k e the second b a r y c e n t r i c s u b d i v i s i o n of C~L

( n o t i n g t h a t the f i r s t first

and

CZL ÷ e

since the n a t u r a l map these spaces•

define spaces EL, *

s u b d i v i s i o n of

s u b d i v i s i o n is the s i m p l i c i a l cone on the ZL,

a s i m p l i c i a l complex).

which we w r i t e as

i

cZ ,

to specify i t

Call t h i s second s u b d i v i s i o n , ( i . e .

as

the f i r s t

s u b d i v i s i o n of

c~') C . Let ~ denote the s i m p l i c i a l r e g u l a r L L L,* neighborhood of the cone p o i n t in t h i s complex. Let ~ be the L,o r e g u l a r neighborhood of the barycenter b of the simplex o in ~L" We i d e n t i f y these spaces with t h e i r homeomorphic images in .

Now we l e t

~

: L,J

L

LVL" (This c o n s t r u c t i o n i s , e s s e n t i a l l y , the c o n s t r u c t n ion of TM I~ ÷ TV made e a r l i e r . ) These l o c a l l y - d e f i n e d maps L(o,M) f i t together c o n s i s t e n t l y to give a global map Y2 ÷ Y1 covering the identity.

Thus

Y2 = Y1 = Yn,k"

Having defined the Gauss map and the natural covering bundle map, we may make some f u r t h e r elementary o b s e r v a t i o n s , p a r t l y to motivate some of the subsequent chapters. In the f i r s t in

n

and

k

There i s ,

place, i t

is natural to look f o r a double sequence

modeled on the f a m i l i a r one f o r standard Grassmannians

÷

G

+

G --~ n,k+1

in f a c t ,

n,k

~ "

G

1,k

+

G n+l,k+l

+

n+

a natural double sequence

54

2.13

,k

+l,k

(3) ,k+l

where

~

and

B

are, in f a c t ,

To define

it

l i n k s of dimension

1,k+l

i n c l u s i o n s of subcomplexes.

s u f f i c e s to define a set map CZ from formal

(n,k;j)

to formal l i n k s of dimension

which is c o n s i s t e n t with face r e l a t i o n s .

(n+l,k,j),

N o t e t h a t a formal l i n k

L

of dimension ( n , k ; j ) is given by data (UL,~L)~ U a (j+k)-plane n+k L in R and ZL an admissible t r i a n g u l a t e d ( j - 1 ) - s p h e r e in SU n+k n+k+l L But under the standard i n c l u s i o n R ~ R , U may be considered n+k+1 L as a ( j + k ) - p l a n e in R , and thus the data (UL,~L) may be viewed as determining a formal l i n k of dimension we denote ( ~ ( L ) .

C l e a r l y C~_: LF ~ ( L )

(n+l,k;j),

which

induces an i n c l u s i o n

+z~.~n k; t h i s is the map a of diagram (3). Of course, ,k +I, must construct a bundle map Y n , k (~E) ~ ÷ Yn+l,k to cover m, i f

~ ~-n

pattern f o r the standard Grassmannians is

V C O-(L)

R n+k+l

Rn+k+l

is j u s t

VLx R,

the

to be f o l l o w e d f u r t h e r .

That t h i s bundle map n a t u r a l l y e x i s t s may be seen as f o l l o w s : that

we

(i.e.,

VL ~)Rn+ k+I ,

Note where

R which is the l a s t summand of n+k+l R1 + R2...+ Rn+k+1 = R Thus TVa ( L ) = TVL (~)~ in a n a t u r a l way.

is a copy of

Since

Yn,k

~(~L ) C e-6~(L), covering

is defined, l o c a l l y on

EL,

as

G*TVL,

and since

t h i s i d e n t i f i c a t i o n induces the desired bundle map

~.

As f o r the d e f i n i t i o n of

B,

we once more r e s o r t to a set map

on the set of formal l i n k s . Given the n+k n+k+l think of R as included in R way as

(n,k;j)

link

L = (U ,Z ), L L in a s l i g h t l y non-standard

R + R ...+ R , a sum of copies of R. Let ~ ( L ) be 2 3 n+k+l given by the data (UL(~)RI,~L);_ ~ ( L ) is a formal l i n k of dimension

55

2.14 (n,k+1;j).

Once m o r e ,

~n

~

induces

an i n c l u s i o n

B~ ~2/ Here, however, since it ,k "~n,k+l n+k+1 as t h e l a t t e r is included in R via n+k n+k+l R C R , and t h a t B(~L) ~ ~ (L)'

is

D

Yn,k+ll~n,k vious

: ¥n,k'

thus

B

is

covered

the it

of

CW

clear

complexes

that

above

V ~(L) inclusion

follows

by a b u n d l e

:

V

L

that map i n

the

ob-

way. Finally,

d i a g r a m (3)

note t h a t

~°B = ~ ° ~ : # /-n,k commutes as does

+

y

+

Yn, k+'l

Y

)

n,k

+ J7t , /~n+1,k+l

so t h a t

÷ n+l,k

(4) (I

(Here we use

~'s

and

corresponding

~'s

A point

which

Gauss map do n o t

B's

and the

+

n+l,k+l

to d e n o t e the bundle maps c o v e r i n g the

B's

in

reader

eral,

continuous.

sents

the

by even very

the

far This

and

e

K dimension

think

formal

of

at

(3).)

may have

all

well

noticed

is

that

~n,k

and t h e

deformation. T h u s , i f we n n+k have a p i e c e w i s e - l i n e a r immersion m : M ÷ R , and d e f o r m i t o through a continuous family m of piecewise-linear immersions (all t convex-linear on s i m p l i c e s with respect to a fixed triangulation), we obtain a family

respond

Y

)

Gauss maps

That link

smallest

n

M ÷ which i s n o t , i n o n,k === perturbing the "solid angle" which

is,

of

to

gt:

a simplex

amount

in

instantly

the

plane

shifts

normal

the

to

the

Gauss map on

gen-

represimplex o

away. suggests

a possible

become c l o s e which

of ~f -n

,k

are (i.e.,

when

close its

retopologization L

in

and

K

are

an i n t u i t i v e

first

barycentric

56

of formal

sense.

~

n,k links of That

subdivision)

is,

so t h a t the

same

we may as

the

e

L

2.15 geometric realization tion

of a s i m p l i c i a l

of/~f

comes from p u t t i n g n,k the s e t o f j-simplices, for all

continuous. realization

set.

a natural j,

The n a t u r a l

n o n - d i s c r e t e t o p o l o g y on

such t h a t

Thus we may o b t a i n a s i m p l i c i a l

retopoligiza-

face o p e r a t i o n s a r e

space whose g e o m e t r i c

puts a s m a l l e r t o p o l o g y on the u n d e r l y i n g p o i n t s e t o f

21 than t h e o r i g i n a l CW n,k e x h a u s t i v e l y in §7 b e l o w . We may n o t e ,

complex.

construction

n o n e t h e l e s s , t h a t even though the ~

r e g u l a r homotopy o f least

PL

and Gauss

n,k d e f i n e d do n o t behave w e l l

map c o n s t r u c t i o n s as o r i g i n a l l y

cordance, at

We examine t h i s

i m m e r s i o n s , t h e y do behave w e l l

under

under con-

if

we may e x t e n d the c o n s t r u c t i o n s a b i t . Given n n+k two PL immersions o f M into R , say f and f , we s h a l l o 1 c a l l t h e two c o n c o r d a n t i f and o n l y i f t h e r e i s a PL immersion n n+k -1 n+k n F: M ×I + R xl, with F (R x{i}) = M ×{i}, i = 0,1, so t h a t n F]M x { i } = f . Note t h a t we do n o t assume t h a t the t r i a n g u l a t i o n s on i e i t h e r end c o i n c i d e . We s h o u l d l i k e t o be a b l e t o c o n c l u d e

2.3 Lemma. I f f and f are c o n c o r d a n t immersions o f n+k o 1 n f, R w i t h r e s p e c t i v e Gauss maps g ,g : M +J~n, then o 1 o k homotopic to (N.B.

~°gl

n M ~og

into o

is

in~ c~ o w -L(o,W) may e a s i l y be seen, d e t a i l s being l e f t to the

Moreover, i t

~

r e a d e r , t h a t the maps g may be chosen to be c o n s i s t e n t w i t h i n c i n ° dence r e l a t i o n s in M and +l,k" I.e., if r < o, then L(o,W) < L(T,W)

and

hg

: g O

sion

on

o ,

T

where

h

is

the face i n c l u -

W

cZC(~,w ) + ZC(T,w ). As to lemma 2 . 3 ,

its

p r o o f may e a s i l y be d e r i v e d from t h i s

con-

struction ance f

(and s l i g h t m o d i f i c a t i o n s t h e r e o f ) . I . e . , given a c o n c o r d n n n n+k n+k n+k F: M x l ; M ×O,M ×1 + R ×I;R ×0,R × I of immersions

n n+k ,f : M + R , one e a s i l y o I G: M xI;M x0,M x l ÷ ; n+l,k

n n n ~

to the Gauss maps

produces a Gauss map

~ Z, '

g 'gl o

for

,k f

o

,k

,f

i

59

.

which r e s t r i c t s

on e i t h e r

end

3.1 3.

Some v a r i a t i o n s of the

A,k

construction

Before we go on to prove some r e s u l t s c l a r i f y i n g s i g n i f i c a n c e of the s p a c e ~ n , k map of an immersion, we s h a l l

and what we have c a l l e d the Gauss

p o i n t out t h a t analogous c o n s t r u c t i o n s

are p o s s i b l e in a v a r i e t y of geometric c o n t e x t s . to those we s h a l l e s t a b l i s h f o r spaces as w e l l , but we s h a l l

the geometric

~n,k

Results analogous

in §4 are a v a i l a b l e f o r these

content ourselves with a mere o u t l i n e of

the c o n s t r u c t i o n s . First,

we take note of a "Grassmannian" t h a t a r i s e s n a t u r a l l y

when one considers normal block bundles of t r i a n g u l a t e d submanifolds of t r i a n g u l a t e d Euclidean space ( o r ,

PL

slightly

more

g e n e r a l l y , normal b l o c k - b u n d l e s of immersions which are s i m p l i c i a l maps w i t h respect to a t r i a n g u l a t i o n of Euclidean space). Historically,

t h i s was the f i r s t

to deal with the theory of

PL

"Grassmannian" devised by the author immersions.

In f a c t ,

it

was t h i s

space which was used to prove a p r e l i m i n a r y version of Theorem 1.6 on the e x i s t e n c e of l o c a l c o m b i n a t o r i a l formulae f o r c h a r a c t e r i s t i c classes.

(C.

Rourke's sharpening of t h i s

c o n s t r u c t i o n f o r the

purposes of d e a l i n g with c h a r a c t e r i s t i c class questions led to the present version of the r e s u l t as e x p o s i t e d in Chapter 1 and [ L e - R . ] . ) Nonetheless, the block bundle version of the worth studying in i t s appeared in ~n

[Le].

which w i l l

,k

own r i g h t .

Grassmannian i s

Some r e s u l t s concerning i t

have

These are p r e c i s e l y analogous to the r e s u l t s f o r be developed in the subsequent c h a p t e r .

The d e f i n i t i o n of the Grassmannian A

(N.B.:

PL

~

•

This space was denoted ~

in n,k

n

The general idea i s to produce a map

M

for PL blockbundles n,k [Le ] ) proceeds as f o l l o w s : ~ 1

÷ ~n,k

n

whenever

M

is

n+k embedded ( o r , more g e n e r a l l y , immersed) in R as a subcomplex of n+k a t r i a n g u l a t i o n of R whose s i m p l i c e s are convex l i n e a r l y

60

3.2 embedded w i t h r e s p e c t to the s t a n d a r d l i n e a r This map i s ,

of c o u r s e ,

to have a n a t u r a l

map from the normal b l o c k bundle of b l o c k bundle

y

n,k A formal b - l i n k

(UL, TL' t i o n of

ZL)

the u n i t

n+k R

c o v e r i n g by a b l o c k - b u n d l e

the embedding to a c a n o n i c a l

21

over

where

s t r u c t u r e on

. n,k of dimension

L UL

(n,k;

j)

is a triple n+k R , TL a t r i a n g u l a -

i s a ( j + k ) - p l a n e of

S C-.U and ~ i s a subcomplex UI L L of T which i s a t r i a n g u l a t e d ( j - l ~ - s p h e r e . T i s , of course, to L 2 be a d m i s s i b l e in the sense of §2. (Thus, i t is c l e a r t h a t t h e r e i s a "forgetful

(j+k-1)-sphere

(U L, T L, rL ) ~

map"

(U L, ZL)

from formal b - l i n k s

to

formal l i n k s . With

n,k

assumed to be f i x e d ,

formal b - l i n k of dimension

L j

we denote the dimension of a

by a s i n g l e parameter and a v e r t e x

v

of

b-link

j.

Given such a b - l i n k

L

s , we o b t a i n the d e r i v e d L The c o n s t r u c t i o n i s c l e a r l y

L having dimension j - 1 . v analogous to t h a t of §2 f o r formal l i n k s .

That i s ,

we l e t

U L V from 0

the ( j + k - 1 ) - p l a n e of

be

U o r t h o g o n a l to the segment p L Let U' be the a f f i n e plane g o t by t r a n s l a t i n g U so t h a t i t L V passes through the m i d p o i n t of p, and l e t S' be some small (j+k-2)

pahere in

by l e t t i n g

U'

a typical

a simplex of

~k(v,T

L the g e o m e t r i c cone on

centered at t h i s m i d p o i n t . s i m p l e x be of the form );

T(~)

This process p r o v i d e s simplicial E'C.. S' of

S'

S'

complex, to

by l e t t i n g such t h a t

T = o*v

~'

w i t h the o r i g i n

We t r i a n g u l a t e

S' ~ P(o)

is a simplex of of

with a t r i a n g u l a t i o n

to

where:

o

v.

S' is

st(v,T

); P(~) is L as cone p o i n t .

U L i s o m o r p h i c , as a

~k(v,T

). F u r t h e r m o r e , we o b t a i n a subcomplex L be the union of those s i m p l i c e s P(o) ~ S'

). ~' i s thus i s o m o r p h i c to ~k(v,~ ) L L under the obvious isomorphism between S' and ~ k ( V , T L ) . We may then send

S'

o C ~k(v,~

h o m e o m o r p h i c a l l y onto the u n i t

by the obvious t r a n s l a t i o n - f o l l o w e d - b y - d i l a t i o n

61

sphere

S of U UL Lv v p r o c e d u r e , and we

3.3 o b t a i n t h e r e b y an a d m i s s i b l e t r i a n g u l a t i o n

has, triple

in

as a subcomplex, (U

,

T

,

z

k k V V Extending t h i s

fact

k

.

As in h(L,~):

over w i t h

0

Here,

s u b d i v i s i o n of

V

thus d e f i n e d by the

p r e c i s e a n a l o g y to

for

in

* TL,

is

L

analogs in

§2.

which

is

a

from the

~ L

a map,

a homeomorphism onto i t s

the cone on the f i r s t

derived

way to the dual

cell

This

isomorphism,

~ L

to the c e l l a

s t r u c t u r e of

~ . With r e g a r d t o these L the same c o n s i s t e n c y p r o p e r t i e s hold as f o r t h e i r

C-W

for having one j - c e l l n,k That i s , we have, c o r r e s p o n d i n g t o

complex ~

b-link.

which may be t h o u g h t of as

image under

h(L,o).

obtain a block-bundle assemble the t o t a l

in

on

cell

each j - d i m e n s i o n a l formal

its

EL,

o

i s o m o r p h i c in a n a t u r a l

We thus may form a

the j - c e l l

L

an c - d i m e n s i o n a l s i m p l e x o f

S dual to T . U L L the subcone on the s u b d i v i s o n of

the dual

h(L,o),

§2 above, we may

the comparable c o n s t r u c t i o n in

we need o n l y note t h a t T

is

any s i m p l e x

the c e l l - s t r u c t u r e

~

~

stressing that

0

in

to

whenever

0

face maps

with

is

which

no d i f f i c u l t i e s .

c EL

moreover, c a r r i e s

L,

V

V

§2 we o b t a i n ,

c TL ,

image.

dual

L v

S , UL

).

L

We omit d e t a i l s ,

~

z'.

of

obtain a derived (j-c-1)-dimensional b-link

§2 c a r r i e s

to

L

= image

p r o c e d u r e , in

j-dimensional b-link L

z

T LV

¥n,k

we i d e n t i f y

c

L

C o n c u r r e n t l y , we a l m o s t a u t o m a t i c a l l y of

fibre-dimension

k

over

n,k"

We

by t a k i n g the union of " b l o c k s " n,k o f the form D = u n i t disc of U where L i s as usual a formal UL L b-link. This union i s taken modulo o b v i o u s i d e n t i f i c a t i o n s . That,

as we have n o t e d ,

space of

c ~ ; L

and i s a homeomorphism onto L i t s image. Of course c T to be i d e n t i f i e d w i t h D in an UL o b v i o u s way so t h a t the t o t a l space of y i s to be d e f i n e d as n,k c T with c T i d e n t i f i e d w i t h i t s image in T C.. c T under L L L L L h(L,o).

takes

By abuse of

c T

y

L is

to

T

n o t a t i o n , we l a b e l

62

this

total

space

Yn,k,

3.4 noting that

~/~ i s n a t u r a l l y i n c l u d e d as the " O - s e c t i o n " o f a n,k k-dimensional block-bundle. I f we l e t e denote the c e l l of L ~nn which i s the image o f c ~ , d w i l l denote the r e s t r i c t i o n ,k L

Yn,k

of

It

to is

n M

when

eL,

i.e.,

then q u i t e

dL

is

the

image of

straightforward

c T L.

to c o n s t r u c t the Gauss map

an n - m a n i f o l d which i s a subcomplex o f a f i x e d l i n e a r n+k n t r i a n g u l a t i o n of R As in §2, we l e t M = Uo*, where ~ i s a 0 n simplex of M n o t c o n t a i n e d in ~ M. For any such a of dimension r

is

we a r e g o i n g t o d e f i n e a b - l i n k

all,

o

n+k-r

lies

i n an a f f i n e

n+k-r

b a r y c e n t e r of U'

plane p a r a l l e l Let

o

centered at

letting

orthogonal to

of

U d e n o t e the L U' Y. Next, l e t

be

to

We o b t a i n a t r i a n g u l a t i o n

~'

By v i r t u e U'

to

triangulation

of

S°

by

of

of U L of

b-link

a Gauss map

fact,

also note,

g:

as i n

x,

the usual

translation

anmd

to

L(o),

S' --

is,

of

S' /~ (~*T)

course, a

PL

call

S , UL it T

and d i l a t i o n

trick,

we may

o b t a i n i n g t h e r e b y an a d m i s s i b l e --

and w i t h

it,

corresponding

L EL C _ T L .

The d a t a

as i n t e n d e d .

M ÷ 0 §2,

those s i m p l i c e s

n-r-1.

S UL a subcomplex

s',

consisting

T C ~ k ( o , Mn ) C ~ k ( o , R n + k ).

sphere o f d i m e n s i o n

(In

we l e t

First

U and p a s s i n g t h r o u g h the L be a s u i t a b l y small ( n + k - r - 1 ) - s p h e r e i n

the b a r y c e n t e r .

sub-complex

such t h a t

formal

Y;

n-r.

s i m p l e x be o f the form S' ~ ( ~ * T ) , where T i s a n+k simplex of ~k(o, R ). This t r i a n g u l a t i o n o f S' contains

a certain

send

S'

of dimension

a typical

typical

to

r-plane

plane through the o r i g i n

an a f f i n e

L

,k glo*

(UL'

J u s t as in

TL' §2,

~L)

we may now d e f i n e

which has the p r o p e r t y t h a t

factors

as

however, t h a t

o* + c ~L(~)

define a

g:

~* ÷ e L ' ~ '1' {

+ eL(o)')

We may

o* + c E ÷ e e x t e n d s n a t u r a l l y and L(o) L(o) without difficulty t o a map ~ ÷ D = c T ÷ d where o-UL(o) L(o) L(~) now d e n o t e s the n+k-r c e l l dual to ~ in the t r i a n g u l a t i o n of n+k . Note t h a t , a c c o r d i n g t o R Thus we o b t a i n a map g: ~ ~ ÷ y o n,k

03

3.5 [R-S],

ILj~ . i s a r e g u l a r neighborhood of Mn and represents the 0 o n n normal block bundle of the embedded M ( r e s t r i c t e d to M ). That 0 i s , each ~ is the k - b l o c k of the normal block bundle over the c e l l d*.

It

is not a t a l l

difficult

b l o c k - b u n d l e map covering

to see t h a t

~

is,

in f a c t ,

a

g.

To summarize, then, by using formal b - l i n k s ,

instead of formal

links,

and x ~ , instead of ~ , as our candidate f o r the ~ n ,k " ~ n ,k "Grassmannian" we o b t a i n a space and a block bundle which i s the

n a t u r a l c l a s s i f y i n g space f o r normal data of submanifolds of a t r i a n g u l a t e d Euclidean space.

It

should be added t h a t the Gauss map

c o n s t r u c t i o n works e q u a l l y well f o r an immersion of a t r i a n g u l a t e d m a n i f o l d i n t o a t r i a n g u l a t e d Euclidean space provided t h a t the immersion i s s i m p l i c i a l with respect to these t r i a n g u l a t i o n s . Other v a r i a n t s of the basic idea of §2 also bear b r i e f mention. One such a r i s e s in connection w i t h t r i a n g u l a t e d homology manifolds ( w i t h r e s p e c t to a p p r o p r i a t e c o - e f f i c i e n t s ) . group.

Let

A

be any a b e l i a n

Consider s i m p l i c i a l complexes which are homology n-manifolds

w i t h r e s p e c t to homology with c o - e f f i c i e n t s in

A.

The a p p r o p r i a t e

analogy to denote

~L~ in t h i s instance is a complex which we s h a l l n,k A ~ ,nk " The s p a c e A~/I-Vn,k is the n a t u r a l t a r g e t of a Gauss

map from an A-homology n - m a n i f o l d embedded or immersed in Euclidean n+k space. In t h i s c o n t e x t "immersed" means t h a t the map i n t o R is l i n e a r on s i m p l i c e s and an embedding on the s t a r of any simplex. To c o n s t r u c t

we f i r s t s t i p u l a t e t h a t a formal l i n k of n,k dimension i i s defined as a p a i r (U L, SL ) where UL i s , as n+k b e f o r e , an ( i + k ) - p l a n e of R and S i s , in t h i s i n s t a n c e , an L a d m i s s i b l y t r i a n g u l a t e d subspace of S which i s an A-homology UL m a n i f o l d w i t h the A-homology type of an ( i - 1 ) - s p h e r e . As b e f o r e , given a l i n k

L

derived l i n k

L o

A~

and a simplex and the map

o

of

s

L

h(L,~): c L

64

we may c o n s t r u c t the o

÷ E L

which i d e n t i f i e s

3.6 c L

with the A-homology c e l l

dual to

~

in

EL.

Taking the union

0

of A-homology c e l l s

c s

L t i o n s by the maps h(L,~)

over a l l

y i e l d s a complex

"A-homology"-cell-complex);this A-homology manifold

PL

links

is A ~ "+k k

immersed in

R

L

and making i d e n t i f i c a (which is n a t u r a l l y an n If M is t r i a n g u l a t e d ( i n the sense given n

above) we obtain a map

g: Mn + A~ n , where M O ,k 0 n of A-homology c e l l s dual to simplices not in ~M .

denotes the union

Despite the n a t u r a l i t y of t h i s c o n s t r u c t i o n , there remains a certain d i f f i c u l t y generalizing

in d e f i n i n g a s u i t a b l e "bundle" over

A~{ n,k X n , k over __'~n,k" This is because the notion of

tangent bundle f o r homology manifolds involves a " b l o c k - b u n d l e " type of c o n s t r u c t i o n .

This d i f f i c u l t y

disappears, however, i f

we seek to

study A-homology manifolds embedded (immersed) s i m p l i c i a l l y in a t r i a n g u l a t e d Euclidean space. N

In t h i s case, we may g e n e r a l i z e on the

model of ~L~ , r a t h e r than ~ n , to obtain a space Ax~n . n,k ,k ,k Here, the idea is to view formal l i n k s as t r i p l e s (U L, TL, ~ ) n+k L where U is an i+k plane in R , T an admissible t r i a n g u l a L L t i o n of U and ~ a subcomplex of T which is an A-homology L L L manifold with the A-homology-type of an ( i o l ) - s p h e r e . We then may form

A ~ , k V n which is a u t o m a t i c a l l y a subspace of

AYn,k,

the

l a t t e r being an A-homology k-block-bundle over the former in the n sense of [M-M]. For an A-homology manifold M embedded(immersed) n+k s i m p l i c i a l l y in a t r i a n g u l a t e d R , we get a Gauss map n

"V

M + A and, in t h i s instance, we obtain as well as A-homology O ,k n block bundle map from the "normal bundle" of M to Ay n,k" A f i n a l g e n e r a l i z a t i o n derives from the r e a l i z a t i o n t h a t "Gauss maps" e x i s t , in an a p p r o p r i a t e sense, f o r piecewise l i n e a r maps of t r i a n g u l a t e d manifolds i n t o Euclidean space which are n e i t h e r embeddings nor immersions.

