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
f~
"S t~
DJ
O
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 .<~-Cell is defined by t a k i n g , as o b j e c t s , isomorphism classes of s - C e l l s and, as morphisms, face i n c l u s i o n s . An ~ - C e l l usual category
set is a c o n t r a v a r i a n t f u n c t o r from i - C e l l C
of sets.
[The reader may t h i n k of the notion of of the f a m i l i a r n o t i o n of Given an ~ - c e l l IQI
of
Q
Given
IAI
× Q(A)
as
to the
set
A-set ( i . e . , Q,
~-Cell
set as a v a r i a n t
s e m i - s i m p l i c i a l set.)J
we define the g e o m e t r i c r e a l i z a t i o n
follows:
an i s o m o r p h i s m = r(A),
where
class
A
of
s-Cells,
Q(A)
is
a set
with
consider the
discrete
topology
form the union
(-~ F(A) and then take the i d e n t i f i c a t i o n ACOb(~-Cell) space coming from i d e n t i f y i n g IBI × ( Q ( f ) ( x ) ) with I f i ( I B i ) × {x} f IAl x {x} whenever B ÷ A is face-morphism of s - C e l l s and x
Q(A).
This d e f i n e s
IQI.
[The reader may compare t h i s
with the standard procedure f o r
forming the geometric r e a l i z a t i o n of a A - s e t . ] Notice t h a t the standard model category f o r A-sets i s a f u l l subcategory of ~ V - C e l l .
I.e.
if
A
has as o b j e c t s the standard
ordered s i m p l i c e s of each dimension and, ds morphisms, order p r e s e r v ing face i n c l u s i o n s , then
AC~Z-Cell as a f u l l
subcategory.
There i s a n a t u r a l f u n c t o r from A-sets to J - C e l l A-set
Q, extend i t
to an % - C e l l
set by
s e t s ; given the
Q ( A ) :~8~ f o r
A~ i
19
Ob(A).
1.10 On the other hand, there is a f u n c t o r from ~ - C e l l A-sets s i n c e , given the ~ - C e l l s-cells.
Thus i t
is
the union of
e s s e n t i a l l y unique way initially
]QI
CW~
Q,
IQ]
is
the union of
(ordered) s i m p l i c e s and thus,
= IQA]
a r i s e s from a A - s e t ,
Let
set
sets to
for
we have
some A-set
QA"
in an
Thus, i f
Q
QA = Q"
denote the category whose o b j e c t s are j - C e l l
sets an~
whose morphisms are homotopy classes of maps of geometric r e a l i z a tions. 1.10 Lemma. CW~
is
n a t u r a l l y e q u i v a l e n t to the category
CW
of
CW-complexes and homotopy classes of maps.
Proof:
We c i t e
equivalent geometric
to
the
A-set
category
IQI
map f r o m
isomorphism
set,
and
isomorphism n.
It
is
make ~ n denotes
the K to
the
classes
of
that
obvious
Canonically,
IKI
over
defines
the
and t h u s
structure
on
map ( o f for
A
is
a natural
defining
regarding
a a
The c o m p o s i t e s bijections
on
Yn
over
Let
of
A
Qn. the
fiber
C
face
maps t o
set.
-cell
is n a t u r a l l y a s u b - s - c e l l complex of
bundle
Qn
over
K.
complexes). s e t y~F/.
Thus,
x + isomorphism class nf K).
foF any ~IA × {x}
Thus we have a
which we may t h i n k of as a map
i
:K + Q . ~
20
of
dimension
an ~ - c e l l
an s - b l o c k
some J - C e l l
n
set
[~nl.
(A x {x}
~
o f maps o f
thus
requisite
we have the assignment
÷
Q,
induce
x (~(A)
transformation 3
there
way o f
bundle
functar)
a canonical
naturally
to each s - C e l l
bundles
c o m p l e x and
I~(~I
set
obviously
s-block
restriction
is
above,
CW + C ~ .
assigns
s-block
be an s - c e l l
is
and m o r p h i s m s .
which
s-cell
CW
standard
we g e t a map
universal
set
define
By our
objects
~-Cell
clear
CW.
a contra-variant
Let We w i s h
of
cited
any ~ - C e l l
CW~ + CW + CW~
classes
that
and h o m o t o p y c l a s s e s
facts
for
to
fact
A-sets
By t h e
CW~
We now d e f i n e be t h e
of
~ IQAI
as an J - C e l l
CW + CW~ + CW
~n
well-known
realizations.
homeomorphism, natural
the
n
1.11 Now be
y
Yn
is
(up to
(A × { y }
is
defined over
Qn
isomorphism f o r
any
a sub-s-cell
Clearly
i
¥n
induces
over
Qn
b l o c k - b u n d l e , hence t h e r e i s map
c:
IQnl
[Remark:
c
is
1.11 is
~n(A), Q ). n from
~
A
any
A × {y}
S -cell.