The c o n s t r u c t i o n i s ,

more s t r a i g h t f o r w a r d than t h a t f o r

65

Vn~,k"

in a sense, even

3.7 We d e f i n e a f o r m a l of data

(ES'

dimension complex

~S'

j-l. n-j. a

~S ) ~

~ is s In a d d i t i o n ,

S'

of

dimension

(n,k;

where

xS

is

a triangulated

j)

to

PL

from the v e r t i c e s

n-j.

of

A

~

consist

sphere of

i s a l i n e a r o r d e r i n g on t h e v e r t i c e s o f S c o n s i s t e n t w i t h t h e s t a n d a r d o r d e r i n g on

~ S a function

and

star

to

the n-j A .

n+k

R

S ~ i s t o have the f o l l o w i n g p r o p e r t y : I f . v denotes s the c o n v e x - l i n e a r e x t e n s i o n o f ~ to a c o n t i n u o u s map An-j. n+k j s : S ÷ R , then (b) = O, where b denotes the S A n-j A b a r y c e n t e r of the s t a n d a r d s i m p l e x a Given a f o r m a l

a vertex

v

of

specified:

star

SS'

~S

S = (s S,

g o t by n o t i n g t h a t

~s)

the d e r i v e d f o r m a l

= ~k(V,Xs);

V

~s,

dimension

star

the o r d e r i n g

the subcomplex

of

(n.k;j)

S is e a s i l y v n-j+1 on a *XS

~S

and

is

n-"

A

J*st(v,~

)C an-J*x v be may .S S n-j+1 n-j identified with A *~k(v,z ). Here, A *v is identified with S n-j+l n-j the standard ~ by u s i n g t h e o r d e r i c l g on ~ *v induced from n-k+1 n-j+1 • At the same t i m e , A *~k(v,~ ) = A *~ also acquires ~S S S V an o r d e r i n g via restriction of ~ and we l e t t h i s be ~ At to S S n-j+l v the a s s i g n m e n t i t a k i n g v e r t i c e s of ~, *X to S S n+k-1 v v S U {0}, t h i s i s o b t a i n e d as f o l l o w s : As b e f o r e , l e t = / n-J.~ d e n o t e the c o n v e x - l i n e a r e x t e n s i o n t o A of ~ . Let b' be S s n " the b a r y c e n t e r o f A -J*v and p = (p'). Then, f o r a v e r t e x w

~

of

a

n-j+l.s t ( v , z

d e f i n e d as

S

),

let

V

to d e f i n e , o

S v

(w)

= i

S

(w)-p'.

Obviously

S v

is

(SS " ~S " ~S )" V

V

A straightforward

simplex

l

for r

assumed t h a t

any f o r m a l

of

Further details

generalization

~S' of

star

S

of

a d e r i v e d formal

this

generalization

the reader is

of

this

c o n s t r u c t i o n a l l o w s us

dimension star will

(n,k;

j)

and any

S with s = ~k(o,XS). o S be o m i t t e d , i ~ b e i n g

by now s u f f i c i e n t l y

familiar

theme from p r e v i o u s examples as t o be a b l e t o f i l l

in

with

this

t h e gaps

without difficulty• The n o t i o n o f

formal

star,

in

turn,

66

leads d i r e c t l y

to the

3.8 c o n s t r u c t i o n of the a p p r o p r i a t e Grassmannian, the

"S"

which we l a b e l

~

in the n o t a t i o n being intended to suggest t h a t

S n,k is

n,k the a p p r o p r i a t e t a r g e t f o r the Gauss map of an a r b i t r a r i l y s i n g u l a r n n+k PL map M + R Of course, such maps, in t h e i r own r i g h t have no p a r t i c u l a r i n t e r e s t as geometric o b j e c t s which might a t f i r s t blush S make the c o n s t r u c t i o n of x)-Z"seem to be an e x e r c i s e in f u t i l i t y . ,k Noting only t h a t such an o b j e c t i o n i s on r e c o r d , we proceed to the c o n s t r u c t i o n , and d e c l a r e the i n t e n t i o n to c l a r i f y , non-trivial

If

aspect of the whole proceedings.

The c o n s t r u c t i o n of ~ ~ n ,k S i s a formal s t a r and ~

goes forward in the accustomed v e i n : a simplex of

s , S w h e r e o*

homeomorphism h ( S , ~ ) : the c e l l

subsequently, the

dual to

triangulation.

o

then there i s a

c ~ ÷ o* C ~ denotes as usual S S in t h e ° c e l l s t r u c t u r e dual to the given

Thus, we may, as usual, form the d i s j o i n t

union of

spaces of the form

c S , where S is a formal s t a r of dimension S (n,k; j) n and k being f i x e d , and then o b t a i n S { by n,k identifying c ~ with o* C ~ under h(S,o). SO S n n+k Quite c l e a r l y , any s i m p l e x - w i s e l i n e a r map f:M ÷ R gives n S n S r i s e to a Gauss map g ( f ) : M ÷ ~ (or g ( f ) : MO + ~/o if n,k /,d" n , k n M has non-void boundary). S Less immediately obvious is the f a c t t h a t ~ carries a S n,k canonical PL n-bundle Yn,k" We sketch the c o n s t r u c t i o n . F i r s t of all,

given a j - d i m e n s i o n a l formal s t a r

C~s' tangent

viz.,

T(A n - j *

bundle.

C SS) I C ~ , S

Denote

this

bundle

S,

where by

¥S"

we have an n-bundle over T

denotes o r d i n a r y We c l a i m

that

if

PL o

is

then the homeomorphism h ( S , ~ ) : c ~S ÷ ~ * C C S o o is c a n o n i c a l l y covered by a PL bundle map h ( S , o ) : YS + yS I o*.

a simplex of

~S ,

Therefore, as we assemble the complex

by g l u i n g t o g e t h e r the n,k we may s i m u l t a n e o u s l y assemble

cells

c ~ via the maps h ( S , ~ ) , S S the t o t a l space of y by g l u i n g t o g e t h e r the various n,k

67

x 's S

by

3.9 the maps h ( S , a ) . We a s s e r t t h a t t h i s c o n s t r u c t i o n does, i n d e e d , S provide ~ w i t h a PL bundle s t r u c t u r e . n,k I t f o l l o w s , moreover, t h a t f o r a given s i m p l e x - w i s e convexn n+k n S l i n e a r map f : M + R , the r e s u l t i n g Gauss map g ( f ) : M + -v,,~Z~/-,k n i s covered by a bundle map TM + n,k A f i n a l word is a p p r o p r i a t e on the m o t i v a t i o n f o r t h i s particular

construction.

As we have n o t e d , no p a r t i c u l a r i n t e r e s t n n+k a t t a c h e s to the set of maps M ÷ R , per se. But c e r t a i n l y i f we e n v i s i o n more r e s t r i c t i v e local

sets of mappings, i . e .

geometric c o n d i t i o n s , i t

c o r r e s p o n d i n g subcomplex

is

mappings

satisfying

not hard to see t h a t a

of

w i l l be d e f i n e d by t h i s nn ' k n+k restriction. That i s , a map f : M + R which s a t i s f i e s the n S whose presumed r e s t r i c t i o n w i l l have a Gauss map g ( f ) : M ÷ ~ n,k image l i e s in ~ / . Conversely, if g(f)(M n) C ~ ,d, f will s a t i s f y

n

the r e s t r i c t i o n . (As an example, we might c o n s i d e r immersions of M n+k into R ; the c o r r e s p o n d i n g subcomplex i s , r o u g h l y speaking, identifiable

w i t h the o r i g i n a l ~ of ~ n ,k To f u r t h e r c l a r i f y the m o t i v a t i o n f o r

§2.) introducing

a n t i c i p a t e the r e s u l t s of the n e x t s e c t i o n , where we w i l l i n t e r e s t e d in

k' we n, c h i e f l y be

/~

and i t s subcomplexes. We s h a l l show n,k zf (Theorem 4 . 1 ) t h a t f o r a p p r o p r i a t e subcomplexes ~ f / c ~ , that if n,k n n a non-closed m a n i f o l d M admits a map M ÷ covered by a bundle n n n+k map TM ÷ Y I~, then t h e r e w i l l be an immersion f : M ÷ R n,k whose Gauss map has image in ~ / . The immediate p o i n t is t h a t the

~

same s o r t of r e s u l t

: For n,k s u i t a b l e subcomplexes )~//c ~ / ' ~ ( c o r r e s p o n d i n g to l o c a l g e o m e t r i c ~ v n ,k n p r o p e r t i e s of maps), the e x i s t e n c e of a map M + / - ~ covered by a n S bundle map TM ÷ ~ I~ w i l l guarantee the e x i s t e n c e of a map n n+k n,k f: M + R such t h a t ~ ( f ) (Mo)C (i.e., f w i l l s a t i s f y the local

h o l d s , m u t a t i s mutandis, f o r

g e o m e t r i c c o n d i t i o n to which ~

the scope of

corresponds.) I t

the p r e s e n t work to s t a t e t h i s

68

is

beyond

r e s u l t p r e c i s e l y or to

3.10 prove i t .

Suffice i t

to say, the methods of proof of theorem 4.1 can

be s u i t a b l y a l t e r e d and generalized and the reader may f i n d t h i s to be an i n t e r e s t i n g exercise.

The p o i n t , then, is that the ~ S n,k construction is not the gratuitous abstraction i t might i n i t i a l l y have appeared to be.

B9

4.1 4.

The immersion theorem f o r

subcomplexes of

/~n,k

n n+k §2 above, given a PL immersion M ÷ R n n t r i a n g u l a t e d manifold M , where M is c o m b i n a t o r i a l l y t r i -

As we have seen in of

the

a n g u l a t e d and the map

n

g:

M

+

this,

is

a

in

on s i m p l i c e s , we o b t a i n a Gauss n

covered by a bundle map

,k

o

of

immersion l i n e a r

a simpleminded sense i s

TM + x . o n,k

easily

The converse n

s e e n to be t r u e .

If

M

PL

m a n i f o l d and g: M + i s covered by a bundle map n , k n+k then M immerses in R One s e e s t h i s by n o t i n g TM ÷ Yn,k' n+k t h a t the t a u t o l o g i c a l map G : ~ + R i s covered by a f i b e r w i s e n+k "~n , k n injective map y + TR thus TM becomes a subbundle of n

(Gog).TRn+ k

n,k

A p p l y i n g the H i r s c h - P o e n a r u immersion theorem [ H - P ] ,

we o b t a i n the d e s i r e d immersion. n

M

to

be

This

section

aims

at

considerably

example, t h a t a map

certain restrictions

Gauss map s a t i s f i e s

4.1

we must assume

result.

i s g i v e n which s a t i s ,k from the p o i n t of view of geometry. n n+k M + R so t h a t the r e s u l t i n g

natural

M

this

÷

the same r e s t r i c t i o n ?

To make m a t t e r s more p r e -

c o n s i d e r the f o l l o w i n g

Definition.

A subcomplex ~

to be g e o m e t r i c i f

and o n l y i f

of

L

K

dimension f o r

happen) then

and o n l y i f

to

and

is

complex ~ - -, k

V = V K L

said

implies

L

r e p r e s e n t s the same l o c a l and

K

e

is L the subcomplex.

n e e d n ' t be of in

the same

a g e o m e t r i c subcomplex

tant

i s in We s h a l l g i v e some i m p o r K examples of g e o m e t r i c subcomplexes f o l l o w i n g the p r o o f of the

main

result

of

e

CW

links).

geometry as a n o t h e r , this

(note

the

e C/~/ L

e C/~, (where L,K are formal K To p a r a p h r a s e , i f one l i n k

if

k = 0

strengthening

g:

Can we then produce an immersion

cise,

if

nonclosed.)

Suppose, f o r lies

(N.B.:

[P],

this

Assume t h a t is

either

open or

section. n

M

First,

however, we s t a t e t h a t

has no c l o s e d components ( i . e . ,

has a n o n - v o i d b o u n d a r y ) .

70

result.

each component

4.2 4.2

Theorem.

map

g:

be a g e o m e t r i c subcomplex o f , , , ~ k " Suppose n ~. n t h e r e i s a map f : M + ~ c o v e r e d by a bundle map TM ~ y n n+k n,k Then t h e r e i s an immersion M + R such t h a t the r e s u l t i n g Gauss M ÷ o

has i t s

,k

imaqe i n

R e c a l l , from §2,

Proof: L.

Let ~

In p a r t i c u l a r

recall

L~) ~ = ~/ eLcT~z L

is

tion

Thus,

retract.

subcomplex o f n map from M

that

.

the d e f i n i t i o n if

~

is

a subspace o f

of

TL ,

for

each l i n k

a subcomplex o f ~ / then n,k' containing ~ as a d e f o r m a -

n,k in

the case a t

hand, where ~

the h y p o t h e s i s o f 4 . 2 , to

we may as w e l l

is

the g e o m e t r i c

regard

f

as a

~2.

An easy i n d u c t i v e argument making r e p e a t e d use o f c o - d i m e n s i o n one

PL

arguments shows t h a t , a f t e r a s l i g h t d e f o r m a -i t i o n of f, we may assume t h a t f (~) = M ~ M i s a codimension L L n zero submanifold of M . Note t h a t the i n t e r i o r s o f the v a r i o u s n manifolds M a r e d i s j o i n t from one a n o t h e r . I t i s , moreover, L e a s i l y seen t h a t M~ ( U M ) may be assumed to be a codimension-O L K O. Then the c o l l e c t i o n {E } t o g e t h e r with L XL,

covers

VL.

Therefore,

X , t h e r e must be a L interior p o i n t of q. V L(~,M 1 ) that

= V , LT

T

if

such

But

that

in

and a g a i n ,

no i n t e r i o r

point

of

o

goes i n t o

E c o n t a i n s the image o f T case i t i s immediate t h a t

this

by the

fact

that

~f/is

an

geometric,

we see

eL(~,MI ) Thus,

for

Gauss map f o r This

for

o

of

M , e c ~ ~/, and i t f o l l o w s t h a t 1 L(o,M I ) triangulation and i m m e r s i o n has i t s image

given

give

the

proof

of

the in

4.2.

some a p p l i c a t i o n s

of

this

result,

but

first

we

some o b s e r v a t i o n s .

(i)

Even a s s u m i n g

immersion ponents

the

completes

We s h a l l pause

any

is

seems

positive,

the

unavoidable

seems

to

be t h a t

union

of

codlmension-O

n-manifolds

that

V . L

the

the

restriction if

the

immersion

To g e t

codimension

proof

these

that is

n n+k M ÷ R

immersions

of

small

74

k n M

to is

pieces

of

the

have

putative

no c l o s e d

com-

go t h r o u g h .

The c a t c h

constructed n of M into

as

immersions,

we m u s t

the

the avoid

various

4.6 n-handles. In f a c t ,

This u l t i m a t e l y accounts f o r the a f o r e s a i d r e s t r i c t i o n .

our r e s u l t is c e r t a i n l y "best p o s s i b l e " in t h i s sense.

may see t h i s by the f o l l o w i n g counterexample:

be a r b i t r a r y , n+k and l e t W denote an n-dimensional v e c t o r subspace of R Let '~// L_~ ~/ n = e . is c l e a r l y a geometric subcomplex. Now i f M is V : WL L a connected manifold immersed so t h a t the Gauss map has image in , n i t is c l e a r t h a t the image of M must l i e in some a f f i n e n-plane n+k R

of other

parallel

hand,

any

to

W.

closed

Hence,

M

n

parallelizable

may

not

manifold

Let

We

be n M

k

closed. will

On t h e admit

a map

( e . g . , the t r i v i a l map) to ~}~/ covered by a bundle map n TM + ¥n,k I~{. Thus, 4.2 cannot p o s s i b l y hold i f the r e s t r i c t i o n

that

the m a n i f o l d in question have non-closed components be removed. In t h i s

r e g a r d , we may consider a s l i g h t l y

more i n t e r e s t i n g

counterexample.

Let ~ / ' ( j ) denote the s m a l l e s t geometric subcomplex 1~n, k such of ~ c o n t a i n i n g the j - s k e l e t o n of / ~ n ~nJ) = ~ e n,k ,k" ,k L that V = V f o r some j - d i m e n s i o n a l l i n k K. The image of the L K n n+k c~(j) Gauss map of an immersion of M in R w i l l l i e in ~ pren,k n c i s e l y when no p o i n t of M has a neighborhood which is " c r i n k l e d " by the immersion worse than neighborhoods of p o i n t s in the i n t e r i o r s of

(n-j)-simplices.

For example, l e t n = 2 and k : I . An immersion whose Gauss 4(0) map goes to ~ must have i t s image in some 2 - p l a n e ; i f the image (1) 2,1 l i e s in x~ 2,1, then the immersion admits edges where, l o c a l l y , two planes meet at an a n g l e , but there are no "sharp" p o i n t s . We claim t h a t 4.2 cannot hold f o r the subcomplex ~/=~}J),--1~n ,k n < k, in the absence of the requirement t h a t M h a v e non-closed

j

components.

This comes by way of the f o l l o w i n g i n t e r e s t i n g f a c t :

Let us d e f i n e a l i n k V

L

,

V

K

for

all

K

L of

of dimension dimension

< n.

75

(n,k;n)

as "s~arp" i f

4.7

4.3

Lemma

(D.

Stone).

n n+k M + R

Let

is closed. T h e n t h e r e are a t l e a s t n M such t h a t L(v,M) i s sharp.

Proof:

Let

Y

to r e f e r

to

and

y ~ Y.

the t a n g e n t cone of

E u c l i d e a n cone c o n t a i n i n g a l l is

some s u f f i c i e n t l y

W sufficiently is

(n+2)-distinct

vertices

n R

be a p i e c e w i s e - l i n e a r subspace of

necessarily a manifold),

small

small,

this

Y

We s h a l l at

y,

vectors

(w-y)

v

of

(not

use the n o t a t i o n

i.e.

neighborhood of cone i s

n M

be an immersion where

the s m a l l e s t

where y

T Y Y

in

w c W and Y.

independent of

W

Note t h a t f o r

W and hence

T Y Y

well-defined. Now l e t

n n+k M ÷ R

f:

some t r i a n g u l a t i o n (n,k;j)

let

Note t h a t

L

Now l e t jection for

all

is

of

of

denote the

U

on

simplices

triangulated

A

for

all

A.

as

L

o

of

n M .

a

PL v

of

xIY

U

so

Now l e t

of

dimension

the cone

V . L

dim Y = O. L and x orthogonal pro-

L(o,M) A ~ U be t h e

Let

subset n of M .

least

a link

respect to

that C

+ U

is

nonsingular

image ~fv

denote

(n+2)-extremal points

xfM.

is the

in

A

may

a vertex convex

C,

of

hull

i.e.

points

which admit t a n g e n t cones

A,

i.e.

T C ~V c extremal sion,

Generically

vertices

T C c o n t a i n i n g no n o n - t r i v i a l c Now these e x t r e m a l p o i n t s must, in f a c t , be p o i n t s

v e c t o r subspaces. of

For

with

l a r g e s t v e c t o r space in

U.

There are a t

c ~ C

closed.

n - d i m e n s i o n a l and sharp i f n+k be an n+1 plane in R

n+k R

be

of

Y L

be an immersion l i n e a r

n n M , M

c : ~fm,

m E M,

~ ~YL(~ L(o,M) ,M) m is not in the

and

hence

cannot

be

positive,

vector

space at

m

must for

xY

are

V'S,

as r e q u i r e d f o r

where

each

interior

of

then

T C c Thus, for such

such

m ~ int

be a v e r t e x

. L(v,M) least (n+2)

there

for

~.

any v.

C.

Thus,

simplex

contain

each

c,

4.3.

76

there

since of

Moreover,

would

c,

However,

v are

must at

is

positive dim

the

C

Y

dimen-

L(v,M)

nontrivial be

least

sharp. (n+2)

Since sharp

4.8 Thus,

t h e r e can be no immersion o f

whose Gauss map has image in --~/AJ) ,k H (2)

The r e q u i r e m e n t in

geometric, rather

or

the subcomplex

L(~,M )

with of

the

a simplex

f o r some s i m p l e x T in T resembles L or L in h a v i n g T

o

M L Rather,

~L"

V = V L(o,M n ) L

the image of whose

subcomplex

i n v o l v e d in

in

were to start out with a non-geometric

geometric

be

k

I

M + V . We see, L L as subcomplexes, t h a t the

M L

L

tain an immersion

is

codimension-O immersions

n M n

link

that

the p r o o f o f n n+k t h a t even though we g e t an immersion M ÷ R

the union of

upon t r i a n g u l a t i n g formal

< n.

than m e r e l y a r b i t r a r y ,

4.2 t h r o u g h the f a c t which i s

4.2

j

n+k R

a c l o s e d m a n i f o l d in

containing

On t h e o t h e r hand, natural

is

not n e c e s s a r i l y n L(~,M ) only or

V

subcomplex ~ ,

Gauss map lies

Thus,

L

L,

if

we

Twe would obin the smallest

~.

the c o n d i t i o n

plex is

a rather

as w e l l

as the p r e v i o u s examples ~ "

that

~/

be a g e o m e t r i c subcom-

one, as may be seen from some examples below

(j) class

of

PL

. If n,k immersions where r e s t r i c t i o n s

geometry o f

t a n g e n t cones,

restriction

corresponds n a t u r a l l y

we want t o r e s t r i c t

to a

depend o n l y on the l o c a l

and n o t on t r i a n g u l a t i o n s ,

then such a I

(3)

to a g e o m e t r i c subcomplex o f ~ f

,k

-

~

+l,k

,k+l

naturally gous l i m i t not

n,k

The d i a g r a m

+l,k+l

suggests the c o n j e c t u r e t h a t

lim n,k be p o s s i b l e

G = BO in nDk to prove this

l i m /}~x = BPL as the a n a l a n , k " n .k the c l a s s i c a l case. However, i t w i l l by o u r

77

methods,

and

it

is

probably

not

4.9 true.

Of

arising

course,

there

Yn,k"

It

b ~ [ K , ~ ] + [K,BPL]

a stable

bundle of [Wa].) with nal

PL

some

Of

easily

is

surjective,

map

g:

K ÷ BPL

weakness in that

in

is

M

+~2 / . Thus n ,k up t o h o m o t o p y . in

proving

7

any f i n i t e b

complex

K,

induces epimorphisms K

to

BPL,

r e p r e s e n t e d by the t a n g e n t

bog

b,

BPL

the n a t u r a l

homotopy e q u i v a l e n t to n+k in R for sufficiently

n

The d i f f i c u l t y

for

n o t e t h a t a map

K,

manifold n M immerses

lim = n,k ,k + BPL(k) of

(and so,

To prove t h i s

n M

PL

map ~ n,k that,

seen,

bundle o v e r

course,

Gauss map

maps

is

homotopy g r o u p s ) .

i.e.

a natural

from the c l a s s i f y i n g

bundles

of

is

is,

K.

large

essentially,

a bijection

arises

(See

the

k, origi-

from a c e r t a i n

the c o n c l u s i o n o f

4.2,

4.2. The a l e r t r e a d e r w i l l have noted n g: M + ~f/ o f the immersion o b t a i n e d i s

the Gauss map

n o t a s s e r t e d to be homotopic i n ~ / ( n o r even i n ~ ) to the o r i g i -n,k n nal map f : M + ~r/ o f the h y p o t h e s i s of 4 . 2 . The reason f o r the failure

of

this

homotopy o f maps to emerge is

r e a s o n s , adduced above, f o r metric.

That i s ,

the union of

if

closely

related

i m p o s i n g the r e q u i r e m e n t t h a t

to

the

~ / be geon n+k M ÷ R ,

the immersion produced by the p r o o f , i

codimension-O immersions

M + V , a l l o w e d of a t r i a n g u L L n i e , l a t i o n of M so t h a t , under t h e Gauss map g, M is sent to L L then i t would f o l l o w t h a t g i s homotopic t o f. As we have a l r e a d y r e m a r k e d , however, we do n o t q u i t e

get t h i s .

The a r b i t r a r y

i m p o s i t i o n of a t r i a n g u l a t i o n ,

perturbs,

so to

association

and i t

difficult

M' with L even up to homotopy. This

of

point

We s h a l l

will

briefly

complexes of / ~ n , k " on p i e c e w i s e l i r e a r

e , L

be f u r t h e r look at

is

considered at

some f u r t h e r

They come by way o f

PL

the n a t u r a l

to r e c o v e r t h i s

the end of

examples o f

§7.

g e o m e t r i c sub-

D. S t o n e ' s work [ S t 3 ,

m e t r i c s on c o m b i n a t o r i a l m a n i f o l d s .

such a m e t r i c may be d e f i n e d more a b s t r a c t l y , mind come from

speak,

St 4]

Although

the examples to keep in n immersions ( o r embeddings) o f a PL m a n i f o l d M

78

4.10

in

E u c l i d e a n space.

For each p o i n t in such a m a n i f o l d , Stone d e f i n e s

the n o t i o n of c u r v a t u r e , or r a t h e r two n o t i o n s which c o r r e s p o n d , roughly, case of

to minimum and maximum s e c t i o n a l c u r v a t u r e in the c l a s s i c a l n smooth Riemannian m a n i f o l d s . Given a t r i a n g u l a t i o n of M

w i t h r e s p e c t to which the given immersion i s l i n e a r on s i m p l i c e s , n these p a r a m e t e r s , K+(p), K_(p) for p ~ M depend only on the simplex

~

in whose i n t e r i o r n the formal l i n k L(~,M ). We may d e f i n e

First

of a l l

K+(L),

K+(L),

Given an a r b i t r a r y union of a l l ~L"

Let

we set

K_(L) L,

infinite

FL

p

lies.

K_(L) = 0

In f a c t ,

for for

they only depend upon

a formal l i n k L

of dimension

c o n s i d e r the E u c l i d e a n cone rays in

U L

from the o r i g i n

denote the maximal v e c t o r subspace of

K+(L),

K_(L)

= O.

If

not,

L

as f o l l o w s .

O

or

I.

r

which is the L through p o i n t s in FL.

If

FL = r L '

let

U be the o r t h o g o n a l compleL ment of FL in U and l e t ~ = S^ r . (We do not i n s i s t on L' L ^ UL L any p a r t i c u l a r t r i a n g u l a t i o n of ZL.) For x ~ 2EL, l e t ~_~°(x) be ^

the set of p o i n t s

Y ~ Z such t h a t t h e r e are a t l e a s t two paths L from x to y in Z of minimal l e n g t h ; l e t ~V(x) be the c l o s u r e L of~_V(x). Define K+(L,x) K-(L,x) as f o l l o w s : I f ~_~(x) K+(L),

K_(L)

Finally, It

= O.

If

~.~(x) ~9~. then

K+(L,x)

=

K+(L,x)

:

max y~(x}

(2x-cos

~iJin

(2x-cos

ycdx)

let

K+(L)

=

min K+(L,x) xaZ L

t u r n s out t h a t

K+(L)

) K-(L).

-1

-i

and

(x.y))

(x.y))

K_(L)

=

max K _ ( L , x ) . x~ L

Let a ) 0 • b. We may d e f i n e a c e r t a i n g e o m e t r i c subcomplex ^. n+k S of ~ c o r r e s p o n o i n g to immersions ef n-manifolds into R b "vn ,k such t h a t a t each p o i n t a • K+ • K_ • b. That i s , we l e t e be L a

79

4.11 a

in

the

S b

subcomplex

if

a )

K+(L),

a > K+(L

for

all In

simplices

q

this

our

case,

4.4

Corollary.

that

at

each p o i n t

f:

n a M + S b

map

The

A further the

First y

of

the

K_(L

and

) < b

theorem g i v e s

non-closed a ~ K+(p)

manifold ~ K_(p)

can

4.2.

It

immerses

if

and o n l y

n TM + y

be o b t a i n e d

concerns

n M

~ b

c o v e r e d by a b u n d l e map

of

the

),

< b

ZL"

corollary

statement,

particular

of

K-(L)

from

n,k

the

IS

a b

in

n+k R

if

there

is

a

•

proof,

characteristic

such

rather

classes,

than

and,

in

L-classes. all,

we n o t e

rational

that

since

characteristic

carries

n ,k

"

classes

for

any

PL

the PL

bundle n

manifold

M

n,k will

be i n d u c e d

from

the

corresponding classes

for

y

n

Gauss

map

g:

simplices real

of

cellular

Lk( ~

n,k structure

) ~

M

÷ ~:~ of any "-n , k some t r i a n g u l a t i o n . co-chain

Suppose, tion

T

of

to

position, are,

the

that

we h a v e ,

that

original cells

transverse

It

that

class

of

Pick

the the

if

to

by t h e

cellular

each

is

following of

to

and

the

in

the

cell-

Lk(M).

M

triangulaas

subspaces

also

and

the

a regular with

general

simplices

of

T

other. on

T*

then

we can

respect

to

formula:

Let

~*

all

for

(on

given

of

PL

are

a

say

representing

are

a cocycle

with

we p i c k

co-chain

addition

cells

on t h e

if

representinQ,

CW-complex

i H (M;R),

same c l a s s

linear

cell-decomposition

the

a

[a]E

orientations

in

M

particular,

g#c

these

of

of

in

triangulation

pair-wise, follows

So,

on x ~ ~ - n ,k we h a v e t h e

an a d d i t i o n a l

the

immersion

triangulation)

We assume

representing

S.

now,

i.e.

cohomology

virtue

n, to the

M,

CW-complex. respect

c

4k~f k ; R ) , H (

dual

, n,k

80

s

representing

get

the

a cocycle

cell

a

structure

be an o r i e n t e d

(n-i)-simplices

some

S

i-cell o

of

of T.

by

4.t2

Thus the i n t e r s e c t i o n

number

s.a

and the value of

d e f i n e d w i t h o u t a m b i g u i t y as to s i g n . a: d e f i n e s an i - c o c h a i n

a

a(~*)

become

T h e n the assignment

s + Z (S.o) a (~*)

on

S

which is

c l e a r l y a cocycle r e p r e s e n t i ing the same u n d e r l y i n g cohomology c l a s s in H (M;R). We now note Cheeger's work in L-classes.

defining a local

formula f o r

This was a c c o m p l i s h e d , as was mentioned in

ing the A t i y a h - P a t o d i - S i n g e r

n-invariant

the

§1, by e x t e n d -

from smooth Riemannian

m a n i f o l d s to more general s t r a t i f i e d

spaces.

In p a r t i c u l a r ,

c o n s i d e r a formal

i

L = (U , :~L ), then the L ~L ) gives r i s e to an

value of

n

on

~L

oriented co-chain sents, is

in f a c t ,

link

of dimension

= 4k,

( w i t h an o r i e n t a t i o n Ci

on n ~ - , k

corresponding o r i e n t a t i o n .

of

we

which is a c o c y c l e and which r e p r e -

the c h a r a c t e r i s t i c

d e f i n e d on an o r i e n t a t i o n

on

if

class

the c e l l

L (y ) ~ H i (C-" ~ ;R). i n,k ~, k i e L as nZL' ~L having the

Some p r o p e r t i e s of

~.

i n c l u d e the

1

following (i)

~ (e) depends only on V , and not on L i t s e l f . 1 L L the m e t r i c geometry of ~ c o m p l e t e l y d e t e r m i n e s the value of L the c o m b i n a t o r i c s of ZL being i r r e l e v a n t . (ii)

I.E., ~,

Suppose

V : V for dim K < dim L. Then ~ ( e ) = O. L K i L This i s because the assumption t h a t V = V t o t some l o w e r dimenL K ~L admits an o r i e n t a t i o n - r e v e r s i n g sional l i n k K means t h a t symmetry. (iii) t,

Suppose

0 < t < c,

i

There is a number

< n.

there is

an

L

e > 0

so t h a t f o r any

with

~ (e ) : t ( f o r an a p p r o p r i a t e i L orientation). The p r o o f w i l l not be g i v e n . n Thus, i f the t r i a n g u l a t e d m a n i f o l d M is immersed s i m p l e x - w i s e n+k l i n e a r l y in R via f, we have a w e l l - d e f i n e d c o - c y c l e , ~.(M,f) 1

representing

L (M). I f , in a d d i t i o n to the t r l a n g u l a t i o , 1 , M i given a t r a n s v e r s e r e g u l a r c e l l d e c o m p o s i t i o n S, we o b t a i n

81

is

4.13 ~ . ( M , f ) E Z (S;R) I 4.5 Lemma.

representing

The cocycle

~ (M,f) depends only on the immersion i t r i a n g u l a t i o n of M.

and not on the s p e c i f i c Proof:

L (M). i

Suppose w~ are given two t r i a n g u l a t i o n s

such t h a t the given immersion is b o t h , and t h a t the c e l l s to both.

of

Consider a c e l l

S 4i S

T, T'

simplex-wise linear

f

of

M

w i t h r e s p e c t to

are in general p o s i t i o n w i t h r e s p e c t of

S,

with a preferred o r i e n t a t i o n .

The value

~ ( M , f ) (s) is seen to be determined as f o l l o w s , using T i as the t r i a n g u l a t i o n in q u e s t i o n : There are a f i n i t e number of n-4i n-4i p.~ ~ , ~ a s i m p l e x of points Pl ' ' ' ' ' p m # S such t h a t J J J n T, and ~ (e ) ~ O, where L denotes the f o r m a l l i n k L(~.,M ). i Lj j J m

i

(M,f)

(s)

is

thus seen to be

m

j =~l n ( ~ L j )

: j=l~ ~ i ( e L j ) '

where each

(or e ) is given the o r i e n t a t i o n induced by t h a t of S. If ZLj Lj we examine the a l t e r n a t i v e t r i a n g u l a t i o n T', we see t h a t , f o r each ~n-4i p., p.E ~ where ~ is a s i m p l e x of T' This f o l l o w s J J J J b e c a u s e , by g e n e r a l p o s i t i o n p. cannot lie in a simplex of T' of J dimension < n-4i. M o r e o v e r , i f i t l a y ~n a s i m p l e x T of dimension

> n-4i,

then,

therefore

setting

n(~L j ) = 0.

(n-4i)-dimensional (for ~i J

orientations

eL,

j

L'j

: Z~ e i L J J

n-4i-simplices i n d e p e n d e n t of

~

: L(T,M),

So,

in

fact

we w o u l d pJ

lies

have in

VLj

: VLj,

and

some

with

V = V and hence L'j Lj c o n s i s t e n t w i t h t h a t on S). It

~ (e ) : ~ {e ) i L'j i Lj follows that

(for

p o i n t s of

there w i l l

be no a d d i t i o n a l

s

in

^

T

of

T'

with

~ (e ) ~ 0). i L(T,M)

Hence

~ (S) i

is

triangulation.

We now come to the main a p p l i c a t i o n of the t e c h n i q u e used in

the

immersion theorem. 4.6

Theorem.

Let

n

M

and suppose t h a t t h a t

be a n o n - c l o s e d m a n i f o l d f o r n+k M immerses in R via f.

82

which Let

L (M) : 0 i S be a

4.14 regular cell

d e c o m p o s i t i o n of

immersion

n+k : M ÷ R

f

i

such

M.

Then

that

the

f

may be m o d i f i e d to an

induced

co-cycle

~.{M,f ) 1 i

vanishes. Proof: in

We l o o k

general

position

which

f

is

if

s

is

a 4i-cell

meets

~,

if

at

the

with

Gauss map

respect

to

simplex-wise l i n e a r . of

at a l l ,

S at

and

n ÷ _~_n, k MO

g(f):

a triangulation We may assume,

~

"

We p u t

T for

of

~.

for

simplicity

an ( n - 4 i ) - s i m p l e x of

the b a r y c e n t e r of

M

S

T,

Given t h i s

that

then data,

s we

have the 4 i - c o c y c l e

~.(M,f) on S which, by a s s u m p t i o n , i s a coI boundary. In p a r t i c u l a r choose c so t h a t ~ . ( M , f ) = ~c, 4i-1 i c E C (S;R). Let r be a ( 4 i - 1 ) - c e l l of s, r C M-@M, so t h a t c(r)

= v

r

) 0

(for

some o r i e n t a t i o n

of

r).

Pick a p o i n t

x

in

r

r

and l e t

D be a small n - d i s c about x. Pick a d e f o r m a t i o n r r e t r a c t i o n of D to x w i t h the p r o p e r t y t h a t the d e f o r m a t i o n r keeps ~ ~ D inside ~ ~D for a l l closed c e l l s a to which r r r is incident. Picking x ~ D together with t h i s deformation rer r traction for all r, we have a d e f o r m a t i o n r e t r a c t i o n ~ D to r r U{x } (we may assume the D are m u t u a l l y d i s j o i n t ) . We may extend r r r t h i s to a d e f o r m a t i o n of the i d e n t i t y on M, a n d , i n f a c t , r e q u i r e further

that

furthermore, of

M,

and

over a l l

this is

deformation preserves closed c e l l s

the c o n s t a n t d e f o r m a t i o n on:

(b)

points

U ( g ( f ) ) - 1 y,

a neighborhood o f y

in

(a)

where

y

is

of

of

the

S 4i-2

where

the form:

y

and, skeleton ranges

cone p t of

,k CZL ~ e L , the l a s t n

M0 +

L

a formal

stage of

this

4J-dimensional l i n k

of

dimension

4i.

If

we l e t

d e f o r m a t i o n , we o b t a i n a map

, and o b v i o u s l y

,k The d e f o r m a t i o n w i l l

k

link

be K r (for

g'

g(f).

rel

g'

We now f u r t h e r

M - UD • For each r, r and a p o s i t i v e i n t e g e r k r

m

denote

= g(f)

deform

° m: ,

g .

p i c k a formal so t h a t

~ (e ) = c(r) one o r i e n t a t i o n of e ). We now deform i Kr Kr on D rel D , keeping the image of the d e f o r m a t i o n w i t h i n the r r 4 J - s k e l e t o n , to a map g w i t h the f o l l o w i n g p r o p e r t i e s : r

r g'

83

4.15 o

(i)

q (D) r r

(ii)

Let

~ e

:~.

L

O(s,r)

for

denote

all q

4i-links

=I

°

(e

ro + e

with

) ~ S

for

r

(ii)

is

understood in

has been i m p l i c i t l y

orientations

on a l l

been g i v e n f o r

s

is

for

all

4i-cells

K r S

t r a n s v e r s e to t h e " c o n e -

kr

the f o l l o w i n g

chosen (so t h a t

incident

o t h e r than

Kr

r C as. Then q IO(s,r) o r Kr point" of eKr p with m u l t i p l i c i t y Point

L

to

r.

sense:

c(r)

As w e l l ,

> O);

an o r i e n t a t i o n this

induces

an o r i e n t a t i o n

has

, v i z ; the one which makes k ~ (e ) = c(r). It Kr r i Kr i s w i t h r e s p e c t t o these o r i e n t a t i o n s t h a t the degree of q IO(s,r) r i s to be - k . That the proposed d e f o r m a t i o n to q can be made i s r r a s t r a i g h t f o r w a r d consequence of the f a c t t h a t ~ is connected. n,k S i n c e , on each D , the d e f o r m a t i o n has been c o n s t r u c t e d r rel D i t i s c l e a r t h a t the union of a l l such d e f o r m a t i o n s o v e r a l l r D with c ( r ) ~ 0 is e x t e n d e d in a t r i v i a l way t o a d e f o r m a t i o n o f r g'. Denote the f i n a l stage of t h i s d e f o r m a t i o n by g". It for

now c l e a r

an a r b i t r a r y

finite e . L ~(P)

is

e

collection Now f o r

that

g":

M+~ n,k s of

oriented 4i-cell of

points

each such

point

(where p,

*L

has the f o l l o w i n g p r o p e r t y : -1 S: (g") ~* ~ s is a L L now denotes the c o n e - p o i n t o f

we g e t a number

X(p)

d e f i n e d by

= ~i

(e ,0 ) (where e i s the unique 4 i - c e l l o f J-nX~ with p r p ,k g " ( p ) E e ) and 0 is the o r i e n t a t i o n i n d u c e d by t h a t o f s near p r p. The i m p o r t a n t p r o p e r t y i s t h a t Z X(P) = O. (Note: w i t h the P o r i g i n a l Gauss map g(F) i n p l a c e of g" used t o d e f i n e ~ ( p ) , we would have had

= ~ (s)). i Now, we s i m p l y proceed as f o l l o w s :

fied

in

Z~(p)

the p r o o f of

4.2,

working with

is

using t h e p r o c e d u r e s p e c i the map

g"

(which,

perforce,

c o v e r e d by a bundle map TM ÷ y ) we produce an immersion n+k n,k f : M + R , which s u p p o r t s a t r i a n q u l a t i o n T w i t h r e s p e c t to 1 i which f i s a l i n e a r homeomorphism on each s i m p l e x . It further 1 f o l l o w s , from t h e method e l a b o r a t e d i n t h e p r o o f of 4.2 f o r

84

4.16 constructing of

T 1,

this

complex t h a t ,

whose f o r m a l

link

given

with

(n-4i)-dimensional

simplex

r e s p e c t t o t h e immersion i s

o

L , 0

then

V L0 some

with

either

the image of

~.(e I

" d e g e n e r a t e , " in

the sense t h a t

V where dim K < dim L or K l i n k of d i m e n s i o n 4i such t h a t

formal in

is

) = O.

g"

In the f i r s t

it

coincides

V = V , where K Lo K * = cone p o i n t of K as we have noted

case,

Therefore, given a 4 i - c e l l

s

of

S',

is e

a is

K

in o r d e r t o

L

determine s

~i(M,fl)(S)

w i t h each

priate

~

in

the l a t t e r

intersection c e n t e r of

of

s

S 1

l

o

of

by the a p p r o -

has been i s o t o p e d t o and so t h a t occurs a t

the

the b a r y -

a.]

each such

~

*K

g"

Eim

let

of

of

K(~)

this

kind,

be chosen such

let

us group them,

with

*K ~

e

, for all a L the i n t e r s e c t i o n number o f

im g " ,

with

to

V = V Lo K(~)" t h e n o t a t i o n o f 4 . 2 i n mind,

fact f : ~ + X where X is n+k 1 K(o)' K(a) R d e f i n e d i n §2. Now, i f we o r i e n t K

i.e.

that

in

4i-dimensional

orientation

~

More u n a m b i g u o u s l y , w i t h

C M and, K(o) ( n - 4 i ) - p l a n e of all

T

w i t h an ( n - 4 i ) - s i m p l e x

C o n c e n t r a t i n g on

for

c a t e g o r y and m u l t i p l y

~ (e ). [ H e r e , we may assume t h a t i LG in g e n e r a l p o s i t i o n w i t h r e s p e c t t o

put i t

(o)

we need m e r e l y count up the i n t e r s e c t i o n s

K

the l i n e a r s

and

e

K

we o b t a i n t h e r e b y an

= K.

We c l a i m t h a t

the

0

w i t h a l l such ~ i s equal t o -1 This i s because we may the a l g e b r a i c m u l t i p l i c i t y u of (g") K" K s ~ M ÷ V hits X isotop S so t h a t under t h e immersion f K K K 1' From t h i s i t f o l l o w s t r a n s v e r s a l l y w i t h i n t e r s e c t i o n number I~K • (m,f)(o) = Z~ = o. i m m e d i a t e l y , since ~. (e ) = ~ (e ) that i 1 p I Lo I K(o) Thus the theorem i s p r o v e d . sum o f

In c o n c l u s i o n , we n o t e t h a t purely differential-geometric

s

Theorem 4.6 suggests

conjecture:

Suppose

n M

the f o l l o w i n g is

a smooth

manifold with of /smi

n

M , : 0

L (Mn) = O. Given a smooth r e g u l a r c e l l d e c o m p o s i t i o n i n does t h e r e e x i s t a Riemannian m e t r i c on M such t h a t for

all

4i-cells

s,

where

85

~i

is

the c l o s e d form

4.17 representing

L in deRham cohomology corresponding to t h a t i p a r t i c u l a r metric?

86

5.1

5.

Immersions e q u i v a r i a n t w i t h r e s p e c t to o r t h o g o n a l a c t i o n s on

In t h i s n-manifold

Rn+k

c h a p t e r we s h a l l study the problem of immersing an n n+k M in R , w h i l e r e s p e c t i n g c e r t a i n geometric r e s t r i c -

t i o n s much l i k e

those in

locally-smooth

PL

§4,

where, in a d d i t i o n t h e r e is

a c t i o n of

the f i n i t e

group

R

given a n M , and an

on

( n + k ) - d i m e n s i o n a l o r t h o g o n a l r e p r e s e n t a t i o n of R, and where the n n+k immersion M ÷ R is to be e q u i v a r i a n t w i t h r e s p e c t to these actions. We f i r s t it

wil|

note t h a t ,

a c t on the

We d e s c r i b e t h i s

~

has a given

O(n+k) r e p r e s e n t a t i o n ,

c o n s t r u c t e d in §2 above. n,k given an element m E O(n+k) and a

Grassmannian

action briefly;

j - d i m e n s i o n a l formal m.L

PL

since

link

L = (UL,

~L ),

m

acts on

L

to produce

by: m.L = (U m-L'

U = m.U where t h e a c t i o n on t h e m.L L O(n+k) on t h e s t a n d a r d G r a s s m a n n i a n

Zm.L

is

T ) "m.L right

is

the

m .

It

is

action

of

G j,n+k-j"

the ( t r i a n g u l a t e d ) sphere which is

under the homeomorphism

standard

the image of

immediate t h a t t h i s

~L

serves to

d e f i n e a d i m e n s i o n - p r e s e r v i n g O ( n + k ) - a c t i o n on the se t of formal links.

Furthermore, i t

t i o n s and of m.(L

linearity

is a s t r a i g h t f o r w a r d consequence of d e f i n i that if

a

is a simplex of

) = (m.L)

sponds to

then

(where m~ is the simplex of Z which c o r r e mo m.L o). In o t h e r words, t h i s O(n+k) a c t i o n preserves face

r e l a t i o n s among formal l i n k s . It

~L'

f o l l o w s t h a t we s h a l l

find

the diagram

87

5.2

cZ

h(L,o)

Lo

~

cZ

mI

L

tm h(m.L,m.o)

c ~ m.L

to be s t r i c t l y

c Z m.L

commutative and to p r e s e r v e obvious cone s t r u c t u r e s .

Now the c e l l s CZL

) m,o

eL

of

Vn~

mod c e r t a i n i d e n t i f i c a t i o n s

o b s e r v a t i o n s above, f o r

are to be t h o u g h t of as the spaces

,k

on the b o u n l a r y .

m ~ O(n+k)

w i t h those i d e n t i f i c a t i o n s ,

i.e.

By d i n t of the

m:c~ ÷ c~ is c o n s i s t e n t L m.L induces a homeomorphism of

m

closed c e l l s

@ :e + e and t h i s is c o n s i s t e n t w i t h face m,L L m.L i.e. @ I e = ~ Thus m produces a c e l l u l a r

relations,

m,L

L m,L~" Cm,L + L'~ ~ ~ZPn , k ~n of a u t o m o r p h i s m s {0 }

automorphism that

the

O(n+k)

@m

set

=

~n

on

extends to an a c t i o n of case

m.L

be the r a d i a l

on on

Q

J.

= roll

L

.,

a group

= (mU)

L

/

PL

action

of

O(n+k)). O(n+k)

on

U = m.U m.L L and l e t t i n g

~n,k In t h i s (under m.L

The r e s t of the conunchanged.

bundle

y

is an n,k w i t h the d i s c r e t e

V : mV (where V m ~ O(n+k), L' m.L L n+k a s s o c i a t e d to the l i n k n - p l a n e in R

that for PL

and,

since

m,L

checked

GL(n+k;R).

G j,n+k-j

O(n+k)

This f o l l o w s since by d e f i n i t i o n ,

onto

group

SUm.L.

the

is acted upon by

Note f i r s t

Vm.L : Qm.L (~) X,n,. L '

L

m(#L)

to show t h a t

as b e f o r e denotes the

mX

linear

above then goes through e s s e n t i a l l y

topology).

L

the general

p r o j e c t i o n of

O(n+k)-bundle ( i . e .

Q

yields

t h a t the a c t i o n of

GL(n+k); R)

We wish, as w e l l ,

L.)

trivially

would have to be s p e c i f i e d by l e t t i n g

the s t a n d a r d a c t i o n of

struction

is

m m~O(n,k) ( w i t h the d i s c r e t e t o p o l o g y on

,k paranthetically,

We n o t e ,

It ,k

m

while is

m

is

V = QL (~) XL' L o b v i o u s l y a homeomorphism of

o r t h o g o n a l , we must a l s o have

±

= U = X m.L ~,.L

88

5.3 Moreover, r e c a l l

map

the t a u t o l o g i c a

~{

n+k + R Direct n,k is an e q u i v a r i a n t

G:

i n s p e c t i o n of t h i s c o n s t r u c t i o n r e v e a l s t h a t G ~/ n+k O(n+k). Recall now map + R w i t h r e s p e c t to t h e a c t i o n o f n,k , e L a formal l i n k . the decomposition of ~ into subspaces m L - v n ,k Clearly ¢ (~) : ~ .L' so t h a t t h i s d e c o m p o s i t i o n i s p r e s e r v e d m

under defined

locally

strictly

m

of

O(n+k).

as

bundles

( G I ~ L ) * T V L.

@mI ~ L ÷ ~ m.L , TV

÷ TV

L .

recall

m of

such l o c a l

~L

~m ~

~m. L

VL

m ' "3

Vm. t.

K < L

(or

L < K),

G:(~ L /'I ~K ) ÷ i n t identified L given

is

of

itself

identified

homeomorphism bundle map a s s o c i a t e d to

~n,k'

i.e.,

G

TVL

and

~

(q ~ . K over

of

L G*TV K

with

TVK

~m K

m.L

m.K

m

int(V

is

~

we have

L

,k

t h a t the union

a w e l l - d e f i n e d g l o b a l bundle map

and thus

over the image under

m E O(n+k)

¥n

¢ . To see t h i s , we merely compare d e f i n i t i o n s m ~ ~ ~ . I f t h i s is not v o i d , we must have L K so, assuming t h i s , we have

(V L #-)VK)

naturally

~ ' L

I~ + Yn J~m n,k L ,k .L d e f i n e d by the map on t a n g e n t

which is

bundle maps is

+ covering n,k Yn,k on spaces of the form

over

a n a t u r a l bundle map

induces by the (PL) m.L remains to check t h a t t h i s

y

that

Since

+ V It m.L c o n s i s t e n t l y d e f i n e d over a l l

L is

G*TV

Finally,

commutes, t h e r e is

covering

mJV

L

the action

L

n

V ) K

~

int(V

B9

m.L

moK

are n a t u r a l l y

Hence the bundle ~

L

f~ ~ . K

Thus,

5.4

and so,

over

equally

well

covering

~L ~

~K

the map

Yn,k [~L :~ ~K + ~ n , k

be v i e w e d as a r i s i n g

the

homeomorphism

m: V

from

TV

L

+ V ) L m.L

l~m. L ~

÷ TV or

~m•K

may

(naturally

m.L from TV

these two maps being i d e n t i c a l over closed subsets of

+ TV . K ' m

K

i n t ( V L ~, VK)•

Hence, as s t a t e d , the c e l l u l a r automorphism

~ is n a t u r a l l y covered m ~m: Yn,k + Yn,k" That t h i s makes

by a bundle map which we may c a l l y

i n t o an n,k immediate.

O(n+k)-bundle ( d i s c r e t e topology on

Again, we digress b r i e f l y As we have seen, morphisms.

this

O(n+k))

to consider the case of

is

GL(n+k;R).

l a r g e r group acts on

~;/ by c e l l u l a r auton,k Note, however t h a t the t a u t o l o g i c a l map G i s no longer

e q u i v a r i a n t with respect to t h i s enlarged a c t i o n , nor are the manifolds y

n,k Yn,k

V e v e n preserved• Nonetheless, i t is s t i l l p o s s i b l e to view L as a GL(n+k;R) bundle. Recall the a l t e r n a t i v e d e f i n i t i o n of from §2 above.

This i n v o l v e s f o r any

L,

and then making i d e n t i f i c a t i o n s on the t o t a l which c o l l a p s e

TVLIbL

space to cover those

e C 1~n~L~ . (Recall t h a t b is merely L ,k L the polyhedron whose v e r t i c e s are those of cT and whose s i m p l i c e s :L n+k are the convex h u l l s in R of the simplex-spanning sets of vert i c e s of

bL

considering

onto

C ~ L . For the purposes of t h i s c o n s t r u c t i o n , i t

i e n t to consider image of

e .) L

bL + bm. L"

bL, For

r a t h e r than the isomorphic m E GL(n+k;R)

CTL,_

i s convenas the p r e -

we consider the induced map

To cover t h i s by a bundle map

TVLIbL ÷ TVm.LIbm. L,

we

may proceed as f o l l o w s : TV = TQL ~ X' where X' i s the subbundle n+k L L L of TR of vectors p a r a l l e l to XL • The given map m: b ÷ b m.L L extends to a homeomorphism in QL + Qm.L in a completely obvious way, and thus

TQLIbL ÷ TQm.Llbm. L •

is w e l l - d e f i n e d .

there i s a bundle map taking Whitney

On the o t h e r band,

X'Ib + X' Ib d e f i n e d , f i b e r w i s e , by L L m.L m.L X'(y) + x' (my) bv X ' ( y ) + m * ( X , ( y ) ) ~ X' (my). The L m.L " L L m.L sum of these two s t i p u l a t e d bundle maps i s the d e s i r e d bundle

90

5.5 map TVLIb L

TVm.Llbm. L.

We c l a i m t h a t

once more c o m p a t i b l e w i t h o b t a i n over and hence,

~ : m n,k the GL(n+k;

face r e l a t i o n s

÷

use of

attention

to

R)

(locally

~

is

smoothly in of

M,

representations

section,

links

and thus

however, we s h a l l

GL(n+k;

a finite

R)

and s h a l l

+ ¥n

,k

make no

c o n f i n e our

group a c t i n g on the

the sense of

[Br]

and,

for

PL

manifold

some a p p r o p r i a t e

simplicially),

we wish to c o n s i d e r o r t b o g o n a l n n+k and e q u i v a r i a n t immersions f: M ÷ R

IT + O(n+k)

a triangulation

simplicial

on formal

of bundle maps i s

O(n+k).

triangulation

If

this

the l a r g e r group

Therefore, if n M

family

the bundle map ~ : ¥ n,k m n,k ( d i s c r e t e ) s t r u c t u r e on njk

For the remainder of further

this

be chosen

and so t h a t

f

n M

for

is

so t h a t

a linear

the

~-action is

embedding on each s i m p l e x ,

we

look at

the Gauss map g ( f ) : M ÷ , along with i t s covering o ,k A n n n bundle map g ( f ) : TM + y . Note t h a t f o r BM ~ ~,, M o n,k o n is a ~-invariant subspace of M . We may t h u s s t a t e

5.1

Lemma.

map o f the

~-bundles,

bundle

above

The Gauss map

by

y the

This directly

n,k given

that

the

induced

from

n

, .

TM , M + y , ~ o o n,k

R-structure

no p r o o f

from

n

g(f):

representation

following

on the

the

space

is

n,k

~J

and

,k

O(n+k)-structure

a

defined

E ÷ O(n+k). beyond

definitions,

some r o u t i n e which

observations

we l e a v e

to

the

reader

results

of

§4 t o

as

exercise.

seeking

equivariant of

is

lemma n e e d s

an e l e m e n t a r y In

where

g(f),

to

case,

generalize it

is

the

clearly

immersion

natural

to

consider

the

subcomplexes

~/

H

which are g e o m e t r i c , in the sense of Def. 4.1 and which, n,k a d d i t i o n a l l y , are i n v a r i a n t under the H - a c t i o n on j~ZZn, k a s s o c i a t e d i

to

the r e p r e s e n t a t i o n

~ ÷ O(n+k).

C o r r e s p o n d i n g l y , t h e r e are r e s t r i c t i o n s above, we s h a l l

_

assume t h a t

the a c t i o n of

91

~

n M . n on M

on

As s t a t e d is

locally

5.6 smooth.

8redon's d e f i n i t i o n

[Sr]

is

for

the case of

topological

a c t i o n s of compact groups, but w i t h ~ - f i n i t e and a c t i n g s i m p l i c i a l l y n on H , we s h a l l s t i p u l a t e t h a t an a c t i o n i s taken to be l o c a l l y n smooth i f and o n l y i f , f o r each o r b i t R.x, x ~ M , t h e r e e x i s t s an O ( n ) - r e p r e s e n t a t i o n of the i s o t r o p y group n , and a PL R-embedding x n n n t : ~× D + M taking n× D onto a c l o s e d ~ - t u b u l a r neighborhood ~x ~x n of ~.x with t~× {0} = ~ . x . Here, of course, D i s given the ~x - s t r u c t u r e of the r e p r e s e n t a t i o n . x Some useful First,

suppose

type f o r M(0)

facts is

e

the a c t i o n .

follow

from t h i s

n o t i o n of

local

smoothness.

a subgroup of q such that R/e i s an o r b i t e Let M be the s e t of p o i n t s f i x e d by B, and

= p.M e,

while M = {x C MI~ = e} and M = p.M . Thus (e) e x (e) e M C M , M ~ M Local smoothness of the a c t i o n i m p l i e s t h a t e (e) n the K-components of M are a l l PL s u b m a n i f o l d s of M . e

Clearly,

M(O)

=

Consider a (topologically) fold. is

P

(e) L~ M 2e (~)' ~-component n M ,

in

(e) M

and so P

of

and l e t

r

is

M

iI-r D -bundle

embeds e q u i v a r i a n t l y embedding i s

~

over

denote i t s

P

a c t e d on by

on the O - s e c t i o n of

Bierstone condition [Bi], condition,

s u i t a b l y adopted to

i n t r o d u c e d by B i e r s t o n e i n

e q u i v a r i a n t g e n e r a l i z a t i o n of usual (or

Hirsch-Poenaru) r e s u l t s . Consider the s e t of

Consider now an element minimal Clearly,

a space

P

of

this

set with

is

in

PL N.

The

the s o - c a l l e d PL

case a t

study of

stands in

hand.

the p l a c e of

the

c o n d i t i o n as f o l l o w s :

(e) M ,

partially

r e s p e c t to

d e s c r i p t i o n is

92

P

is

We f o r m u l a t e t h i s

This

which

there

the n o n - e q u i v a r i a n t Gromov

~-components of

p o s s i b l e i s o t r o p y groups.

That i s ,

~

the

his

Gromov t h e o r y ,

"no c l o s e d components" c o n d i t i o n of

closed

~. n M

A new c o n d i t i o n which we now impose on

is

dimension as a mani-

onto a c l o s e d neighborhood of

the i d e n t i t y

stratified.

and assume t h a t

(o)'

then has an o p e n e q u i v a r i a n t n e i g h b o r h o o d .

a block -

This

naturally

a

e

r a n g i n g over a l l

o r d e r e d by i n c l u s i o n . this

ordering.

R-component of

some

5.7 M( e ) , of

with

n = e for a l l x E P, and thus, P i s a n-component x as w e l l , so t h a t P is a m a n i f o l d t o p o l o g i c a l l y closed in

M

(e)

n

M . The Bierstone C o n d i t i o n , then, may be s t a t e d : non-closed as m a n i f o l d s .

(I.e.,

All

such

P

e a c h ~ o p o l o g i c a l component of

are P

i s a handlebody with no t o p - d i m e n s i o n a l h a n d l e s . ] ( I n p o i n t of f a c t ,

B i e r s t o n e ' s o r i g i n a l f o r m u l a t i o n of the con-

d i t i o n i s somewhat d i f f e r e n t ;

the c h a r a c t e r i z a t i o n given above i s

subsequently shown to be e q u i v a l e n t ) . Note t h a t the Bierstone c o n d i t i o n a u t o m a t i c a l l y guarantees t h a t n M

itself

is

non-closed.

We may now s t a t e the main r e s u l t of t h i s

s e c t i o n , an e q u i v a r i a n t

c o u n t e r p a r t to 4.2. 5.2

Let ~ be a geometric subcomplex of - ~ , invariant n,k n be a PL l o c a l l y - s m o o t h ~ - m a n i f o l d s a t i s f y i n g Let M

Theorem.

under

n.

the Bierstone c o n d i t i o n . n Suppose f : M ÷ ~ map

f:

is an e q u i v a r i a n t map covered by a F-bundle

n

TM

+ Y I~. n,k Then there i s a B - e q u i v a r i a n t immersion

the e q u i v a r i a n t Gauss map Proof: t h e r e f o r e of

~

~,

upon _~

n,k

~L"

contains ~

R-invariant.

Mn ÷ / ~

n,k

n n+k M + R

has i t s

As p r e v i o u s l y observed, the a c t i o n of

i n t o the subspaces then

g(f):

f:

viz,

so t h a t

image in T~.

O(n+k), and

preserves the decomposition of ~

Cm

L

: ~

m.L

.

Let

~

as a deformation r e t r a c t and is

As in the proof of 4.2,

=

n,k

~ ~ ; eLc~2j L

itself

we wish to deform the map

f:

M ÷7~/c/~/z , staying w i t h l n , ~ / , so t h a t the new map f ' : M + ~ / -I n has M = f ' (~) a codimension-O submanifold of M . Moreover, we L L wish to have t h i s deforma*ion n - e q u i v a r i a n t . The analogous step in the n o n - e q u i v a r i a n t case 4.2 was v i r t u a l l y

93

5.8 trivial,

based on general p o s i t i o n c o n s i d e r a t i o n s .

group a c t i o n p r e s e n t , f u r t h e r argument is

However, with a

needed because, as is

well-known, the problem of e q u i v a r i a n t l y deforming an e q u i v a r i a n t map so as to put i t

i n t o general p o s i t i o n with respect to some i n v a r i a n t

s u b v a r i e t y of the t a r g e t space may meet some n o n - t r i v i a l obstructions.

Let us rephrase the problem somewhat:

Let

~y~= L < L . . . < L 1 2 r ordered by face r e l a t i o n s , e space of fact,

~

with a t r i v i a l

{~J}

be a sequence of formal l i n k s l i n e a r l y r c ~/ T~en ~ : /r~ ~ i s a subLi i =i Li r - l - d i s c bundle neighborhood. So, in

is a s t r a t i f i c a t i o n

of ~/z,

and we may t h e r e f o r e say

t h a t i f a map, e . g . f : M ~7~/ is transverse to {~ -I_ -1 {f e~ } w i l l be a s t r a t i f i c a t i o n of M, with f ~ of codimension map

f

r-l,

r = #.~.

may be e q u i v a r i a n t l y deformed to

The key o b s e r v a t i o n is Q

Thus, we s h a l l

then

a submanifold

show t h a t the given M +~/,

f'

of t h i s type.

the f o l l o w i n g general p r i n c i p l e :

is acted on by the f i n i t e

vided with a s t r a t i f i c a t i o n

f':

},

group

r,

and

Q

i n v a r i a n t under

r.

is,

Suppose

f u r t h e r m o r e , pro-

As we have noted

above, the problem of e q u i v a r i a n t l y deforming an e q u i v a r i a n t map $: W + Q,

(W

nontrivial.

a

r - s u b m a n i f o l d to a transverse map i s ,

However, the f o l l o w i n g r e s u l t gives a s u f f i c i e n t

c o n d i t i o n on the 5.3

r-space

Proposition.

p r o j e c t i o n map

If

itself

W~ is

Q + Q/r

Q

f o r o b s t r u c t i o n s to vanish.

the o r b i t s p a c e Q/r

Q ÷ Q/r

any e q u i v a r i a n t map so t h a t

in general

is

is

stratified

so t h a t the

transverse to the s t r a t i f i c a t i o n ,

then

~: W + Q may be e q u i v a r i a n t l y deformed to is

transverse to the s t r a t i f i c a t i o n ;

transverse to the s t r a t i f i c a t i o n

on

Q

thus

induced by

projection. The proof of t h i s p r o p o s i t i o n i s r o u t i n e , and in any case, f o l l o w s from the general theory of o b s t r u c t i o n s to e q u i v a r i a n t transversality.

94

¢' ¢'

5.9 To r e t u r n to the case a t hand l e t where ~J~ = LI < L2 . . . ~#

(or of

~p,~

< Lr .

, p e Jl)

{~

a stratification

) of

< pL 2 . . . < p.L 1 r -eR~ ~ ~ / / l l to be the image of

Define

the form

~/IT.

f'

5.4 f o l l o w s by d i r e c t and of

the d e f i n i t i o n

of

We now may r e l a b e l

fold

of

M

m a n i f o l d of

M

+

~/ {~j

is e q u l v a r i a n t l y }.

i n s p e c t i o n of the c o n s t r u c t i o n s f o r

the a c t i o n of f'

as

f,

in p a r t i c u l a r

~

M~

with

M = f L

e

{E L} ,

on / ~ n , k "

f o r all~AO

M

(L)

=

~ M p£ ~ p . L '

L

,

= f

-i__

eJ

now a co-

: #~? ).

(r

a codimension-O submani-

which is an i n v a r i a n t

n,k

codimension-O sub-

M.

~We may assume t h a t the new n f J

--)y

t r a n s v e r s e to the s t r a t i -

n

M.

Set

TM

is

t r a n s v e r s e to

d i m e n s i o n ( r - 1 ) - s u b m a n i f o l d of We we have,

is

Moreover,

AT

with

We c l a i m t h a t ~[i# r-1 x D in IfSo,

( I - o r b i t s of m u l t i - i n d i c e s )

Lemma. The p r o j e c t i o n /Tz ÷ " ~ / ~ _ fication { e n~j } ; t h e r e f o r e the map f : f',

p.L

j/

e ~

( i n d e x e d by a l l

5.4

d e f o r m a b l e to

=

under p r o j e c t i o n .

has a t u b u l a r neighborhood o f in p a r t i c u l a r

P7

f

is

covered by a

~I-bundle map

I~P#/.

Let

V : ~ V where _LL denotes a b s t r a c t d i s j o i n t u n i o n . (L) p~ i~ p . L ' n+k Consider the map G o f : M ÷ R Since MLEI M = ' l ~ unless p.L L = p.L, we see t h a t G o fIM i s f a c t o r e d u n i q u e l y as an e q u i (L) n+k n+k v a r i a n t map f : M ÷ V + R , where 2 : V + R is (L) (L) (L) (L) (L) merely the obvious i n c l u s i o n on each V C V p.L -- ( L ) " ^ AS w e l l , the bundle map fITM(L) + Yn,k d e t e r m i n e s an e q u i v a r i a n t bundle map Yn,k

is,

f

: TM ÷ TV (L) (L) (L} e s s e n t i a l l y , the p u l l b a c k of

The idea of the p r o o f i s

~

~L

= (.} pETI p . L '

TV(L ).

to t u r n each map

e q u i v a r i a n t ccdimension-O immersion. g e n e r a l i z a t i o n of

since on

The t o o l

M ÷ V i n t o an (L) (L) we use ~s B i e r s t o n e ' s

the P h i l l i p s - G r o m o v - H i r s c h t h e o r y r e s t r i c t e d

95

to the

5.10 immersion problem and t r a n s l a t e d i n t o state

the r e s u l t

we a c t u a l l y

the

PL

category.

need as a l e m m a .

First,

however, we

extend the B i e r s t o n e c o n d i t i o n to cover the r e l a t i v e n

Let

N

be a l o c a l l y - s ~ o o t h

~ - m a n i f o l d and

codimension-O s u b m a n i f o l d t o p o l o g i c a l l y Z

n

= ~

is

a l s o an i n v a r i a n t

be the minimal

elements of

case. n

W

an i n v a r i a n t

n N ,

c l o s e d in

We s h a l l

so t h a t

codimension-O s u b m a n i f o l d .

the s t r a t i f i c a t i o n

of

Z

8 i Z J from the

Let

arising

~-action,

8 v a r y i n g over the i s o t r o p y subgroups of ~. We s h a l l i n n say t h a t the p a i r M ,W s a t i s f i e s the r e l a t i v e B i e r s t o n e c o n d i t i o n ei i f and o n l y i f each connected component K of each Z can be obJ t a i n e d from a c o l l a r on K ~ BW by a sequence of h a n d l e - a t t a c h m e n t s not

i n v o l v i n g t o p - d i m e n s i o n a l handles. n

5.5

Lemma.

n

~-bundles

f:

M ,

n

~-manin n f o l d s s a t i s f y i n g the r e l a t i v e B i e r s t o n e c o n d i t i o n s . Let f: M ÷ V n n be an e q u i v a r i a n t map, V a ~ - m a n i f o l d , and assume t h a t flW is an immersion.

Let

W C_ M

Moreover, l e t n n TM ÷ TV ,

be a p a i r

f

locally-smooth

be covered by a bundle map of

so t h a t ,

bundle map induced by the

of

n ~ ,

this

coincides with

deformed,

rel.

W,

over

the

immersion. n

Then variant

f

may be e q u i v a r i a n t l y

immersion

h.

Moreover, the d e f o r m a t i o n from

be covered by a d e f o r m a t i o n through bundle map induced by We s h a l l it

of

not a t t e m p t a p r o o f of that

be r e p r o v e d from s c r a t c h

Hirsch

work as well

to

n - b u n d l e maps of

f

to the

this

We note t h a t

in

in

But then~

Hirsch and Poenaru

h

may

p r o v i n g an e q u i v a r i a n t v e r s i o n of

the

p r o v i n g an e q u i v a r i a n t Gromov the t e c h n i q u e s u~e~ by

t o e x t e n d the smooth Hirsch

96

context

B i e r s t o n e ' s techniques, a

they may be combined w i t h [H-P]

may

the much more l i m i t e d

That i s ,

immersion theorem as they do in

theorem.

lemma here.

the B i e r s t o n e theorem [ B i ]

smooth e q u i v a r i a n t immersions.

fortiori,

f

h.

a r i s e s from the f a c t

certainly

t o an e q u i -

immersion

5.11

theorem [ H i ]

to the

PL

c a r r y i n g through t h i s Turning back,

case.

The s t a t e d lemma i s

program.

then,

This is a l s o s t u d i e d in

to the p r o o f of 5.2 we s h a l l

an i n d u c t i v e argument on the dimension of Recall

the map

bundle map tablishing PL

M(L ) + V(L )

through

TM ~ TV (both (L) (L) n A terminology, let

~-manifold.

GofIM

equivariant). denote

[Mi].

c a r r y through

the formal l i n k s

which

(L)

For

an

involved.

factors, the

arbitrary

of

and

sake

of

the

es-

locally-smooth

~ , x ~ A, has an e q u i X n v a r i a n t t u b u l a r neighborhood of the form ~ x D , f o r some n-dimen~x n n s i o n a l r e p r e s e n t a t i o n of ~ . A codimension-O s u b m a n i f o l d A C_A x i n n s h a l l be c a l l e d a punctured A i f i t c o n s i s t s of A w i t h the i n terior

Recall

the r e s u l t

t h a t each o r b i t

of

such an e q u i v a r i a n t t u b u l a r neighborhood of an o r b i t n n removed. A s h a l l be c a l l e d a m u l t i p l y p u n c t u r e d A if it consists 1 n of A w i t h the ( d i s j o i n t ) i n t e r i o r s of s e v e r a l such neighborhoods of o r b i t s

removed.

We may now s t a t e our i n d u c t i v e h y p o t h e s i s . Hypothesis h(L):

H(j):

(a)

M(L ) + V(L )

Moreover

on

I(L),I(K )

each

covered by M

~

~ - e q u i v a r i a n t maps

n-bundle maps h ( L ) :

M

t

o h

(L) (K)' (L) are the obvious maps V ( L ) , (b)

MiL ),

There are

a multiply

For

punctured

dim L < j , M(L),

=

(L) V(K ) h

~

TM(L ) + TV(L ). ° h

where

(K)n+ k (K) + R

(L)

is an immersion on

and the bundle map

h(L)IM~L )

is

merely t h a t d e f i n e d by the immersion• (c) h(x) : (j-l) M

The map

n

h: M

n+k + R

d e f i n e d by

i

oh (x), x E M , i s an immersion on (L) (L) (L) = L.) Mdim L<j (L) To begin w i t h , we o b v i o u s l y have H ( O ) s a t i s f i e d ,

with

^

f(L)

'

f(L)

clauses ( b ) ,

a~ d e f i n e d above p l a y i n g the r o l e of (c) are vacuous f o r

j

97

= 0).

h(L ), h

(because (L)

We must t h e r e f o r e show

5.12 that

H(j-I}

implies

a typical that of

[j-l)

h(L) (j-2)~

is

M

H(j).

link

L.

Without

an immersion

M (L)"

H(j-I),

loss

M

consider

of g e n e r a l i t y ,

on an i n v a r i a n t (j-2)FI

Note t h a t

collar

M (L)

V

for

(L)

we may assume n

neighborhood

C

~ M ~ M . pE~ p.L K K d+l,

to

course,

this

by

The = n,

fj

a deformation

inductive

the

our

to

we may

original

~

extends,

f~ ^'

from

deformation

step,

Like-

^ ii

of

relabel

inductive

TMIM~*

.fl' ~:

to

on

F~

to

F~.

F"

as

F

This, and

hypothesis,

with

d+l.

final i.e.,

Mj,

maps f r o m

covers

we have a t t a i n e d

replaced

each

dim M

of

deformations).

I

step, our

then,

consists

assumption

length~

~ 2.

F

= FIM f o r each L L may no l o n g e r a p p l y Rather,

we c o n s i d e r

of

each

component.

us

that,

keeping

L

is

of

now t h a t

So we c o n s i d e r

with

c 8. L Hirsch-Poenaru

the

considering

e

L

fixed

F'IM ° L L

F'

is

an

the

problem

case

where

immersion of

on

deforming

Since

dim M : dim V : L L theorem automatically.

M° = M with a disc L L The p r e v i o u s inductive aM

the

removed step

from

the

argument

may be d e f o r m e d

to

n

we

interior

now a s s u r e s an

immersion

n

F

into

L It

F

II

:[-) o

F

L

V . L is

It

is

easily

seen

that

this

may be done I

a global

immersion

on a l l

of

M

L

so t h a t

o

:

UM . B u t , as L L M , so t h a t M :

in

§2,

I

note

that

(collar). closed F,

(Here,

we o b t a i n

Now,

W.

is

in

This

e ~ B, L

to

a copy

we use

components.)

homotopic

F

we may f i n d

the

once

M , i

of

M

more

the

hypothesis

Re-parameterizing immersion

in

M , i

F:M + W.

It

as

is

that M

M

and

obvious

has

M U i no

F"IM o I

that

F

as is

Gof.

we a s s u m e , general

perhaps

position

may be d o n e ,

after

with

retaining

we h a v e a c o d i m e n s i o n - O

a slight

respect the

to

the

condition

submanifold

110

further given that

M of L

deformation,

that

triangulation

of

for

M (that

each is,

L with

6.11 M ~ M in 1 L immersion. -I M = F ~, plex of B

the o l d n o t a t i o n ) Suppose, t h e n , ~

M,

~M

in

M + V C W a codimension L L we t r i a n g u l a t e M so t h a t each

that

a simplex of

a¢

with

W,

this

F:

is

a subcomplex.

triangulation,

be the s m a l l e s t s i m p l e x o f

W

such

with

Let

a

0

be a sim-

E M and l e t L' a C M . We must show 8

that

b

gF(a*) ( B. We s k e t c h

this

fact

as f o l l o w s :

s i o n simple~: and

K

the d e f i n i t i o n

the Gauss map

in it

e . K

of

the f o r m a l

Let

link,

~ > ~

e C ~7~ (a) ( c o n s t r u c t e d in K "~n ,k so t h a t a* ~ a has image

g)

Since a r e g u l a r n e i g h b o r h o o d o f

follows

that

argument i n

§2, e i t h e r L'

some f a c e

e LC - ~ n , k ( ~ ) " ~

L

~

Thus, ~ : V

eL , c ~

< L,

K

b

goes to

a

on t h e model of

~

(a).

be a maximal dimen-

~

or

V

under

L

F,

the a n a l o g o u s

~

n ~ : ~ ~ ~ for L' K (Perhaps a n o t h e r way o f s a y i n g the

, k

same t h i n g standard ~ then Thus,

is

a)

e

L' (LB.

6.1

As i n

locally

we t a k e ~

n,k etc.

C_ B K im gF C B,

a fixed

We s h a l l

smooth ( s e e ,

in

their

acts

on

V

we assume t h a t but with

L'

§2))

< L.).

B.

Hence

manifolds n+k W

§5 t h a t

e.g.,

~n,k(W)

is

a

PL

the added h y p o t h e s i s

W as a group o f

assume as i n [Br]).

the

o l d sense (as i n

E q u i v a r i a n t immersions i n t o

~

with

which was to be p r o v e d .

triangulation,

group

t o be i d e n t i c a l

f o r some L' by the g e o m e t r i c i t y of

e

Thus

(a)

with

the proceeding s e c t i o n ,

the f i n i t e

morphisms.

C B,

Addendum:

manifold with that

If

and use VL, V n,k K is seen t o be i d e n t i c a l

V K since

gF(a*~

this:

the a c t i o n

simplicial of

~

auto-

on

W

is

t h e r e w i t h becomes a

n-space as f o l l o w s : Let cial

u ~ ~,

homeomorphism (u) ~ O(n+k)

~,B

~

n+k u:

a simplex of

W

with

u(a)

= B.

The s i m p l i -

c l e a r l y d e t e r m i n e s an e l e m e n t ntk n+k (u) (A ) = a , c h a r a c t e r i z e d by

a + B

~Ith ~,B

111

6.12 -i (u) ( x ) = @ o u • ¢ ( x ) , ~B B ~ n+k the homeomorphism ~ + A

n+k x ~ ~ (resp.

Here, n+k 6 + a )

¢

(resp. ¢ ) is ~ 6 of the p r e v i o u s

section. 6.6

Definition.

u ~ ~, u(L)

u(e

:

L

) = e

(e~,B(u)

It

Let

Q. eLC / ~ n ,k (m), where

u(L)

(UL),

u(L)

~a,B(u)

should be c l e a r t h a t ,

r e s p e c t i v e l y of morphism

u: e

L

We c l a i m ,

where is

the formal

6.7

link

Then, f o r

given by

(TL)). since we t h i n k of

C~L,_ C~u(L),_

the map

~

~p8

e , e as images, L u(L) (u) induces a homeo-

+ e

u(L)" l e a v i n g the e l e m e n t a r y task of v e r i f i c a t i o n

r e a d e r , t h a t the map thus d e f i n e d , v i z , globally

L = (U L , ~L ).

to the

u:x~/ (~) ÷ / ~ (6) n~k " - n ,k

is

consistent i.e.

Proposition.

If

e ~ (~) ~ (~), with u(~) = B, L n,k k u(~ ) = ~ , then e C (~) A (8), and moreover the map u(L) ,k ,k u: e + e i s determined the same way by ~ (u) as by L u(L) ~,~

(u). ~Im~a

Moreover 6.8

Proposition.

If

e L C ~ n ,k (A)

and

K < L

then

u(K)

< u(L)

and the diagram e

U

K

---~" e

u(K)

U

e

k

~- e

u(L)

commutes. Again, n,k

(W) =

the p r o o f is L_] ~ (~) n-k v n ,k

purely routine.

T h u s since

we see t h a t a g l o b a l homeomorphism

u:~

(W) ÷~2~ (W) a r i s e s . Moreover, when _ ~ (W) is r e a l i z e d n,k n,k n,k as a s e m J - s i m p l i c i a l complex ( v i a the f i r s t b a r y c e n t r i c s u b d i v i s i o n

112

6.13 of

the

cell

It

is

structure

uw

seen

be a

as

PL

Suppose

further

all

variant

action,

last

disc,

triangulated

Z

2 flip

that

that

f

is

in

int

M.

immersion property.

Z

and

supposition

sality

The

well,

n M

now t h a t

points

This

a semisemplicial to

check

of _~

n,k

that

(W).

map. for

u,

Thus

/~

n,k

w ~ (W)

is

~-space.

smooth

at

as

is

self-homeomorphisms

We s u p p o s e

W

u

straightforward,

u ° w : to

{eL}),

is

is

a

f:

M + W

transverse

is

regularly

For

by

manifold is to

no means

homotopic

instance,

as

PL

let

admitting

a locally

a q-equivariant the

given

triangulation

vacuous. to

one

Not e v e r y

with

~ = Z 2 (~)Z 2,

immersion.

this

and

of

equi-

transver-

let

W

be

the

below:

~_~Z a c t i o n i s g i v e n by h a v i n g the g e n e r a t o r o f one copy of 2 the f i g u r e a b o u t t h e h o r i z o n t a l a x i s , w h i l s t the g e n e r a t o r

2 o f the o t h e r i s

the f l i p

a b o u t the v e r t i c a l

axis.

Let

M

be two 1-

s i m p l i c e s w i t h a common v e r t e x B

A I with

-~

C

Z

~ Z a c t i o n where the f i r s t generator acts t r i v i a l l y while 2 2 the second i n t e r c h a n g e s A and C, leaving B fixed. The i n c l u s i o n d e t e r m i n e d by

A ~÷ a,

equally clearly,

is

rot

general position

with

With

B ~ b,

C ~+ c

is

clearly

d e f o r m a b l e to an e q u i v a r i a n t

equivariant

and,

immersion in

r e s p e c t to

the h y p o t h e s i s t h a t

the t r i a n g u l a t i o n . n n f: M + W i s , in f a c t ,

113

t r a n s v e r s e to

6.14

the t r i a n g u l a t i o n , n

(M

+

o that

we examine the Gauss map

(W) n,k has boundary, and c o n c l u d e , by i n s p e c t i o n ,

n

g(f):

M

(W) if M ,k g(f) is a ~ - e q u i v a r i a n t map.

At the same t i m e , in f a c t ,

a

n-bundle.

we note t h a t the c a n o n i c a l bundle This is

p r e v i o u s l y , the bundle

seen by n o t i n g t h a t ,

y

(W) is, n,k l o c a l l y , as seen

(W) is induced by the t a u t o l o g i c a l map n,k _c~ .~ aj+k G from the v a r i o u s t a n g e n t bundles TV . Thus, i f e ~ ( ) W O~ i+k L L j,k but not in ~ / . k(B ) f o r any B < ~, we observe t h a t we may form

L) 7 u(~)

ueTI u(L) ~/I eL

=

v~I' L

( r e g a r d e d as a union of d i s j o i n t

denote the union

~-invariant iant,

y

L) u(e ~) ~ L

subspace o

and moreover

,k

h,k(W).

(W).

It

GwIeL ÷ W f a c t o r s

is

manifolds.)

Note t h a t clear that

naturally

e

L

is

Now l e t a

GW is e q u i v a r -

through a R - e q u i v a r -

i a n t map n G : L

Thus,

~If rr e + ~ , L L

x

k(W)I oR- = (G~) * T# L q_ and thus a c q u i r e s a ~ - a c t i o n as a n, L L ' bundle. I t remains only to observe t h a t these " l o c a l " n-bundles cohere to d e t e r m i n e a g l o b a l

q - a c t i o n on

y

(W). n,k f o l l o w from d e f i n i t i o n s t h a t the c o v e r i n g

Furthermore, i t w i l l n bundle map g: TM ÷ ¥ (W) n,k R-map.

becomes, under these c i r c u m s t a n c e s , a

We may thus proceed to f i n d an e q u i v a r i a n t analogue to the main result

6.5 p r e v i o u s l y s t a t e d .

Naturally,

the c o n d i t i o n t h a t the

m a n i f o l d to be immersed have no closed components w i l l

have to be

r e p l a c e d by the analogous c o n d i t i o n in e q u i v a r i a n t - m a p t h e o r y ,

viz.,

the B i e r s t o n e c o n d i t i o n . First,

a c o n d i t i u n on subcomplexes of / ~{ (W) - v n ,k ing the n o t i o n of g e o m e t r i c i t y {Def. 6 . 4 ) . 6.9

we f i n d

Definition.

A subcomplex

B of ~/~ (W) n,k

114

is

s a i d to be

extend-

PL

6.15 n-geometric i f f

B

is

geometric (in

i a n t under the given a c t i o n of

~

the sense of

on ~

n,k

6.4) and

R-invar-

(W).

6.10 Theorem. Let BC ~ (W) be a R - g e o m e t r i c subcomplex. Let ~Vn ,k n M be a m a n i f o l d w i t h l o c a l l y - s m o o t h PL ~ - a c t i o n , s a t i s f y i n g the B i e r s t o n e c o n d i t i o n (see §5).

If

f TM ~ )

y

t h e r e is

n,k

f M ~

(W) I B

8

then t h e r e e x i s t s an e q u i v a r i a n t immersion induced Gauss map Proof: W),

gF

has image in

Given any l i n k

consider its We s e t

orbit

L,

an e q u i v a r i a n t bundle map

F: M + W such t h a t the

B.

( a s s o c i a t e d to some s i m p l e x

~

of

R.L.

= LL # , which i s , of course a m a n i f o l d a d m i t t (L) Kc~.L K R - a c t i o n , as w e l l as a n a t u r a l ~-immersion ~ : V ÷ W.

ing a

V

(L)

As in borhood

the p r o o f of

we r e p l a c e

B

by i t s

(L)

e q u i v a r i a n t neigh-

Lc~ eL' and r e g a r d f as an e q u i v a r i a n t map i n t o e B Consider the s t r a t i f i c a t i o n of ] given by {~), ~,

~

5.2,

=

= (L1) ,

(L 2)

I

•

L

< L ... < L ; 1 2 r t r a n s v e r s e to t h i s

(L) r and e D

where

C__ B. L. I stratification

mimics the analogous step in equivariantly M(L ) = f

() eL _

§5,

(

)

denotes

The f a c t

that

~-orbit, f

].

and

may be made

via an e q u i v a r i a n t d e f o r m a t i o n i.e.

Lemma 5.4.

Thus,

M

is

decomposed i n t o codimension-O s u b m a n i f o l d s and, as in the p r o o f of 5 . 2 ,

on

M(L),

GW°f

factors

f (L) M ~ ~ ÷ W where the map V + W i s given (L) (L) (L) by the n a t u r a l i n c l u s i o n on each V , K ~ (L). K From here on i n , the p r o o f mimics t h a t of 5 . 2 , e x p l o i t i n g the u n i q u e l y as

immersion lemma 5.5

The r e a d e r may check t h a t d e t a i l s

parallel.

115

are e x a c t l y

7.1 7. 7.1

The Grassmannian f o r

p i e c e w i s e smooth immersions

The s p a c e / ~ c n,k We have h e r e t o f o r e r e s t r i c t e d

ing p i e c e w i s e - l i n e a r

t o problems i n v o l v n n+k manifold M into R

(or

and have shown how the com-

into

our a t t e n t i o n

immersions o f a PL n{k a triangulated manifold W ),

plex ~

( o r the ~ n -"bundle"/4~/~ (W)) and i t s g e o m e t r i c subn,k ,k n,k complexes a r e r e l a t e d to g e o m e t r i c r e s t r i c t i o n s on such i m m e r s i o n s .

In the subsequent s e c t i o n s we s h a l l s w i t c h our focus to p i e c e w i s e n n+k smooth immersions o f m a n i f o l d s M in R ( o r , more g e n e r a l l y , n+k n i n t o Riemannian m a n i f o l d s W ). A l t h o u g h the m a n i f o l d s X will be

PL,

in

structures,

the sense t h a t

they admit u n d e r l y i n g c o m b i n a t o r i a l

piecewise linear

important role. n n+k M + R (or

properties,

will

on such s t r a t i f i c a t i o n s

Xj

i s a s t r a t u m and Sn - j - i n-j-1 f i c a t i o n induced on S

smooth on each s t r a t u m . a certain its

technical

"linking"

by t h a t o f

sphere,

n M

shall

(A s i m p l i c i a l

complex i s ,

stratified

the s i m p l i c e s

t h e m s e l v e s as s t r a t a . )

There w i l l situations

re-

i.e.,

if

then t h e s t r a t i be a s i m p l i c i a l

of c o u r s e , n a t u r a l l y

be a "Grassmannian" and "Gauss map" a p p r o p r i a t e t o

a space ~2Ic such t h a t , g i v e n a n,k n n+k p i e c e w i s e - s m o o t h immersion M + R t h e r e ensues a n a t u r a l map n c n c c g: M ~z~~ n a t u r a l l y c o v e r e d by TM + x where y i s the n,k n,k / - n ,k "canonical" PL n - d i s c bundle o v e r . M o r e o v e r , we s h a l l see n,k n a t u r a l l y correspond to r e s t r i c t h a t c e r t a i n subspaces H o f / ~ n ,k tions

of

We s h a l l

condition,

triangulation. with

n o t p l a y an

R a t h e r , we a r e concerned w i t h immersions n n+k n M + W ) wherein M is s t r a t i f i e d by smooth

m a n i f o l d s and the immersion i s quire

per se,

this

sort,

i.e.,

of a g e o m e t r i c n a t u r e on i m m e r s i o n s .

will

undoubtedly a n t i c i p a t e , is t h a t , at n n folds M , a bundle map Tt.i + ¥ i H n n'kn+k a p i e c e w i s e - s m o o t h immersion M ÷ R

116

The r e s u l t ,

least

for

as the r e a d e r

non-closed mani-

g u a r a n ~ e s the e x i s t e n c e o f whose Gauss map has i t s

7.2 image in

H.

In f a c t ,

r e s u l t s of t h i s s o r t w i l l

stronger than the corresponding r e s u l t s r e l a t i n g

prove to be a b i t PL

immersions to

geometric subcomplexes of -~-, k " Our f i r s t c bundle Yn,k'

task,

then, is to construct

with i t s canonical n,k and to define the Gauss map f o r a p p r o p r i a t e immersions.

Matters are s i m p l i f i e d g r e a t l y by the f a c t t h a t

, as a set, n,k coincides p r e c i s e l y w i t h -~ as p r e v i o u s l y d e f i n e d ;~- c is n,k " - n ,k A merely z ~ with a smaller t o p o l o g y . - ~ n ,k We define t h i s topology on the u n d e r l y i n g p o i n t - s e t as, essentially,

a metric topology.

That i s ,

we specify the

of a t y p i c a l p o i n t in / ~ for ~ > O. "~n ,k i d e n t i c a l to ~ we see t h a t any such ,k i d e n t i f i c a t i o n map of at l e a s t one p o i n t L = (UL,ZL) that

x

Since . ~ c is, pointwise, n,k x is the image under the n+k | x ~ C~LC- R , where

is a formal l i n k of dimension

y c/Q?'~ n,k

is w i t h i n

I

~ CZL, y' ~ CZK of

x

~

of

and

x y

iff

~-neighborhoods

(n,k;j).

We s h a l l say

there are r e p r e s e n t a t i v e s

respectively

(dim L = dim K = j )

such t h a t : i)

U is w i t h i n ~ of U in the standard metric on the L K standard Grassmannian G j+k,n-j so t h a t ii) There is a s i m p l i c i a l isomorphism ¢: ZL +Z K n+k of ~. ¢(v) is w i t h i n ~ of v in R for a l l vertices v L n+k I i ii) y is w i t h i n ~ of x in R As usual,

0c

~-neighborhood of

~

c

is open i f f f o r every n,k x, ~ ( x ) , with ~ ( x ) ~ O.

x ~ 0

Another way of c h a r a c t e r i z i n g the topology of ~ c follows:

the f i r s t

com~iex, t h a t i s , (i.e.,

b a r y c e n t r i c s u b d i v i s i o n of ~ n , k

there is some

is as n,k is a s i m p l i c i a l

the geometric r e a l i z a t i o n of a s i m p l i c i a l set

a s i m p l i c i a l space with the d i s c r e t e t o p o l o g y ) .

Note that

each ~ir,~plex of t.~is p~rticula~" s i m p l i c i a l complex has a n a t u r a l l i n e a r ordering on i t s

vertices.

117

7.3 If

we now r e t o p o l o g i z e t h i s

s i m p l i c i a l space so t h a t the set of

j - d i m e n s i o n a l s i m p l i c e s has a s m a l l e r t o p o l o g y we s h a l l

have, in

passing to the geometric r e a l i z a t i o n , r e t o p o l o g i z e d J ~ / as w e l l . n,k Consider, t h e r e f o r e , a t y p i c a l j - s i m p l e x o of the f i r s t baryc e n t r i c s u b d i v i s i o n of ~ dimension

. There i s a unique formal l i n k ( o f n,k which we denote by L(o) such that into C int e

r)

We t h e r e f o r e d e f i n e an

c-neighborhood of

j - s i m p l i c e s by the f o l l o w i n g : i)

aim L ( o ' )

= aim L(~)

Say t h a t

o a'

L"

in the set of is

within

~

of

~

iff

= r

il)

U is w i t h i n ~ of U in the standard m e t r i c on L(~') L(o) the standard Grassmannian G k+r,n-r iii) There i s a s i m p l i c i a l isomorphism ¢: ZL(~ ) + ZL(o,) such n+k that @(v) i s w i t h i n ~ of v in R f o r each v e r t e x v of

L(o). Thus we o b t a i n a neighborhood basis f o r each element of the set of

j - s i m p l i c e s , and consequently a t o p o l o g y on t h i s

hard to show t h a t ,

with r e s p e c t to t h i s

set.

It

i s not

t o p o l o g y , face o p e r a t i o n s are

continuous m a p s . T h u s we o b t a i n a s i m p l i c i a l space whose geometric r e a l i z a t i o n is We leave i t

n,k to the reader to v e r i f y

t h a t the two d e f i n i t i o n s of

the t o p o l o g y of

coincide. C l e a r l y , the f o r g e t f u l map n~k ÷~3~ , which i s the i d e n t i t y on the set l e v e l , i s continuous. n,k ~ n ,k c Our next task i s to d e s c r i b e the PL n-plane bundle y njk c which i s to play the r o l e of the canonical bundle over ~ Pointn,k" c c wise y c o i n c i d e s with y k" i . e . , the f i b e r of y over x n,k n, n,k i s to be i d e n t i f i e d with the f i b e r of y over x ( r e g a r d i n g x as a p o i n t of

~n

e a s i l y described.

'k ).

n,k The topology of the t o t a l

A p o i n t of

y

space of

c Yn,k

is

l y i n g over x may be s p e c i f i e d n,k ( d c c o r d i n g to one of our c h a r a c t e r i z a t i o n s of y ) as the image of n,k a p o i n t in the tangent bundle to V at x ~ b C_ V where x is L o L L o a pre-image of x. This means, in e f f e c t , t h a t t h i s p o i n t in y n,k

118

7.4 may be described as the image of a p a i r

(x ,y ) where y s V is o o o L close to x . We t h e r e f o r e may c h a r a c t e r i z e the s-neighborhood of o c t h i s p o i n t ( i n the topology f o r ¥ ) as the set of a l l points of n,k i I I y which may be described by pairs x ,y where x ~ b is in n,k o o o L' , , the pre-image of x , (x w i t h i n ~ of x in ) and where n+k n,k ! y is w i t h i n ~ of y in R o o c With respect to t h i s topology, y is c l e a r l y a t o p o l o g i c a l n,k n-disc bundle over ~ nc, k . A s l i g h t a d d i t i o n a l argument must be made

~

in order to v e r i f y t h a t t h i s bundle admits a n a t u r a l structure.

We may see t h i s by f i r s t

closed subsets. 0 < j

< n.

Let

L = (UL,~L)

denote the set of

such that

PT = L~Q)~ C~L.

Here,

A, c

n,k denote a t r i a n g u l a t i o n of the

T

Let ~ T

decomposing

ZL

L

(j-l)

is a b s t r a c t l y isomorphic to

the t o p o l o g y

sphere,

is understood

T.

Let

to be t h a t

induced

c PT ÷ NT

where

NT = L~L-) eL C_~;i . n,k

may be t o p o l o g i z e d by i d e n t i f y i n g the p o i n t

cone p o i n t of of

into certain

j - d i m e n s i o n a l formal l i n k s

"'T by the n a t u r a l map itself

PL(n)

c~ L- in PT"

provided t h a t

G j+k,n-j

and

¢(v)

I.e.,

if

L,K s~-~i

Now ~ T

L ~J~T with the

we have

K

within

U as points of U is w i t h i n E of L K is w i t h i n ~ of v f o r some s i m p l i c i a l isomorof

and a l l v e r t i c e s v ¢: ZL ÷ ~ K f o l l o w i n g is a neighborhood of L ~ :

phism

~L"

In p a r t i c u l a r the

Pick a small neighborhood

I

of

U in L for U ~ Gj + k , n - j . vi

G j+k,n-j" where

Pick a map

O(n+k)

so t h a t

denotes the n a t u r a l a c t i o n of

Denote the v e r t i c e s of

pick a neighborhood ~ i

that i f

s: ~ ÷

~L

in the

by

O(n+k)

V l , V 2 , . . . , v q.

(j+k-1)-sphere

s(U).U on

For each

SUL.

We claim

and the ~ . are chosen to be small enough then I ~ X ~ l X ~ z 2 x . . . xq~ is nomeomorphic te a neighborhood of L in That i s ,

=

L

/~

given

w = (U,y I . . . . y ) ~ ~ x ~1. . . . . q

~q

Gefine a l i n k

T" L

w

be the image of ~L under the UL = U and l e t t i n g ZL w w of the assignment v i ÷ s(U).y i . "geodesic" extension to [L

setting

119

by

U

7.5

Clearly, this

correspondence i s a homeomorphism of

~ x ~ 1. . . . . .

~/~=/?~

i

onto an open neighborhood of

~. But now observe t h a t T i s a smooth m a n i f o l d (of a dimension depending on j and on the num-

ber of v e r t i c e s of f o l d and, in f a c t ,

T).

L

in

Thus, ~

is a t l e a s t a t o p o l o g i c a l mani-

a smooth m a n i f o l d since i t

these l o c a l charts f i t

t o g e t h e r smoothly.

s u f f i c e s to observe t h a t ~

is

thus a

is easy to see t h a t

For our purposes, i t PL

manifold.

It

follows

t h a t the space

P which we have d e f i n e d above, i s , in geometric T terms a PL m a n i f o l d , in f a c t a PL j - d i s c bundle over ~ . (The T PL s t r u c t u r e comes about since P is r e a l l y a bundle over T T with s t r u c t u r e group given by the s i m p l i c i a l automorphisms of T.) c Let YT denote the p u l l b a c k of y to P under the map n,k T ~ r P ÷ N C ~;Z~ YT may e a s i l y be seen to be i d e n t i f i a b l e with a T T n,k" Whitney sum ~@n where ~ i s the PL bundle of "tangents along the f i b e r " of the

j - d i s c bundle

P over ~ and where n i s the T T p u l l b a c k of an ( n - , ) - v e c t o r bundle n over ~w~Z~, v i z ; n is the o I On+k bundle whose f i b e r over L ~ ~L p is the vector space U C R T L Not only does YT admit a PL s t r u c t u r e , but the same i s also t r u e of the bundle from which c Yn,k INT" The p o i n t i s t h a t i f

¥

was o r i g i n a l l y induced, v i z , T c we view YT ÷ Xn,kINT as a q u o t i e n t

map, the i d e n t i f i c a t i o n s which produce i t PL

glue f i b e r s t o g e t h e r by

isomorphisms. Thus,

we have s p e c i f i e d p a r t i c u l a r

various r e s t r i c t i o n s

c Yn,k INT"

PL

Now suppose

s t r u c t u r e s f o r the NTI~

NT2 ~ ~ .

This

only occurs when

T is the l i n k of a simplex T (or v i c e - v e r s a ) . 1 c 2 We claim t h a t the two PL s t r u c t u r e s on y IN ~ N coincide. n,k 1 2 We leave d e t a i l e d v e r i f i c a t i o n of t h i s p o i n t to the reader as a straightforward exercise. specified a

PL

Since

strurture tor

{NT} y

covers ~;f c n,k

we have in f a c t

C

n,k We now wish to analyze f u r t h e r some aspects of the geometric

s t r u c t u r e of the space ~/c . n,k

We have a l r e a d y i n t r o d u c e d the closed

120

7.6 subspace all

N T

and we have noted t h a t union in

cone p o i n t s may be i d e n t i f i e d

N - imageLU L ~T T '

i s a homeomorph of

denotes open cone. of

with ~ T "

Let

the cone

c

image of

~ c'~ . L(~T L

c

"radius"

i

P = (J c ~ T Lc ~T L

int

denote the cone of " r a d i u s " I,

(Thus

N = image L.) c Z T L(~ L T Note a l s o t h a t

and l e t

0

T is a b s t r a c t l y

0 T

where I

of

c

inside

N be the homeomorphic T homeomorphic to P as T

a space and e q u i v a l e n t as a d i s c bundle over C~PT.) Recall t h a t t h e r e i s a s t a n d a r d v e c t o r bundle is

whose f i b e r over L ~T' n+k of R (dim nT = n - j

w h e n dim L = j

O b v i o u s l y , then) t h e r e i s

a n a t u r a l map

Grassmannian eT:

classifying

the v e c t o r subspace for all

eT

nT

over O~PT U L

L ~K~T).

from C~T

to t h e

and e x p l i c i t l y

standard

given by

L ÷ UL ~ G n _ j , j + k.

7.1

f:

Gn_j,j+ k

L = (UL,~L),

nT

Lemma.

e T

Proof:

Let

X +

is a f i b r a t i o n X

with

G

the sense of S e r r e ) .

be an a r b i t r a r y

be an a r b i t r a r y

homotopy

(in

= g.

map,

finite

g =

oT f

We must e x h i b i t

complex and l e t and

g:

a homotopy

F:

a

XxI + ~ I

O

with

X×I * Gn_j,k+ j

F = f and e oF = G. o T Consider the s t a n d a r d f i b e r i n g

V + G of the n-j,j+k n-j,j+k S t i e f f e l m a n i f o l d over the Grassmannian. Choose a c o v e r i n g {A } of l G such t h a t t h e r e are l o c a l s e c t i o n s s : A + V of n-j,j+k i i n-j,j+k x. Pick a s u b d i v i s i o n of X and a p a r t i t i o n O = t

< t < ... < t = 1 of I so t h a t o I q a t l e a s t one i where o i s an a r b i t r a r y X

and

x . ~ = G.

r

< q.

Over each such

,~ssume, i n d u c t i v e l y

~x[t that

Xx{O} L ) X ( p ) X I

r F

,t

~T(ox[t ,t ])C_ A for r r+l i s i m p l e x of the s u b d i v i d e d

] we have ~ = s oG, r+l i ~has been d e f i n e d on:

U X( p + l ) X [ O , t

121

r

].

i.e.

7.7 (Here,

X(p),

X(p+I)

denote the

( i n the s u b d i v i s i o n ) and re-parameterize i t

as

p

r < q.)

~'xl

and

p+1

Consider

w h e r e ~'×{0}

skeleta r e s p e c t i v e l y

ox[t ,t ] and r r+1 is i d e n t i f i e d with

ox{t } C ox[t , t ]. Define F on o ' × I as r r r+l I F(x,t) = (U(x,t),Z(x,t)) f o r each x ~ o , t ~ I procedure: sponds to To define

F i r s t of a l l ,

U = (G(x--~E,)) w h e r e (x,-~E) c o r r e (x,t) (x,t) in the o r i g i n a l p a r a m e t e r i z a t i o n of o x [ t , t r + l ] . r Z{x,t) note t h a t ~(x-'~,t), ~(x-~,O) give ordered o r t h o n o r -

mal bases f o r (in

x

and

U ( x , t ), U(x,O ) t)

we may define cial

by the f o l l o w i n g

r e s p e c t i v e l y , and thus a continuous

f a m i l y of isometries ¢ : U ÷ U . Thus -1 (x,t) (x,t) (x,O) Z(x,t) as ¢ ( x , t ) ( Z ( x , O ) ) (with the obvious s i m p l i -

structure).

Thus,

F

has been defined on

I

~ ×I = o x [ t , t ] r r+1 was a r b i t r a r y , we have extended F

with

e oF = G. Hence, since ~ T to Xx{O} U x ( P ) x I U x(P+1)x[O,t ] and the most r o u t i n e of r+l i n d u c t i v e arguments show t h a t we may extend F to Xx{O} U x ( P + I ) x I

and t h e r e f o r e , f i n a l l y ,

to a l l

of

X×I.

The proof

of 7.1 is thus complete. (In f a c t ,

one might also observe t h a t ~T

f i b e r bundle over determined by fashion as {v I . . . . . Vs} T,

G with p r o j e c t i o n map e whose f i b e r , n-j,j+k T is as f o l l o w s : Order the v e r t i c e s of T in some

,...,v v I ,v 2j+k 1 s ÷

S

and consider the space of maps which induce piecewise-geodesic embeddings of

the topology being induced from the

Sj+k-1

with i t s e l f .

differ of

T,

v's

of

s

0

in f a c t ,

the f i b e r of

denotes the unique " t r i e n g u l a t i o n "

I

then

= image ot

O-skeleton of ~ --

k

i s , i,, f a c t , c in v(Z/ I ~n,k )

a

they

which extends to an automorphism

The i d e n t i f i c a t i o n space i s ,

eT:~V_T ÷ Gn - j , j + k .) We note in passing t h a t i f I

s - f o l d c a r t e s i a n product of

I d e n t i f y two points in t h i s space i f

by a permutation of the

T.

is a l o c a l l y - t r i v i a l

copy of the s t a n d a r d

Grassmannlan

122

G

n,k

embedded

7.8 Y IY is n,k ~ 0 identity.

the s t a n d a r d

We s h a l l

show,

in

geometric s i t u a t i o n s

n - v e c t o r bundle over

the n e x t two s e c t i o n s ,

G n,k

that

and

e o

is

the

t h e r e are c e r t a i n

to a "Gauss map" i n t o ~/c . These n,k n involve "piecewise d i f f e r e n t i a b l e " immersions o f m a n i f o l d s M into n+k R

7.2

LS-Stratified

giving rise

Manifolds

Consider a c l o s e d of

n M

shall

closed,

from or

PL

denote, s p e c i f i c a l l y ,

connected subspaces

1)

Each n M

2)

aX i

3)

If

int @X ~ 1

X i

is

We s h a l l

X i

X 2

X ~ 1 the union of

is

flat)

are d i s t i n c t

stratification n M

a decomposition of

compact

s t r a t a of

Moreover,

A strict

called strata

a (locally

the union of

X , 1

X . 2 @X 2

is

n M .

manifold

such t h a t PL

submanifold of

l o w e r dimension

strata,

then

int

X 1

is

X ~ only i f X C_ @X or 2 1 2 lower d i m e n s i o n a l s t r a t a .

X i s i n c i d e n t to X (notation: j i The symbol < means i n c i d e n t or equal t o .

X ~ X . j i The e x t e n s i o n of the n o t i o n of s t r i c t stratification n folds M w i t h n o n - v o i d boundary or to open m a n i f o l d s i s In the case of

stratification X

compact m a n i f o l d s w i t h

the s t r a t a

n @M ,

n M

of

n ~M

of

where

X

M = codimension o f one i n s i s t s X ~ i

disjoint

say t h a t

iff

that

into

meet

where a t y p i c a l a s t r a t u m of

Y

in

@M.

< X ) i

to maniimmediate.

boundary one merely i n s i s t s

transversally,

is

X j

@X 1

M

stratum with

resulting Y

is

in a

a component of

codimension o f

X

In the case o f an open m a n l f o l d

in n M ,

X are p r o p e r in the sense t h a t i C i s compact f o r a l l compact sub,paces C of M. n We s h a l l , f o r the moment, assume t h a t M i s a compact, s t r i c t l y

stratified

that

n BM

X ~ 2

the s t r a t a

manifold.

Let

X

be a s t r a t u m and

123

X o

any codimension-O

7.9 submanifold of from

@X.

X

If

( w i t h o u t b o u n d a r y ) so t h a t

X < Y

neighborhood in

Y

and

in

dim Y = dim X+I

~

is

o

disjoint

then o b v i o u s l y

the form of a c o l l a r ,

X has a o a homeomorph o f

i.e.

X x I. We s h a l l c a l l such a r e g u l a r n e i g h b o r h o o d " g o o d . " Now o consider a stratum Y, X < Y and dim Y-dim X = q. We s h a l l c a l l r e g u l a r neighborhood

X in Y o inductively-defined condition

following X

x Rq

(where

of

"good" i f viz;

it

RC Y

satisfies is

denotes the s t a n d a r d h a l f - s p a c e ) ,

the

of

the form

and

R /~ Z

is

+

good f o r regular for

Rq

+

0

R

a

any s t r a t u m

Z

neighborhood

all

Y

with

with

R

of

X < Z < f. X o

n M

in

F i n a l l y , , we s h a l l "good"

iff

call

R~ Y

is

a

good

X < Y.

X ; in p a r t i o c u l a r , i t i s c l e a r t h a t such a good r e g u l a r n e i g h b o r h o o d R o f X o is strictly stratified where the s t r a t a c o n s i s t o f the components o f Good r e g u l a r

neighborhoods

clearly

exist

for

any

X and t h e components of R ~ Y for all Y with X < Y. Furthero more, i t i s n o t a t a l l hard t o see t h a t , as a s t r i c t l y - s t r a t i f i e d manifold,

R

has the form

X

×~,

where ~

is

a stratified

disc,

of

t a k i n g the cone on

0

dim~

= codim X

a strict

~F i s

stratification

a small that

and

disc

for

DJ

is

Dj

itself

Dj

X < Y,

thus

itself

Note t h a t c a t e g o r y of

X

of

DJ p

with

at

strata

by l e t t i n g

DJ ~

Y

for

the s t r a t a

X < Y of

the cone p o i n t d e l e t e d ,

as a s e p a r a t e m i n i m a l

this

essence by t a k i n g

p ~ X ~ Dj ( j = codim X) so .o Rq " we have D j ~ Y o f the form +

with

stratified

described in

strictly-stratified

Dj plus

and

be the the

stratum. disc ~

depends ( i n

the

strictly-s,ratified

the p a r t i c u l a r particular

strictly

strictly

by d i n t

~is

dim Y-dimX = q,

stratified is

~.

t r a n s v e r s e to

cones on t h e s t r a t a cone p o i n t

of

stratified

m a n i f o l d s ) o n l y on X as i t s i t s in n strictly-stratified M , and n o t a t a l l on Xo. In

we have a s t r a t i f i e d

h e n c e f o r t h be c a l l e d

the l i n k

(j-l)-sphere of

the s t r a t u m

The most o b v i o u s example o f a s t r i c t l y

124

~F ~ L ( X )

which s h a l l

X. stratified

m a n i f o l d , of

7.10 n M ,

course, i s a c o m b i n a t o r i a l l y t r i a n g u l a t e d m a n i f o l d s t r a t a are the s i m p l i c e s . is

the usual

stratified,

Thus, n

link

ck(o,M )

if

o

which is

is a s i m p l e x , then ~

generalize it essentially,

slightly

a t r i a n g u l a t e d , hence s t r i c t l y

strictly

and speak of

stratified

stratified

m a n i f o l d s which are l o c a l l y

p r e c i s e l y , a m a n i f o l d is

strictly

said to be s t r a t i f i e d

U which are s t r i c t l y i stratifications inherited,

stratified.

iff

it

stratified

strict

r e s p e c t i v e l y , from

In the case of a s t r a t i f i e d

is a stratum i f f

X

is

int

~_~ i n t X U i i boundary modulo i d e n t i f i c a t i o n s

stratified

U

i

covered by U. ~ U ~ j and U

m a n i f o l d , we s h a l l

j

the are

say t h a t

Xn U is the union of i in the a f o r e m e n t i o n e d a t l a s .

X =

X

is a m a n i f o l d - w i t h -

on the boundary.

the d i f f e r e n c e between s t r i c t l y - s t r a t i f i e d

and

m a n i f o l d s by means of a simple example.

Fig. In F i g u r e 7.1

1 M

point

p

1 M

stratified

is

so t h a t on

U i is a m a n i f o l d and

We i l l u s t r a t e

is

More

connected and

same-dimensional s t r a t a f o r each Thus

m a n i f o l d , we

m a n i f o l d s , meaning,

open sets

X

(~)

sphere.

Having d e f i n e d the n o t i o n of

identical.

where the

is

a stratified

( o f dimension

O)

7.1 m a n i f o l d where the s t r a t a are the

and the e n t i r e c i r c l e

( o f dimension 1).

because we have a c o v e r i n g by two c h a r t s as in

F i g u r e 7.2

125

7.11

P

Fig.

LJ

7.1a

Fig.

both of which are s t r i c t l y strictly

stratified

in

However,

stratified.

t h a t the i n t e r i o r

of

7.2b 1 M

itself

is

not

the 1 - d i m e n s i o n a l stratum

which has no boundary {as a m a n i f o l d ) , c o n t a i n s the O-dimensional stratum. It

is

c l e a r t h a t even in

stratified

non-strictly

the case of a m a n i f o l d which is

the l i n k ~ ( X )

of a st r a t u m i s ,

equivalence, a well-defined strictly-stratified t h e r e f o r e now narrow the class of

stratified

c o n s i d e r a t i o n by p l a c i n g r e s t r i c t i o n s be a l l o w e d .

In p a r t i c u l a r ,

linkwise simplicial the sense of (for

all

(abbreviated

strictly

strata

we s h a l l

stratified

up to

manifold.

m a n i f o l d s under

on the l i n k s ~ ( X ) call

LS) i f f

We may

a stratified ~ (X)

is

which w i l l

manifold

e q u i v a l e n t , in

m a n i f o l d s , to a t r i a n g u l a t e d sphere

X.)

The c o n s t r u c t i o n about to be d e s c r i b e d w i l l

be of

tance in

the subsequent s e c t i o n s . Consider an LS n manifold M We s h a l l d e s c r i b e a decomposition of

some impor-

stratified n M into

codimension-O s u b m a n i f o l d s , each of which " t h i c k e n s " a p a r t i c u l a r stratum. X

We s h a l l

denote such a d e c o m p o s i t i o n by ~ =

{M(X)}

where

ranges over the s t r a t a . First

j-1. in

of a l l ,

let

Z

denote a t r i a n g u l a t e d sphere of dimension

Assume, f o r convenience sake, t h a t Z is a d m i s s i b l y embedded j+k-1 S We s h a l l decompose c~ i n t o codimension-O s u b m a n i f o l d s ,

namely one

N

f o r each s i m p l e x

o

c o r r e s p o n d i n g to the cone p o i n t . cone

c'ZC_ c Z.

ductively that

This

For the r e m a i n i n g N

of

~, last

N's,

126

N

is merely the s m a l l e r

l.e.,

has been d e f i n e d f o r a l l

0

plus a s i n g l e

the ~

N ,

assume,

of dimension

in-

7.12 i

< j-1.

first cZ

N

considering the copy (i.e.

simplices in

We must define

~';

~' ~

= @N ~ )" of

let

~.

P

=

of

o'

~

i-dimensional Z

PJ';

Rj+k

U

be the copy of

denote the

IY ~ U}.

and we do so by

let

0

Then

N

o

Rj+k ~

~x{1} in

in

(j-i-1)-cell

be the image of

finally,

a,

parameterized as

denote the ( i + l ) - p l a n e in

p r o j e c t i o n onto c P®P

Z'

Let Let

and the o r i g i n and l e t

{(x,y)

for

~'

for

dual to

~'

determined by

under orthogonal

denote the space is p r e c i s e l y defined as the

closure of the f o l l o w i n g set: 0 h c ( s t ~) -

U N -N dim T

The f o l l o w i n g Figure 7.3 i l l u s t r a t e s t h i s decomposition when j

= 2, k = 0

sides

and

Z

a t r i a n g l e with v e r t i c e s

Sl, s2, s 3.

vI ,

v2, v3

and

v,

g~ Fig.

7.3

We remark t h a t the assumption t h a t l a t e d as a subsphere of some standard z a t i o n of

N

N

~J

a fact that,

Z

was admissib|y t r i a n g u -

sJ+k-1

makes the c h a r a c t e r i -

appear to depend on the embedding.

However, i t

is

O

up to i s o t o p y , the s i m p l i c i a l s t r u c t u r e of

Z

itself

determines t h i s decomposition. Suppose we are given an u n s t r a t i f i e d connected manifold a t r i a n g u l a t e d sphere

~n-r-1

s i m p l i c i a l automorphisms of

Z

Let

Aut(~)

and

n

127

an

r V

and

denote the group of Aut(~)

bundle over

r V

7.13 A s s o c i a t e d to

n

is

a bundle w i t h

fiber

c~,

in

the obvious way,

r and we s i m p l y denote t h i s as a t w i s t e d p r o d u c t V x cZ, the u n d e r l y t ing p r i n c i p a l bundle n being u n d e r s t o o d . Vx c~ is a s t r a t i f i e d t m a n i f o l d in a n a t u r a l way. The minimal stratum i s r r r V x * = V x * ~ V . The o t h e r s t r a t a may be d e s c r i b e d as f i b e r bundles t r (i.e., t w i s t e d p r o d u c t s ) over V . That i s , c o n s i d e r an a r b i t r a r y r v C V ;

point ~l(V,v)

on

given a simplex

~

coming

a

of

~,

from the holonomy

we then have an a c t i o n of ~1(V,v)

+ Aut(~)

of

the

o

bundle

n.

Let

0(o)

action,

and c o n s i d e r

be the o r b i t cO(o)-*

r V x c Z i s of the form r we have a neighborhood Moreover, g i v e n the a c t i o n of

o

in

Xl(V,v)

of

the subspace

= X C_ c~.

A typical

o

under t h i s

stratum of

o

r V x X . F u r t h e r given N C__ cZ, we see t h a t to * r r V x N of V , which we denote N(V). t* Z, let P(o) denote the o r b i t of N under o and i t

becomes c l e a r t h a t we may d e f i n e a

r subtwisted product V x P(o) t we take a t y p i c a l stratum of denote i t any

o

by

X,

such t h a t

r of V x 9. As a m a t t e r of n o t a t i o n , i f t '~ r r V x c~ (Other than V itself) and t r then we s h a l l use N(X) to denote V x P(o) for t X = X (N(X) i s , of course, i n d e p e n d e n t of the O

choice of

o).

We now d e s c r i b e a way o f

building,

by s t a g e s , more c o m p l i c a t e d r stratified m a n i f o l d s from p i e c e s of the form V x c~, and, i n c i d e n t t a l l y , o b t a i n i n g a c e r t a i n decomposition ~ = {M(X)} of the f i n i s h e d product. The

0

spheres

th

Z.

stage i n v o l v e s t a k i n g and forming the d i s j o i n t

various triangulated union

(J/

I

stratified

manifold

(with

boundary).

have a d e c o m p o s i t i o n of

Mo

one f o r

X

form of

*

the

c~i)

each s t r a t u m o f (for Zi ).

or

cZ.,

i

N .

one of

the

T h e n M(X) (Note t h a t

into

Call

this

(n-1)-

regarded as a

I

manifold

Mo.

codimension-O s u b m a n i f o l d s

by n o t i n g t h a t any such s t r a t u m i s

cZ.) or co-, i i s of the form Mo = 0

is

O

128

(for N,

o (in

We Mo(X), of

the

a s i m p l e x o f one the a p p r o p r i a t e

also a p o s s i b i l i t y . )

7.14 1 The n e x t stage i n v o l v e s t a k i n g 1 - m a n i f o l d s V and t w i s t e d 1 i products V × c~. for corresponding t r i a n g u l a t e d (n-2)-spheres i t i Zi" The d i s j o i n t union of these t w i s t e d p r o d u c t s is a s t r a t i f i e d m a n i f o l d , and, as we have seen, f o r each s t r a t u m have a m a n i f o l d

N(X),

X

of t h i s

union we

Now suppose we are given some p o i n t s in

B(UV ) ( c a l l t h i s set D). T h e n the r e s t r i c t i o n of the v a r i o u s i t w i s t e d p r o d u c t s to the p o i n t s of D gives a s t r a t i f i e d (n-1)-manifold

A.

For each s t r a t u m

manifold

N(Y),

such t h a t ii)

i}

of

A

we have a c o - d i m e n s i o n

given by the unique component X

i s a s t r a t u m of

M(Y) ~ Y ¢ ~8~.

M(Y')

of

with

aM . Now l e t o homeomorphism of A into

tion-preserving

PL

@(M(Y)) C M(Y')

whenever

@(M(p))

~VixtcZi

M(Y)

of

= M(@(p)).

sub-

M(X) ~

A

BM P o be a s t r a t i f i c a -

¢

~M o Moreover, i f

@ ( y ) C y'

0

X ~ A ~Y;

By the same token f o r every s t r a t u m

we have a s u b m a n i f o l d

posit that

Y

Y'

of

so t h a t p ~ D,

then we

We then may form the union

i

M L) ( U V × c~ ) = M which i s , o ~ it i 1 manifold. That i s , the s t r a t a of

M 1 such t h a t the union i s

k

and X

(_JVixtcZi is

in an obvious way, a s t r a t i f i e d

connected:

If

s t r a t a of

N o such a s t r a t u m

X = L) Y set N(x) = U M ( Y ). we thus r r o b t a i n a d e c o m p o s i t i o n of M i n t o codimension-O s u b m a n i f o l d s . 1 th In g e n e r a l , we proceed i n d u c t i v e l y and assume t h a t a t the r stage,

the union of

are unions of

r

< n,

strata

we have a s t r a t i f i e d

manifold

M and a decomposir t i o n i n the form {M(X)}, X r a n g i n g among the s t r a t a of M . Supr pose we now take a d i s j o i n t union L~V ri + l x t c are where the ~i

Zi

triangulated

(n-r-2)-spheres.

As b e f o r e , l e t

D

be a codimension-O

s u b m a n i f o l d of

U V and A the union of the r e s t r i c t i o n s of the i v a r i o u s t w i s t e d p r o d u c t s to the components of D. Just as in the

b u i l d i n g of s t r a t u m of ¢

taking

M1,

we note the d e c o m p o s i t i o n

{M(Y)}

of

BMr,

Y

a

@M . We c o n s i d e r a s t r a t u m - p r e s e r v l n g PL homeomorphism r t ) with A into aM , and form M = M U (tJVixtc~i), r r+l r

129

7.15 its

stratification and d e c o m p o s i t i o n . We may c o n t i n u e t h i s th up to the n stage. ( A l t h o u g h we i n c l u d e the p o s s i b i l i t y say, the f i r s t of

the n u l l

p

s t a g e s , and the l a s t

set.)

Note t h a t ,

if

q

process that,

stages i n v o l v e a d j u n c t i o n

we c o n s i d e r the s t r a t i f i c a t i o n

of

M

= M and decomposition { M ( X ) ) } which has been o b t a i n e d , and set n X = X ~ M(X), i t is s e e n t h a t X is a codimension-O submanifold o o of X such that int X is a s m a l l e r copy of i n t X. Moreover, o M(X) i s a d i s c - b u n d l e over X w i t h s t r u c t u r e group equal to the o s i m p l i c i a l automorphisms of the l i n k of X. We now put f o r w a r d , w i t h o u t d e t a i l e d p r o o f , LS

stratified

above. of

m a n i f o l d may be o b t a i n e d by the procedure d e s c r i b e d

(N.B.:

we do not e x c l u d e the case where an i n i t i a l

s t a g e s , say

0

through

successive a d j u n c t i o n s of

r,

itself,

That i s ,

set.

of dimension

it

is

By way of f i x i n g

involve

r+l.)

It

f o l l o w s t h a t any

{M(X)}.

We put f o r t h

unique, up to ambient

e s s e n t i a l l y determined by the s t r a t i f i c a t i o n

We l e a v e v e r i f i c a t i o n

straightforward, albeit

i.e.

sequence

This corresponds to the case

decomposition is

and not by choices to be made in

procedure.

are t r i v i a l ,

m a n i f o l d admits a decomposition

the f u r t h e r c l a i m t h a t t h i s isotopy.

r < n,

the n u l l

where the minimal s t r a t u m i s LS-stratified

the c l a i m t h a t any

of

the course of the i t e r a t i v e

these claims to the r e a d e r as a

somewhat t e d i o u s e x e r c i s e .

i d e a s , the r e a d e r may want to keep in mind, as

a m o t i v a t i n g example, the s p e c i a l case where the s t r a t i f i c a t i o n on n M c o m e s from a c o m b i n a t o r i a l t r i a n g u l a t i o n , i . e . the s t r a t a are the open s i m p l i c e s . slight

In t h i s

e l a b o r a t i o n of

w i t h one

the c o n s t r u c t i o n above amounts to a n the procedure f o r b u i l d i n g M as a handlebody

r - h a n d l e f o r each

r - s l m p l e x , then core d i s c

case,

Xo,

r-simplex.

In t h i s

case,

if

X

is

an

M ( X ) may be t h o u g h t of as the a c t u a l r - h a n d l e w i t h which is a closed

r-simplex

(i.e.

a " s h r i n k i n g " of

X). The f o l l o w i n g simple example may a l s o prove useful

130

as an a i d to

7.16 visualization.

Figure

tinguished points c o n s i s t s of

7.4

illustrates

( O - s t r a t a ) and arcs

connected 2 - m a n i f o l d s

figure

7.5,

7.4

corresponding s t r a t a

-

_

.

.

Fig.

LS-stratifled

d e s c r i b e a c e r t a i n map

u:

which p r e s e r v e s c l o s u r e s of u: i n t X + I n t X th o r stage we s h a l l th a t the 0 stage, the t r i v i a l

map

on the c l o s u r e o f

The remainder

the r e g i o n s bounded by d o t t e d l i n e s are the

and the i n t e r s e c t i o n s w i t h

Given the

(1-strata).

some d i s -

(2-s~rata).

Fig. In

a 2-manifold with

_

the

X . o

_

7.5 n

manifold

n n M + M ,

M ,

we are now going t o

homotopic t o

the i d e n t i t y ,

s t r a t a and has the p r o p e r t y t h a t

homeomorphically.

We proceed i n d u c t i v e l y ;

have d e s c r i b e d

on

we d e s c r i b e

M(X) + X.

M(X),

u

a t the

L_) M(X). F i r s t of a l l d~m X~r on each M(X), X a O - s t r a t u m , by

u

Now f o r

X

a l-stratum,

u

is

defined

X-X

and o b v i o u s l y extends to a map X + X o o which r e s t r i c t s t c a homeomorphism i n t X ÷ f n t X. This e x t e n s i o n o defines u on X. Moreover we have ~ p r o j e c t i o n 7: M(X) ÷ X , we o we d e f i n e u on M(X) by the (ompocitlor~ Fo~. In g e n e r a l , assume t h a t we wish to e x t e n d

u

u

has been d e f i n e d on

to a t y p i c a l

131

M(X),

where

X

L J M(Y); dim Y
PTI~T,

g e o m e t r i c subspaces

int

recalling

that,

by the d e f i n i t i o n

of

N ~ H = int ~ . We now use an i n d u c t i v e T T r procedure to b u i l d up a s t r a t i f i c a t i o n of H having one s t r a t u m X T r Let Aoc . . . c Ai ~ Ai +~i . . . CA n : H f o r each component T of 6~~T . be a f i l t r a t i o n

of

H

d e f i n e d by

A i

=

L_J ~ . dim T~cord

c o m p l e x , homotopy c l a s s e s o f

liftings

y

a r e in

I

i-1

correspondence with

homotopy c l a s s e s of

liftings

y

in

the

and

B'

is

diagram

y

t

.~COS

/0

~

cBr A

X ~ - ~ - - > ~ ° r d / o ~ AOrd

where

~'

is

the c o m p o s i t i o n o f

~

with

projection

pushdown o f

the

the O - e q u i v a r i a n t map ~. n n Now l e t M be an o r d e r e d t r i a n g u l a t e d m a n i f o l d . Immerse M n+k in R by a p i e c e w i s e smooth ( e . g . l i n e a r on s i m p l i c e s ) map thus n ~cord cord o b t a i n i n g a Gauss map , : M + / ~ n , k ~ Q ~ . I t i s e a s i l y seen that,

up to homotopy w i t h i n

angulation

(within

the

j~cord

PL

o r d e r i n g nor the c o d i m e n s i o n

, ¢

does n o t depend on the t r i n e q u i v a l e n c e class of M ) nor on t h e k.

176

8.17 n ord @': M + A

Now i f of

~

and

cos .

to

cBr A ,

to

Cl OS y

But since

n M

follows that

is

smoothable; in

induces a p a r t i c u l a r

lifting

bundle r e d u c t i o n of

TM

however, t h a t

of

have a l i f t i n g

~,

the g i v e n l i f t i n g

is

easily

n ord M ÷ A

~'

stable vector n smoothing of M . Note,

identified

up to homotopy

defined at

locally-ordered triangulated manifold. c o m e s from such a l i f t i n g ,

of

hence a s p e c i f i c

thus a p a r t i c u l a r

n ord @': M ÷ A

the s t a n d a r d map

fact

s e c t i o n , which depends merely on the l o c a l

n M

we s h a l l

has a v e c t o r bundle s t r u c t u r e n cord induces the s t a b l e t a n g e n t bundle of M from y it

¢

with

lifts

the b e g i n n i n g of t h i s n s t r u c t u r e of M as a

To see t h a t any smoothing of

we endow such a smoothing w i t h a

smooth t r i a n g u l a t i o n , l o c a l l y o r d e r t h a t t r i a n g u l a t i o n , and immerse n n+k M smoothly in R The Gauss map ~ i s then seen to have i t s image in ~ o s , cBr A

through clearly

and thus

@'

strictly

Thus we have a s t a n d a r d l i f t

the

~'

to

natural on

bundle A

ord A ,

8.11

ord

cord

structure

y

We do n o t

claim

but merely t h a t

Remark.

liftings

cBr A

which

8.1.

R e m a r k . The r e a d e r should note t h a t nowhere d i d

bundle to

of

in a unique way

induces the given smoothness s t r u c t u r e .

This completes the p r o o f of

8.10

factors

Note t h a t

that

over

.~cord

the

PL

is

we c l a i m t h a t

induced

bundle

data

from

a

prolong

the smoothing problem does!

we do not c l a i m a i - i

correspondence between

of

A cBr

If n

and smoothings of

liftings

of

ord

n M ; we merely a s s e r t t h a t

a necessary and s u f f i c i e n t fact,

g

g

condition for

classify

the e x i s t e n c e o f one i s

e x i s t e n c e of the o t h e r . n s t r u c t u r e s on M s o m e w h a tr i c h e r

177

In

8.18 than n M ,

mere

smoothings.

structures

concordant that

each

to

stratum

equivalent two

if

triangulation

triangulated n by L O S - s t r a t i f i c a t i o n s of M , n together with smoothings of M so

is

submanifold,

on the

and

Consider, n M

given

a smooth

only

if

LOS-stratifications

strata

are

smooth

the

are

two

the

given

of

liftings

in

two

are

through

the

ordered

Call

smoothings

concordant

submanifolds

Then homotopy c l a s s e s of

for

such

structures

concordant

a concordance

concordance

of

while

the

whose

smoothings.

the diagram

A cBr

n M are

in

i-1

g

ord > A

correspondence w i t h

e q u i v a l e n c e c l a s s e s of

such

structures.

8.12

Remark. cos

P

of

at

least,

÷

this

Consider the f i b e r cord fiber

cBr f A ~

of

We c l a i m t h a t , has

PL/O

ord A ,

on the l e v e l

as a summand.

of

i.e.

the f i b e r

homotopy groups

To see t h i s ,

consider

the diagram P

where Y1 that

y

classifies

the n a t u r a l t h e r e are

~

os

~

cord

)

PLIO

Y1 -

~

Y

the n a t u r a l

PL

80

8PL

v e c t o r - b u n d l e s t r u c t u r e of i n v e r s e maps

BO

s

t BPL

cord ~ ~/

178

cord y over cord)/~os

bundle y

cord We c l a i m

and

8.19

splitting

and

1"

Of course t h e r e is

a n a t u r a l map

BO +~2 os.c

which r e s u l t s from i d e n t i f y i n g G p o i n t w i s e , w i t h the O-skeleton ord n,k of ~ and hence, t o p o l o g i c a l l y , w i t h the imaqe of t h a t s k e l e t o n n,k cord

in ~

.

However, to d e f i n e

a somewhat d i f f e r e n t

map.

V,

bundle being approximated by PL

sal

stable

of

W.

manifold PL

map, s t a b i l i z e d ,

we must use

the u n i v e r s a l s t a b l e v e c t o r

Now t h i n k of

V

BPL

is

W c o m b i n a t o r i a l l y so t h a t

so t h a t

V

as approximated

Immerse

a submanifold of

V

is a subcomplex

W p i e c e w i s e - s m o o t h l y in

smoothly immersed. T h e n the Gauss /jord ~os g i v e s us a map W,V + , which we may take s

is

and

t.

It

follows that

x,(PL/O)

is a

~,P.

The reader may f i n d

it

instructive

to compare our r e s u l t w i t h

the approach of Cairns and Whitehead [C1, theory for

t,

as approximated by a h i g h -

We may assume t h a t

as an a p p r o x i m a t i o n of summand of

with

TV.

w i t h a smooth t r i a n g u l a t i o n . Euclidean space,

BO

to

W whose t a n g e n t bundle a p p r o xima t e s the u n i v e r -

bundle.

Triangulate

and extend i t

Think of

dimensional smooth m a n i f o l d

by a

s

combinatorial manifolds.

C2, C3; Whd] to smoothing

This approach, i t

will

be

r e c a l l e d , i n v o l v e s the idea of a t r a n s v e r s e f i e l d of k planes on a n n+k manifold M embedded in R The e x i s t e n c e of such a t r a n s v e r s e field

is

shown by C a i r n s , w i t h gaps r e p a i r e d by Whitehead, to n guarantee the e x i s t e n c e of a smooth s t r u c t u r e on M . The problem of n f i n d i n g such a f i e l d , when M i s l i n e a r l y embedded w i t h r e s p e c t {o some c o m b i n a t o r i a l t r i a n g u l a t i o n , extent. The hub of n n M , and st(o,M )

then a n a l y z e d to a c e r t a i n

t h i s a n a l y s i s is is

in

that if

"general p o s i t i o n "

t e r m i n o l o g y , the formal l i n k number of

is

L{~,M n) n

v e r t i c e s of ~k(~,M ) ) , n t r a n s v e r s e to M at a p o i n t s n o f l i n e a r embeddings c~k(~,M ) , *

o

i s a j - s i m p l e x of

(i.e.,

in our

has maximal s p a n equal to the

then the space of k - p l a n e s of

179

~ is n-j R ,*,

homeomorphic to the space d i v i d e d o u t by the

8.20 action

of

the g e n e r a l l i n e a r

group

latter

space up to homotopy, i s n n-j embeddings c~k(o,M , * ) + R n -j -i S ,

divided

Br(~k(~,M

n

out

by

the

GL(n-j;R).

Of c o u r s e ,

the same as the space o f which t a k e v e r t i c e s

action

of

O(n-j).

of

Call

this

linear n ~k(~,M )

this

to

space

).

In our approach to role,

that

s m o o t h i n g , the same space o b v i o u s l y p l a y s a ord given a c e l l e of A , c o r r e s p o n d i n g t o the

is

o r d e r e d sphere

Zj - l ,

its

i n v e r s e image i n

j-1 bundle o v e r the space Br(~ a transverse field locally

h

n M

f

t h e o r e m , 8.1 o f not for

in

Our r e s u l t of

the spaces

the case f o r

-1

for

~

A

r

ord

each dual

cell

o*

of

Mn.

a homotopy s e c t i o n ,

Our and t h u s ,

transverse field.

Br(Z).

Our r e s u l t

renewed i n t e r e s t

in

the t o p o l o g y

R e c e n t l y , D. Henderson has a n a l y z e d c o m p l e t e l y

of d i m e n s i o n 2 [ H e ] . also

T e l e m a n n ' s paper [ T ] In t h i s

g(o*)

g

c8

would seem to i n v i t e

Z

j-disc

the diagram

c o u r s e , m e r e l y asks f o r

a specific

a trivial

In f a c t , i t may e a s i l y be seen t h a t n n+k n t o the t r i a n g u l a t e d M ~ R yields, if M be

ordered, a section

h(~*)

is

).

A

so t h a t

AcBr

invites

comparison w i t h

on t h e " d i f f e r e n t i a l

geometry" of

p a r t of PL m a n i f o l d s .

Telemann c h a r a c t e r i z e s g e o m e t r i c a l l y the PL/O-bundle n o v e r a PL m a n i f o l d M , s e c t i o n s of which are i n i - i c o r r e s p o n d n ence w i t h smoothings of M . His approach seems i n t e r e s t i n g l y analogous

secion,

the f i r s t

to

the

one

adopted

here.

180

9.1 9.

E q u i v a r i a n t Piecewise D i f f e r e n t i a b l e

As we have seen in ally

on

Rn+k ,

§5,

there is

if

E

is a f i n i t e

an induced a c t i o n of

Immersions

group a c t i n g o r t h o g o n n,k " on ~ __

n

i m m e d i a t e l y e v i d e n t t h a t under the r e t o p o l o g i z a t i o n of p o i n t s e t which c o n v e r t s ~ to z~/c , ~ n ,k "~n ,k c o n t i n u o u s . Thus, i t is n a t u r a l to study porting locally

smooth

n-manifolds

remains n M sup-

~ - a c t i o n s w i t h a view towards d e v e l o p i n g some

r e s u l t s on necessary and s u f f i c i e n t w i s e - d i f f e r e n t i a b l e immersions in geometric c o n s t r a i n t s . analogous to those of

is

the u n d e r l y i n g

the a c t i o n of PL

It

conditions for e q u i v a r i a n t piecen+k R which r e s p e c t a d d i t i o n a l

These r e s u l t s w i l l

be in

l a r g e measure

§5.

By way of background, c o n s i d e r an immersion of the s o r t n co n t e m p l a t e d in §7. That i s , M is a smoothly L S - s t r a t i f l e d manin n+k f o l d , and f : M + R is an immersion s a t i s f y i n g the c o n d i t i o n s n+k e, B, y of §7. Now l e t us suppose t h a t ~ acts on R orthogonn a l l y and on M ( l o c a l l y - s m o o t h l y in the sense of Bredon [ B r ] ) so t h a t the a c t i o n is

a group of s e l f - e q u i v a l e n c e s from the p o i n t of

view of L S - s t r a t i f i e d m a n i f o l d s . This means t h a t i f p E ~ then n n p: M + M preserves the s t r a t i f i c a t i o n and i s a d i f f e o m o r p h i s m of each s t r a t u m to i t s is

image s t r a t u m .

t h a t t h e Gauss map

however, the s l i g h t l y

g(f):

The c o n c l u s i o n we o b v i o u s l y want

M n÷ ~ c

n,k subtle point that

speaking, w e l l - d e f i n e d .

It

is e q u i v a r i a n t . g(f)

is

not,

There i s strictly

is

dependent, be i t remembered, upon n choice of the d e c o m p o s i t i o n {MIX)} of M (X r a n g i n g over the n n s t r a t a ) , and the s t r a t u m p r e s e r v i n g map u: M ÷ M . However, i t e a s i l y be a s c e r t a i n e d t h a t the c o n s t r u c t i o n s of be d o n e so as to r e s p e c t the a c t i o n of and

pM(X) = M(pX)

for

p

~.

This,

~

(i.e.,

in t u r n ,

{MIX)} u

and

u

may can

is e q u i v a r i a n t

renders

g(f)

equivar~ant. Moreover, a f i n a l variant

{MIX)}

and

n i c e t y is u

differ

to note t h a t two choices of e q u i by e q u i v a r i a n t ambiant i s o t o p y , thus

181

9.2 the r e s u l t i n g Gauss maps f o r homotopic, b u t ,

in

fact,

f

are not o n l y e q u i v a r i a n t l y

made so by t h i s

isotopy.

On the bundle l e v e l , i t w i l l be a p p a r e n t t h a t the a c t i o n of c c Again, on ~ f u r t h e r extends to a continuous a c t i o n on ~ . n,k n,k c since the t o t a l space of y c o i n c i d e s on the set l e v e l w i t h t h a t n,k of Yn,k' the a c t i o n of q on the t o t a l space i s i m m e d i a t e l y s p e c i c fled. C o n t i n u i t y may be checked r o u t i n e l y . Thus y acquires a n,k c n-bundle s t r u c t u r e over the .q-space n,k o

9.1 P r o p o s i t i o n .

If

f:

n n,k M ~ R

is an e q u i v a r i a n t immersion ( w i t h k c r e s p e c t to the a c t i o n of ~, the bundle map TM + x covering n,k is e q u i v a r i a n t . the Gauss map g ( f ) ÷ ~ nc, k We omit the p r o o f . The problem which w i l l section is

occupy us f o r

the remainder of t h i s

t h a t of d e r i v i n g the analog of Theorem 5.2.

Let

H

be a

which i s i n v a r i a n t under the a c t i o n of geometric subspace of X~/c k; n, n ~. Let M be an open PL m a n i f o l d w i t h a l o c a l l y smooth a c t i o n by s a t i s f y i n g the B i e r s t o n e c o n d i t i o n . 9.2 Theorem.

n

If

h: M

+ H

i s an e q u i v a r i a n t map covered by a

~-bundle map then t h e r e i s an e q u i v a r i a n t so t h a t ,

w i t h r e s p e c t to t h i s

LS

stratification

of

n M

there e x i s t s a piecen n+k w i s e - d i f f e r e n t i a b l e and e q u i v a r i a n t immersion f : M + R (satisn c f y i n g (~) , (B) and ( y ) of §7) s u c h t h a t g(f): M ÷ ~ n , k has i t s image in

H

and

We s h a l l l i n e of

is

e q u i v a r i a n t l y homotopic to

g i v e a p r o o f which,

in i t s

h

in

H.

e s s e n t i a l s , f o l l o w s the o u t -

7.5 w i t h a p p r o p r i a t e m o d i f i c a t i o n s to deal w i t h the a c t i o n of

as needed. however t r i v i a l trivial

g(f)

stratification,

in

arguments.

There i s , in

however, one p o i n t in

the p r o o f of

the absence of a group a c t i o n , is

the p r e s e n t case. We are in

fact

This

7.5 which,

c l e a r l y not

has to do w i t h t r a n s v e r s a l i t y

referring

182

to t h a t s e c t i o n of

the p r o o f of

9.3 7.5 which, by way of p r e l i m i n a r y , induces an L S - s t r a t i f i e d s t r u c t u r e n on M . i t w i l l be r e c a l l e d t h a t t h i s argument, in t u r n , was e s s e n t i a l l y drawn from Lemma 7.4. the h e a r t of

the m a t t e r is

t h a t the map

v e r s e , s i m u l t a n e o u s l y to a l l h

(keeping

M

in

H).

Upon e x a m i n a t i o n , one sees t h a t

T 's

the

Our problem i s

in an e q u i v a r i a n t c o n t e x t .

h: M ÷ H

v i a a small d e f o r m a t i o n of to r e c a p i t u l a t e t h i s

argument

T h u s the e q u i v a r i a n t t r a n s v e r s a l i t y prob-

lem must be a n a l y z e d away b e f o r e the r e s t of

9.1

can be made t r a n s -

the p r o o f may proceed.

Equivarlant transversality The key r e s u l t we s h a l l

need i n v o l v e s f i n d i n g c o n d i t i o n s s u f f i -

c i e n t to a l l o w e q u i v a r i a n t t r a n s v e r s a l i t y arguments to go t h r o u g h . In our case, we s h a l l

be d e a l i n g w i t h maps e q u l v a r i a n t w i t h r e s p e c t

t o the a c t i o n of

( t h e domain being a

in

turn,

n,

contains a

PL

manifold).

The range,

~ - i n v a r i a n t subspace w i t h an i n v a r i e n t n e i g h b o r -

hood, and the aim w i l l

be to deform the map e q u i v a r i a n t l y so as to

become t r a n s v e r s e to the subspace. Let a

PL

B

be a

~-space and

disc bundle).

= p*(p), Now l e t

There is

and, since

M

be a

PL

~ - e q u i v a r l a n t map.

p

p:

E + B

a

~-bundle over

a n a t u r a l bundle over

is e q u i v a r i e n t , t h i s

m a n i f o l d on which Suppose f u r t h e r t h a t

H

f:

satisfies

(i.e.,

viz.

is also a

acts and

M

E,

B

R-bundle. M÷ E

a

the B i e r s t o n e

condition. 9.3 Lemma. Suppose F: TM ÷ ~ (~)~, variantly

o

where

deformable to

over, there w i l l Proof:

B

be a

: M ÷ E

is

~

~-bundle over

is a

covered by a

q-bundle map E;

then

f

some dimension

by i t s

f

: M + E with g t r a n s v e r s e to 1 -1 ~-bundle map T(g B) ~ aIB.

Assume, w i t h o u t loss of g e n e r a l i t y t h a t

~-manlfold (of original

f

g).

E.g.,

B

B.

is a

one m i g h t r e p l a c e the

e q u i v a r i a n t r e g u l a r neighborhood f o r

183

is e q u i -

some

More-

9.4 R-embedding in Also,

r e g a r d a map

Since

TM,

we may t h i n k of

the sphere of an o r t h o g o n a l r e p r e s e n t a t i o n of

in

f:

M + E

our case,

is

as a s e c t i o n

the sum

f*~ as a sub-PL-bundle o and we denote i t s f i b e r a t x by

TM,

of

Now, g i v e n any

T

( n o t a bundle map)

arising

¢:

from

TM + Tl~

= M x B C M x E = ~.

¢

~ = M x E + M.

e

x" M ÷ E,

f:

of

TM = f * ( ~ ( ~ ) 5 ) = f * = ( ~ ) f * ~ , o o o ~ ( e q u i v a r i a n t under ~)

covering

we say t h a t

~

T.

is

consider a

X'

nice

if

for

how t h i s

Given the is

T,

hand,

~,

"FE

projection ÷ ~

a natural into

i.e.

map

DT:

T~IT(M).

f

is

niceness of maps

x E M ,

and w i t h is

inde-

TM ÷ T~,

Clearly,

t r a n s v e r s e to ~

is

if

essentially D7

is

nice,

T

B.

an o p e n c o n d i t i o n .

In the

c o n s i d e r T ; we have a r - b u n d l e map TM + ~ ~)TE. o denote p u l l b a c k to ~, and note t h a t t h e r e i s a

~ ~'FE ÷ "[7[.

covering

T . o by the e x t e n s i o n of Phillips

is

~,

for

done.

T~(M)

t r a n s v e r s e to

case a t Let

there

i n c l u s i o n of

Note t h a t

is

map

every f i b e r

¢19

pendent of

PL

Let

i s t r a n s v e r s e to ~. This makes sense, s i n c e , X TE may be i d e n t i f i e d w i t h a small neighborhood of - f , Tx x such a neighborhood small enough, the c o n d i t i o n on ~19 x 9

q.

theory,

~

T h u s there is

a composite

@ : TM ÷ ~ ~'FI~ o our sense. Thus,

Note t h a t

¢ i s n i c e , in o Bierstones e q u i v a r i a n t version [ B i ] is

o

equivariantly

of

Gromov-

deformable to a s e c t i o n

T

1

such t h a t

D~ i s nice and the c o r r e s p o n d i n g map f : M + E has the 1 _11 required transversality property. Moreover, on f B = V the t a n 1 gent bundle i s g i v e n by the "complement" o f 8 a t each p o i n t x. x More f o r m a l l y i f clear that

the normal

t a n g e n t bundle TV

is

we c o n s i d e r

TV

identified

of

with

T M I V and

bundle of V.

But,

f*~IV 1

:

O

in

by t h i s f*~!V. o

184

81V TM

as a subspace, i t restricts

description,

on it

V is

is

t o the seen t h a t

9.5 9.4

Remark. There i s ,

as w e l l , a r e l a t i v e version of 9.3.

replace the hypotheses of 9.3 by the assumption t h a t

We

: M . E is o a l r e a d y transverse to B on the codimension-O submanifold M , and o t h a t M, M s a t i s f i e s the r e l a t i v e Bierstone c o n d i t i o n . In t h i s o case we obtain an e q u i v a r i a n t deformation, r e l M , to the desired o transverse r e g u l a r map f 1" The usefulness of 9.3 and 9.4 appears in the f i r s t stage of the proof of 9.2.

f

It will

be r e c a l l e d t h a t the analogous f i r s t n the proof of 7.5 i n v o l v e d showing t h a t the map h: M + H,

stage of (H

a geo-

metric subspace of ~ c ) induced an L S - s t r a t i f i c a t i o n of M. The n,k proof came about by appeal to t r a n s v e r s a l i t y , completely unproblema t i c a l in the case where no group action is i n v o l v e d .

We wish to

make the analogous argument in the presence of the a c t i o n of ~ on c M, .~ and i t s subspace H. n,k We merely sketch the proof. The i n d u c t i v e aspects merely f o l l o w the pattern set by the analogous step in 7.5 (which rests in turn on the argument of 7.4).

In the f o l l o w i n g , we understand t h a t

M may

be replaced, at need, by a m u l t i p l y e q u i v a r i a n t l y punctured v e r s i o n , since the argument o u t l i n e d w i l l , desired

produce the

L S - s t r a t i f i e d s t r u c t u r e on such a punctured m a n i f o l d .

then f i n d a

E-homeomorphic copy of the o r i g i n a l

a fortlori

M

We

i n s i d e which i s ,

L S - s t r a t i f i e d in the desired manner. Assume, t h e r e f o r e ,

that for a l l r < j

in the long run,

triangulation-classes T

we have ( f o r

-1_

(i)

h

(2)

h

-1

~T (~I) T

__~T

= ~T

~ H

~T

'

is a codimension

r

of formal l i n k s of dimension NT ~ H: submanifold.

is a bundle neighborhood with

h-1('~T)

+

l~T

h-~T

+ ~T

185

9.6

a bundle map ( i n f a c t , of n-bundles). Not l e t T be of dimension -1 j. We l e t M = h (~). O m i t t i n g arguments, we claim t h a t M may T T T be taken to be an i n v a r i a n t codimension-O submanifold of M. We may assume

h

is c

TMT ÷ Yn,k JNT

map is

the canonical

and where

{

is

bundle i t s e l f . it

near

transverse to ~ T is,

U ~ . dim U<j U

We note the bundle

in e f f e c t a map to the sum

~ ~ ~,

where

r - v e c t o r bundle over ~

r e s t r i c t e d to ~ T : ~ hH T T the p u l l b a c k under p r o j e c t i o n P: ~T +~T of t h i s

r4 e q u i v a r i a n t l y i f necessary, to make T s a t i s f y the Bierstone c o n d i t i o n , r e l a t i v e to a region near

U N dim U<j U

We puncture

and away from ~ i

.

The Lemma 9.3 and i t s

9.4 then apply and we make

h

transverse to ~ T

r e l a t i v e form

by an e q u i v a r i a n t

deformation i n s i d e

~ . The deformation i s constant o u t s i d e a T co-dimension-O submanifold of M . The l a s t step is to f u r t h e r T -i_ e q u i v a r i a n t l y deform h so t h a t a bundle neighborhood of h ~T in

MT

"expands" to f i l l

being pushed o u t s i d e of

all ~

T

of

9 ,T

with the remainder of

MT

w i t h o u t , however, i n t e r s e c t i n g

UN dim U<j U The remainder of the proof of 9.2 may now be c a r r i e d through c l o s e l y mimicking the proof of the n o n - e q u i v a r i a n t v e r s i o n . be r e c a l l e d t h a t there are two f u r t h e r p r i n c i p a l steps: stratification first

It

will

Smoothing the

and a c t u a l l y c o n s t r u c t i n g the d e s i r e d immersion.

The

is c a r r i e d out as in §7 using Lashof-Rothenberg e q u i v a r i a n t

smoothing theory [La-Ro] in place of Cairns-Hirsch since each i n v a r i a n t stratum of

M

with l i n k

T

produced by the argument j u s t above has

a tangent bundle induced from some ~ - v e c t o r bundle ( i . e . canonical

~ - v e c t o r bundle over ~PT).

from the

The second ~ay be done by

using the Bierstone r e s u l t in place of the : l i r s c h immersion theorem. Note t h a t to make these argumertts run, ure

M

we may have to f u r t h e r pur.ct--

e q u i v a r i a n t l y ( i n order to make each e q u i v a r i a n t stratum

186

9.7 s a t i s f y the Bierstone c o n d i t i o n r e l

a neighborhood of i t s

boundary.)

So we thereby immerse a m u l t i p l y e q u i v a r i a n t l y - p u n c t u r e d M. has been p o i n t e d out b e f o r e , a M

But, as

~-homeomorphic copy of the o r i g i n a l

may be found i n s i d e t h i s . Finally,

we note t h a t the main r e s u l t 9.2 of

t h i s s e c t i o n has

been s t a t e d f o r o p e n ~ - m a n i f o l d s ( r a t h e r than manifolds w i t h bounda r y ) s a t i s f y i n g the Bierstone c o n d i t i o n . tion will

A f u r t h e r technical condi-

a l l o w the extension of the r e s u l t to manifolds with bound-

ary ( e s s e n t i a l l y by a l l o w i n g the t r a n s v e r s a l i t y arguments to go through on the boundary).

We w i l l

not pursue t h i s

this point.

187

elaboration at

10.1 10.

Piecewise D i f f e r e n t i a b l e Immersions i n t o Riemannian Manifolds In §6 we considered how to c o n s t r u c t a complex __~n,k(W)

triangulated into

W,

In t h i s

(n+k)-manifold

W in order to study

by way of extending r e s u l t s on

PL

for a

PL

immersions n+k immersions i n t o R

s e c t i o n we perform an analogous extension f o r piecewise

smooth immersions. u l a t i o n of

It

is n a t u r a l t h a t the r o l e played by the t r i a n g -

W in §6 i s now assumed by a smooth Riemannian s t r u c t u r e

W.

on

We f i r s t

show how a space ~ c (W) may be b u i l t which plays a n,k r o l e analogous to / ~ (W) in the p r i o r study. n+k n,k With W , as i n d i c a t e d , a Riemannian m a n i f o l d , c o n s i d e r , f o r any p o i n t

w E W,

the tangent space T W with i t s m e t r i c . An n+k w T W with R a l l o w s us to consider formal l i n k s of

isometry of

w

dimension

(n,k; j),

formal l i n k U

is a

L

0 ( j

( n,

of dimension

j

in

such l i n k s of dimension

union of a l l

l i n k s of a l l

T W. That i s , a w T W is a p a i r (U,Z) where w an admissibly-triangulated

in

T W, and w ( j - l ) - s p h e r e in the u n i t sphere

of a l l

j+k-plane

associated to

Z

S . As we have seen in §7, the set U j has a n a t u r a l topology so t h a t the

dimensions becomes a s i m p l i c i a l space.

now t o p o l o g i z e the union of a l l

j - d i m e n s i o n a l l i n k s over a l l

This is done in the obvious way.

First

m e t r i c on

TW.

sion

j,

W n a t u r a l l y metrizes

We

w

W.

note t h a t the Riemannian Moreover, f o r a f i x e d dimen-

the c l a s s i c a l Grassmannian bundle

G .(W) = ~ G T W also acquires a n a t u r a l m e t r i c . j+k,n-j w j+k,n-j w L = (U L, ZL ),

L'

= ( U ' L , , ZL" )

s p e c t i v e l y to the p o i n t s s i m p l i c i a l isomorphism (1)

for all

(2)

U

is

be

w, w' ~ W. ¢: ~L

vertices c - c l o s e to

÷

v U'

j - d i m e n s i o n a l l i n k s associated r e T h e y are

Z' such that L of Z, Cv i s in

188

Let

~-close i f

c - c l o s e to

G (W). j+k,n-j

there i s a

v

in

TW.

10.2 As usual we form the set all

points

w E W,

for a simplex

U C~L, over a l l

identifying

CZL,o with i t s image under h(L,a)

~ c-.ZL. The image of

We topologize so that two points

C~L

x,y

L

is

~-close to

i s , as usual, denoted

are close i f

under the i d e n t i f i c a t i o n map of points t i v e l y where

l i n k s associated to

~, ~

K and

x, y

in

CZL, C~K

are

eL•

they are images, respec-

~-close in

TW.

Call the r e s u l t i n g space ~ nc, k ( W ) . Note the abvious fact that ~nn,~(W) =

~ .j~.c ( w ) w h e r e ~ c ( w ) i s a copy of X c arising w E W n,k n,k n,k n+k c under the isometry TwW ~ R In fact ~ (W) is a f i b e r bundle n,k c over W with f i b e r )-/ i

n,k

We n e x t _

over

c

construct the canonical

PL

n-disc bundle

t"

~ n ,k

(W).

y (W) n,k

I t is obvious that we want to have

yc (W) = ~ yc (x) where yc (x) is a copy of yc over /~n (x). n,k x n,k n,k n,k ,k However, for our purposes, i t is most useful to use the following c picture of the PL bundle structure of y (W). n,k Again, l e t T denote an isomorphism class of (j-1)-dimensional triangulated spheres. space whose points are of

W)

based a t

such that x.

Thus

We now use ~T(W)

to denote the topological

j-dimensional formal l i n k s

~L ~ T. (x)

We l e t --c~(x) is

a copy

of

L

(at a l l

points

be those l i n k s of --~T(W) d~Cf'n,k

thus n a t u r a l l y embedded in

(W). Recall also the space N the n%k T' image mod some i d e n t i f i c a t i o n on the sphere bundle boundary of the

natural

PL

j - d i s c bundle

P over O~DT. NT, i t w i l l be recalled T N (w) denote the is n a t u r a l l y included in --/~",k" Thus, i f we l e t T we may define N (W) as LJ N (w). corresponding copy in ~L~ (w), ,k T w~W T c I t w i l l be seen that Nl(W) is n a t u r a l l y the image in . ~ (W) of a n,k PL j - d i s c bundle over ~T(W). T h a t i s , i f we take L_~ c)Z L = P (W), with the appropriate topology we have ~ T L E~T(W T

189

10.3 embedded as the union of a l l of a

PL

j - d i s c bundle.

cone p o i n t s , making ~

NT(W)

is the image of

the O-section T P(W)T under the

CZL + e

for a l l L. L Recall now the natural map --~T ÷ Gn_j,k+ j of §7, defined by the i map L ~ U , L a j-dimensional l i n k . The n a t u r a l g e n e r a l i z a t i o n L shows t h a t there is a map ST:O~bPT(W)__ ÷ G (W). In analogy to 7 n-j,k+j we have q u o t i e n t map which takes

10.1 Lemma. The map

ST:~PT(W) + G (W) i s a f i b r a t i o n . n-j,k+j Note t h a t under the map PT(W) ÷ NT(W) two points may be i d e n t i -

f i e d only i f projection over PL

NT(W)

PT(W) ÷~T(W)

and

Yn,kI(NT(W).

n a t u r a l way to glue

It

~~ 6 B

of the canonical

C

T h u s there i s a

Therefore we may define a bundle

as the pullback of

k+j(W), is

YT

over the same. p o i n t of ~Z~T(W).

NT(W) +~T(W).

bundle

Gn_j

they l i e

is

over ~T(W)

where

YT ~

is the

the p u l l b a c k , via yT(W) ÷

( n - j ) - v e c t o r bundle over

Gn_j

k+j(W).

is s t r a i g h t f o r w a r d to show t h a t there is a

yT I

to

YT2

over

NTI(W) Q NT2(W) f o r two

t r i a n g u l a t i o n classes

T , T (of d i f f e r e n t dimensions). This 1 2 c YTiIN (x) = y (x)IN (x) so the g l u i n g i s w e l l Ti n,k Ti

f o l l o w s , since

defined by the manner in which the f o r each

YTiINTi(X)

x.

Consider, t h e r e f o r e , a smoothly f:

are glued together

n M ÷ W be an immersion

L S - s t r a t i f i e d manifold

n M .

piecewise smooth in the sense of ~7.

Let In

l i n e w i t h our usual procedure, we define a Gauss map .C c g(f): M +~ k(W) covered by a bundle map TM ÷ y (W). Much as in n, n,k §7, we consider the decomposition of M i n t o subsets M(X), one f o r each stratum that

g(f)

Oi,.~e~slon

X

of

M.

As in §7 we proceed i n a u c t i v e l y .

has been defined f o r < j,

~M(X)

we look at a t y p i c a l

with the understanding t h a t

M(X)

over a l l

strata

Assuming X

j - d i m e n s i o n a l stratum

of X,

i s parameterized as a t w i s t e d

190

10.4 h Z =~(X),

product

X x N,, o purpose, as X x a

0

x E X,

let

c Z

(the n o t a t i o n here i s as in §7).

y = f ( x ) C W. x × cZ

its

In f a c t ,

s t r u c t u r e of

Now, i f

T

as mapping image.

o r , more c o n v e n i e n t l y f o r our immediate

~,

to

The " d i f f e r e n t i a l "

TyW

this

and i s ,

df

in f a c t ,

a

may be understood

PL

homeomorphismto

image, t o g e t h e r w i t h the given s i m p l i c i a l

d e f i n e s a formal l i n k

of dimension

n-j

in

T W, Y which we t h i n k of as a p o i n t in ~T(Y),To = [ ~ ] . By J

denoted

L(x,f),

t h i s p o i n t w i s e d e f i n i t i o n we o b t a i n a continuous map

h: X + C~XT (w).

By composing with the map U: X ÷ X we o b t a i n h o u on i n t X . o o Recall t h a t ul i n t X is covered by an obvious bundle map from the o ° normal bundle of int X to the normal bundle of X. Since h i s o o covered by a bundle map from the normal bundle of X to the canonical bundle o b t a i n a map extension of

PT(W) over ~T(W)

in a canonical way, we

int

X x c Z + PT(W) + NT(W), o T g(f) to M{X).

which defines the needed

c TM + ¥n,lW} over

D e f i n i t i o n of the covering bundle map

also done by g e n e r a l i z i n g m i l d l y the techniques of §7. in t h i s map

r e g a r d , t h a t on

TX +~ vX o o

X o ÷ nT (~)PT(W)

g(f)

Be i t

is

noted,

t h i s bundle map s p l i t s as a d i r e c t sum where

( j - v e c t o r ) bundle induced by

sT :

n denotes the canonical T TXo ÷ nT T(W) . Gj , n + k - j (W).

is

a v e c t o r - b u n d l e map, with r e s p e c t to the v e c t o r bundle s t r u c t u r e on TX o

coming from the given smooth s t r u c t u r e on the stratum

X.

The next order of business i s to g e n e r a l i z e the notion of geometric subspace to the present c o n t e x t . batim from the corresponding d e f i n i t i o n ~

10.2

Definition.

The subspace

in

We adapt i t

nearly ver-

§7.

C

H cj~rn,k(W)

i s said to be geometric

under the f o l l o w i n g c o n d i t i o n s o

(1)

If

H ~ N (W) ~ ~Qv then t h i s i n t e r s e c t i o n i s an open disc T bundle ( w i t h f i b e r cT - T) over H O ~ ( W ) . T (2)

Hn~T(W)

i s open in ~ T

191

and i t s

i m a g e ST(H n ~ ( w ) )

is

10.5 open in

~T(W)

G (W) n-j,j+k

+ G n-j,j+k (3)

Here,

the l i g h t

§7,

s

and

T

is

the n a t u r a l map

is

a fibration.

the most n a t u r a l examples of g e o m e t r i c subspaces, in

l e c t i o n of e q u i v a l e n c e classes of

~ N (W) where ~ is T~y T t r i a n g u l a t i o n s where, i f

and

[ ~ k ( v , Z ) ] e~/'.

v

is

of

= dim T

(W).

s IH~ ~ T T

As in

j-1

10.1, are of

a v e r t e x of

the form

Z,

The main r e s u l t of t h i s 10.3

Theorem.

Let

H

then

some c o l ZE

T~If

s e c t i o n i m i t a t e s 7.5.

be a geometric subspace of ~ n ,k(W).

Let

Mn

be a

PL m a n i f o l d w i t h no closed components. Suppose t h a t the map n n c n h: M ÷ H is covered by a bundle map TM + y (W)IH. Then M adn,k mits a s t r a t i f i c a t i o n as an L S - s t r a t i f i e d manifold such t h a t , with r e s p e c t to t h i s

stratification, there is a piecewise d i f f e r e n t i a b l e n n+k n c immersion f : M + W so t h a t g(f): M ÷~ (W) has i t s image ,k in H and, moreover, g ( f } i s homotopic to h in H. There i s ,

as w e l l , a r e l a t i v e

v e r s i o n analogous to 7.6 whose

s t a t e m e n t we o m i t . The p r o o f of briefly

10.3 proceeds much in

indicate it

the manner of

7.5.

We

here, paying p a r t i c u l a r a t t e n t i o n to such m o d i f i -

c a t i o n s as may be needed. If is a H

we adopt the n o t a t i o n ~

PL

disc bundle over ~ T "

can be " s t r a t i f i e d " The f i r s t

: ~ ( W ) ~ H, then ~ = N (W) T "T T T As in the p r o o f of 7.5 we see t h a t

w i t h one s t r a t u m f o r each component of ~ T "

m o d i f i c a t i o n of

the given map

h

lies

in deforming i t

so as to be t r a n s v e r s e to t h i s s t r a t i f i c a t i o n . We t h e r e b y induce a n stratification on M wherein a t y p i c a l s t r a t u m Y is a component of

the i n v e r s e image of a s t r a t u m of Smoothing of

in

7.5 w i t h l i t t l e

this

stratification

modification.

192

~. a l s o f o l l o w s the model se t down

10.6 The main i d e a , t h e n , i s t o t a k e the g i v e n smooth LS s t r a t i f i n c a t i o n of M and c o n s t r u c t , w i t h r e s p e c t t h e r e t o , an immersion n n+k f: M + W which i s p i e c e w i s e - d i f f e r e n t i a b l e and whose Gauss map g(f)

is

homotopic t o

Note t h a t ,

h

given

in

h,

H. n n+k : M + W o W, and thus f is 0 note that for f a

there is

an o b v i o u s map,

f

s i m p l y because ~ (W) is a f i b r e bundle over h " v n _, k_ Dr01" g i v e n by M--) H c~ c (W) ~ W. (In passing n,k piecewise-differentiable immersion, f is recovered

from

map i n

hand,

precisely

this

way.)

s i o n t o be produced s h a l l As u s u a l ,

of

M.

Assume,

t h e case a t . o Assume t h a t

we proceed i n d u c t i v e l y .

neighborhood is i-stratum;

in

be h o m o t o p i c t o

s i o n on a n e i g h b o r h o o d o f

strata

Clearly,

= M

furthermore, that

the Gauss map o f

Gauss

the immer-

f

(i-i)

U M(Z) dim Z

its

h

,

f

i s an immero ranging over the

Z

restricted

the i m m e r s i o n .

to

Now l e t

this Y

be an

clearly,

our goal i s t o deform f , rel a slightly (i-I) o s m a l l e r neighborhood of M , so as to make i t an immersion on a neighborhood of at

M(i-1)U

M(Y).

t h e same t i m e be h o m o t o p i c t o

hood, M(Y)

Moreover, i t is

will

in

h

succession,

the i n d u c t i v e

on a l l

the m o d i f i c a t i o n for

each

have c o m p l e t e d the i n d u c t i v e

t h e same s m a l l e r n e i g h b o r -

the d e f o r m a t i o n of

h

near

of

of

M.

h

we then append t h e

and o f

i-stratum step,

If

Y,

i.e.,

f

o we s h a l l

replaced

can be c a r r i e d see t h a t

(i-i)

M

by

we

(i)

M

hypothesis.

To proceed t o

the d e t a i l s

i-dimensional stratum which

rel

the immersion w i l l

s t r a t u m p r e s e r v i n g , and thus e x t e n d a b l e to a s t r a t u m -

observation that out,

h

be seen t h a t

p r e s e r v i n g d e f o r m a t i o n of

in

The Gauss map o f

Y.

Let

of

this

c o n s i d e r the ( i 1) be the n e i g h b o r h o o d o f M on

N

inductive

step,

f

is a piecewise-smooth immersion; l e t N' be a s l i g h t l y o s m a l l e r neighborhood with ~' N. (We may t a k e N, N' t o be manifolds-with-boundary if

we l i k e . )

193

Let

Y , Y 1 2

be

Y ~ N',

Y~ N

10.7 respectively, Y . 1

and we may assume t h a t

N o w , over

Y C Y o

we have the

YI' = YI ~ Yo' Y'2 = Y2 ~ Yo' ity

Y-Y

C Y is bounded away from 2 o PL-disc bundle M(Y). If

we may assume,

w i t h o u t l o s s of g e n e r a l -

that

M(Y)IY' 1 I choice of N, N ).

N, M ( Y ) I Y ' c N'. ( T h i s may i n v o l v e m o d i f y i n g the 2 | T h u s we o b t a i n an immersion of M(Y)iY . 1 We want now to make an assumption on t h i s immersion. The s t a t e -

ment i n v o l v e s r e c a l l i n g

that,

if

Y

has

T

as l i n k ,

then

h:

Y ÷3_ and, moreover, the map of the t a n g e n t bundle o c TM + y (W) i s the " d i r e c t sum" of two maps, v i z : n,k (I)

TY + ~, where { i s the v e c t o r bundle over o from the c a n o n i c a l map ~P_ + G (W) i-ih+k+i (2) group)

The map of

PL

~_ I

coming

disc-bundles (with discrete structure

M(Y) ÷ ~ ( T ) .

Thus,

if

Y E Y' we have a f i b e r Dy of M(Y) over y, and 1' i t is i d e n t i f i e d w i t h an (n-i)-disc D' in the f i b e r d i s c of c h(y) (W) over h(y). But, s i n c e , i f w = f y = p r o j h(y) ~ W, the n,k c o f i b e r of Yn,k over h(y) i s viewed as a subspace of TwW, we have essentially in

identified

Dy

itself

w i t h a s u b - d i s c of

the u n i t d i s c

T W. W

Now we may c l a r i f y

the assumption we wish to make:

t

Y E Y sufficiently f a r from Y {i.e., c l o s e to 1 2 the immersion f IDy c o i n c i d e s w i t h the c o m p o s i t i o n o exp ' Oy ~ Dh{y ) C_ T WW

it

must be e x p l a i n e d t h a t

n e n t i a l map f o r

- ~

and

the Riemannian s t r u c t u r e on

t h a t rays of l e n g t h We c l a i m ,

> 0

s

is

that l

for

Here,

It

j

Y - Y ) o 1

W

exp W,

denotes the expor e p a r a m e t e r i z e d so

in

T W go onto geodesics of l e n g t h W w i t h o u t d e t a i l e d p r o o f , t h a t wc may modify ~

~.s. and

f 0

on

N,

rel

N'

so t h a t t h i s

assumption i s

194

realized.

The p r o o f is

10.8 by use of the standard t u b u l a r neighborhood theorem f o r the smoothlyI

embedded m a n i f o l d

Y . 1 Proceeding, t h e r e f o r e , on the basis of t h i s assumption, the

remainder of the proof of 10.3 f o l l o w s f a i r l y r a p i d l y . F i r s t of a l l , h we note the map Y ~ ' ~ ' T ÷ G i , n + k - i ( W ) ' which i s covered by a v e c t o r bundle map from assumption on

TY M

to the

i-flag

bundle of

immersion theorem [ H i ] , Y2'

( T h e non-closed

i s used, as in §7, to produce t h i s bundle map along

with the piecewise-smooth s t r a t i f i c a t i o n

of

W.

we may deform

to an immersion

fl"

of f

o

M). on

Y,

By the Hirsch rel

a neighborhood

Moreover, t h i s deformation i s covered

by a deformation of the bundle map from

TY

to the canonical i - p l a n e

bundle over

G (W), such t h a t the t e r m i n a l stage of the deformi,h+k-i a t i o n c o i n c i d e s with the map a r i s i n g from the d i f f e r e n t i a l of the smooth immersion.

A d d i t i o n a l l y , we may, by v i r t u e of G r o m o v - P h i l l i p s

theory [P] i n s u r e , as w e l l , t h a t the e n t i r e deformation of the map Y ÷ G (W) stays w i t h i n the open set im ( ~ ) . Since, by the o i,n+k-i -'T d e f i n i t i o n of geometric subspace, ~ T i s f i b e r e d over i t s image in Gi,n+k_i(W),

we o b t a i n a deformation of

by a deformation of

hiM(Y) ÷ ~ ( T ) ,

stage of t h i s d e f o r m a t i o n on both

hlYo +-~-'-'-I

and l e t Y

and

h

i M(Y).

Now cover t h i s

denote the t e r m i n a l For each

Y

in

Y - Y , we have, as we have seen, an i d e n t i f i c a t i o n of Dy w i t h o 2 D' and thus with a disc in T (W). Composing with exp hl(Y) fl(y) c (c

v a r y i n g c o n t i n u o u s l y with

extending the given one on this still

NI;

y)

we o b t a i n an immersion on

M(Y)

an easy argument s u f f i c e s to extend

f u r t h e r to an open neighborhood of

N' U M(Y).

obvious from the c o n s t r u c t i o n t h a t the Gauss map of t h i s

It

is

immersion i s

h

(i.e. h on N' and h on M(Y)) and i t i s t r i v i a l t h a t the 1 1 deformation of h to h (over an open neighborhood of N' U M(Y)) 1 extends to a stratum p r e s e r v i n g deformation of h on M. Thus,

the key i n d u c t i v e step is

195

concluded and, by v i r t u e of our

10.9 earlier

remarks, the main r e s u l t 10.3 has been proved, at l e a s t in

outline. We conclude t h i s of a l l , finite

it

s e c t i o n with some b r i e f g e n e r a l i z a t i o n s .

i s n a t u r a l to consider Riemannian manifolds

group

T

a c t i n g by i s o m e t r i e s .

In t h i s

First

W with a

case the obvious

e q u i v a r l a n t g e n e r a l i z a t i o n s of the d e f i n i t i o n s and r e s u l t s of t h i s s e c t i o n are e a s i l y a c c e s s i a l b e . i s o m e t r i c a c t i o n of

~

We see immediately t h a t given the

on

W the space ~ c (W) becomes a R n,k c space and the canonical bundle y (W) a bundle with R - a c t i o n .

nn, k Furthermore, given a m a n i f o l d

M

with

R-action and an e q u i v a r i a n t

piecewlse smooth immersion M + W, we o b v i o u s l y o b t a i n a Gauss map c M ÷~L~ (W) which i s E- e q u i v a r i a n t . Now consider geometric ~ n ,k subspaces H of --~'C~w) i n v a r i a n t with respect to R. We then have n, k an obvious g e n e r a l i z a t i o n of previous r e s u l t s : 10.4 Theorem.

n M

n - m a n i f o l d s a t i s f y i n g the Bierstone c c o n d i t i o n and l e t H be an i n v a r i a n t geometric subspace of ~ (w). n,k n Suppose t h a t there is a E - e q u i v a r i a n t map h: M + H covered by a n c n ~-bundle map TM + y I H, Then M admits an e q u i v a r i a n t n,k L S - s t r a t i f i c a t l o n such t h a t , with r e s p e c t to t h i s s t r a t i f i c a t i o n , n n+k there i s an e q u i v a r i a n t immersion f : M ÷ W whose e q u i v a r i a n t Gauss map in

g(f)

Let

be a

has image in

PL

H

and i s ,

in f a c t

~-homotopic to

h

H. The proof combines techniques of t h i s s e c t i o n with those of the

previous s e c t i o n §9 where e q u i v a r i a n t immersions i n t o l i n e a r a c t i o n s on Euclidean spaces were considered. A concluding remark concerns a f u r t h e r g e n e r a l i z a t i o n which w i l l not even s p e l l out in any d e t a i l . t a r g e t manifold

W,

but only a smoothly

Suffice i t

to say t h a t , f o r our

we need not take a smooth Ricmannian m a n i f o l d L-S

stratified

m a n i f o l d whose s t r a t a ~r~

equipped w i t h Riemannian m e t r i c s which make face i n c l u s i o n s

196

10.10 isometries.

It

is an exercise in combining the constructions and

techniques of t h i s section with those of §6 to formulate a precise c (W) d e f i n i t i o n of.4~n,k

in t h i s case, to specify the kinds of immer-

slons f o r which Gauss maps are defined, and to prove r e s u l t s analogous to 10.3, and, in the e q u i v a r i a n t case,

197

to 10.4.

A.1

APPENDIX:

Glossary of Important D e f i n i t i o n and Constructions REMARK

PAGE 1.3

l o c a l ordered formula and l o c a l formula f o r charact e r i s t i c classes.

1.7-9

s - c e l l complex; s-block bundle; s - c e l l ; ~ - c e l l set.

" ~ _ - c e l l sets" is a category e q u i v a l e n t to CW complexes but more useful in f a c i l l i t a t i n g c o n s i d e r a t i o n of ordered combinat o r i a l subdivision.

1.10

Qn; IQn]; ~n

The ~ - c e l l set ( t o g e t h e r with i t s geometric r e a l i z a t i o n and canonical s-block bundle) serving as u n i v e r s a l object f o r n-dimensional s-block bundles.

I . 13

~: T*

Gauss map c l a s s i f y i n g the tangent s-block bundle of a l o c a l l y ordered t r i a n g u l a t e d n - m a n i f o l d .

1.17

qA(n)', T: T* + qA(n)

c l a s s i f y i n g space, analogous to Qn, for A-homology s-block bundles; Gauss map f o r A-homology n-manifolds with l o c a l l y - o r d e r e d triangulation.

1.23

o(n)

CW complex of n-dimensional Brouwer s t a r s .

1.23

B(n)

s i m p l i c i a l space of Euclidean c o n f i g u r a t i o n s of Brouwer stars n a t u r a l l y mapping to e(n).

1.23

B(M); gM: Mn+ +e(n)

If Mn is a l o c a l l y - o r d e r e d Brouwer t r i a n g u l a t e d m a n i f o l d , B(M) is a c o n s t r u c t i o n analogous to B(n) over e ( n ) ; the Gauss map gM: Mn + en is covered by B(M) + B(n).

1.24

Nk

space of normal planes to the g e n e r a l - p o s i t i o n embedded Brouwer star K.

2.1

formal l i n k (n,k;j))

2.4

2.6

+

Q

n

(of dimension

~n,k

represents possible l o c a l geometry of a simplex-wise c o n v e x - l i n e a r immersion of an n-manifold in Rn+k near an ( n - j ) simplex. PL Grassmannian, with one j - c e l l f o r every formal l i n k of dimension ( n , k ; j ) . canonical Gauss map from an n-manifold M immersed in Rn+k.

g: M + ~ n , k

198

A.2 PAGE

REMARK

2.7-2.9

O, G: ~ n , k

2.9

Yn,k

+ Rn+k

tautological map of ~ n , k to Rn+k. canonical

3.1-3.3

n,k; Yn,k

3.5-3.6

A~n,k ( A ~ n , k )

Grassmannian for s i m p l i c i a l immersions of triangulated n-manfolds in triangulated Rn+k, together with canonical k-block bundle. Grassmannian for simplex-wise convex-linear immersions ( s i m p l i c i a l immersions) of A-homology n-manifolds in Rn+k ( t r i a n g u l a t e d ) . A n admits a canonical Ahomology block-bundle A~n,k.

s

3.8

4.1

6.2-6.3

PL n-bundle over ~ n , k.

Grassmannian for a r b i t r a r y simplex-wise convex-linear maps from triangulated n-manifolds to Rn+k, together with i t s canonical PL bundle.

,k , Yn,k

geometric subcomplex of ~ n , k

~

Grassmannian "bundle" for immersions of n-manifolds into the triangulated (n+k)-manifold W; analogous to the Grassmannian bundle Gn k(W) of a smooth (n+k)-mani~old.

n,k(W)

6.5

GW

tautological map from ~ n , k ( W ) into W.

6.5

Yn,k(W)

canonical Xn,k(W).

6.6-6.7

g ( f ) : Mn +/Z~n,k (W)

Gauss map corresponding an immersion f : Mn + W n+kt°.

PL n-bundle over

Grassmannian for piecewise-smooth immersions of triangulated manifolds and more general s t r a t i f i e d manifolds.

7273

c

PL n-bundle over

c

7.3-7.4

Yn,k

canonical

n,k-

7.s

-YT

Space of formal l i n k s with a given combinatorics T; ~ T is a subspace of n,k with a natural neighborhood NT, i t s e l f the image of a disk bundle PT-

199

A.3 PAGE 7.8-7.11

7.13

REMARK

strict stratification; s t r a t i f i c a t i o n ; linkwisesimplicial stratification

codimension-O decomposition of M corresponding to a s t r a t i f i c a by strata {X}.

{M(X)}

~'=

7.18

BLS

C l a s s i f y i n g space for l i n k w i s e s i m p l i c i a l s t r a t i f i e d structures on manifolds.

7.29

Gauss map of piecewised i f f e r e n t i a b l e immersion

map Mn + A , c n,k defined for a c l a s s of i m m e r s i o n s s a t i s f y i n g ~, 8, ¥ of page 7.29.

7.32

Geometric subspace of

c

double l i m i t

7.34 a n a l o g o u s to

L., C lim ~'#-n,k, n,k+~ BO.

7.35

ei

group o f e q u i v a l e n c e c l a s s e s LS s t r a t i f i e d i-spheres.

8.1

AOrd

classifying space f o r l o c a l l y ordered triangulated manifolds.

8.2

ABr

complex o f Brouwer s t r u c t u r e s on ordered triangulated spheres: [N8: ABr is s i m i l a r to e(n) of §1].

AcBr

retopologization

8.3 8.4

~st st n,k; ~ , k

/ / •n,ko r; ~X~Zn,k dc o r d

8.6

~

ABr.

subcomplex o f ~ n k arising from " s t r a i g h t " f~rmal links; together with its retopologization

8.5

of

of

as a subspace of 21 c n,k.

of o~r~ ~

slight generalization to take account of local ~ ings; together with r e t o p o l o g i z a -

t i o n analogous to

os ~,cos n,k ; /~ n,k

n,k.

subcomplex, a r i s i n g from

2,~ rd

" s t r a i g h t " l i n k s , of jk; together with i t s r e t o p o l o g i z a ~cst cord ~ c o s t i o n [Note: /~d-n,k, ~Jn,k ,/~n,k are used in the proof of Theorem 8.1 but are not themselves of primary i n t e r e s t . ]

200

A.4

PAGE 10.1-10.2

REMARK

~

c n,k - bundle over a smooth Riemannian m a n i f o l d W, t o g e t h e r with i t s associated canonical

c

Yn,k( W)

PL n-bundle, n,k(W) i s the t a r g e t f o r the Gauss map of a piecewise d i f f e r e n t i a b l e immersion i n t o W of an LS s t r a t i f i e d n-manifold.

201

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