Yn.
has the s t r u c t u r e
of an o r d i n a r y
a h o m o t o p i c a l l y unique c l a s s i f y i n g
a homotopy e q u i v a l e n c e , i . e .
not e s s e n t i a l
to
the p r o o f o f
IQnJ 1.6,
BPL(n).
but is
worth
passing.]
Proof:
Let
K
be a s i m p l i c i a l
concordance c l a s s e s of i-1
the b l o c k o v e r
÷ BPL(n).
1.11 P r o p o s i t i o n .
n o t i n g in
yE
complex of
: K + Qn
Forgetfully,
by l e t t i n g
complex.
We must show t h a t
n - d i m e n s i o n a l b l o c k bundles o v e r
correspondence with
[IKJ,
JQnl],
which w i l l
K
suffice
are in
t o prove
the p r o p o s i t i o n . Let
~
be a b l o c k bundle o v e r
we may t r i a n g u l a t e original
the t o t a l
s i m p l i c e s of
triangulation,
K
we make
induced by a map
space
of
E
fiber
so t h a t
are subcomplexes.
~
into
1~:Ka ÷ Qn
complex whose c e l l s
K
Ka
~a.
Thus
ordering,
so as to agree w i t h
either
~:Kb
end (making
a r e s u b d i v i s i o n s of
+ Qn
IKI
exLend~ng
n
into
an s - c e l l
I~I
over
L
s - c e l l - c o m p l e x maps f o r i~,
d e f i n e d up t o
i B.
Thus
homotopy.
i~ I.e.
is s-cell
the s i m p l i c e s of of
E,
E × I
:
Jl~l
complex
× I, and,
~
with suitable
21
on
whose c e l l s
a simplex of
K.)
IKI
+ IQnJ
[IKI,
is
well
BPL(n)] ÷
Thus
of
s - b l o c k - b u n d l e s , a map ~n:
we have a map
[IKJ, jQnJ].
L
by the c a n o n i c i t y
I~I:
K.
and
the g i v e n o r d e r e d t r i a n g u l a t i o n s
the v a r i o u s
we g e t an s - b l o c k bundle classifying
we may t r i a n g u l a t e
× I
~a
denotes a s u i t a b l e
M o r e o v e r , g i v e n a n o t h e r such o r d e r e d t r i a n g u l a t i o n map
Then
the b l o c k s o v e r t h e
are o r d e r e d s u b d i v i s i o n s of
resulting
n.
Upon o r d e r i n g t h i s
an s - b u n d l e
where
dimension
L + Qn
1.12 To show t h i s is a b i j e c t i o n we need the f o l l o w i n g p r o p o s i t i o n . A Qn denote the s - c e l l - s t r u c t u r e on Qn coming from i t s
Let
simplicial
structure.
1.12
Lemma. There i s an s - c e l l s t r u c t u r e A on one end and Qn on the o t h e r . Proof: inductive IQnl
We prove t h i s
h y p o t h e s i s is
( k)
x {0} ~
structure on IQn~ )'~
IQn
on IQnl × {1}.
ICI
o b v i o u s l y holds f o r a given
k,
and w i t h
We f u r t h e r
end a r e such t h a t
× I
which is
for
structure
the o r i g i n a l
the s i m p l i c i a l
× I
k = -I.
For a ( k + l ) - d i m e n s i o n a l its
an s - c e l l
assume t h a t
= ILl
we see t h a t
triangulate
t h e r e is
which agrees w i t h
× {0}
IQnl
by i n d u c t i o n on the s k e l e t a of
that
i × I
on
Qn.
some c e l l
C
s-cell
n o t in e i t h e r This
Assuming the h y p o t h e s i s t r u e ,
then,
cell
L
of
as w e l l
Qn,
of
for
k+1
consider
ILl
We t r i a n g u l a t e
We o r d e r t h e t r i a n g u l a t i o n
of
ILIxI
ILlxl
itself
consistent
with
× I;
we
of
as c b d y ( I L I x I ) . the e x i s t i n g
orderings, IQnl
and then o r d e r the t r i a n g u l a t i o n of (k+l) x {0} ~ IQn I × I consistently with all
for
as f o l l o w s .
boundary by t a k i n g the g i v e n t r i a n g u l a t i o n
I~I x l ~ I L I x { O}U I L I x { 1 } •
(k)
Qn
Qn-
we may show i t
L
Our
on
s u b d i v i s i o n of
the c e l l s
Qn
orderings. L×{O},
Thus,
~ x 1,
structures
on
as an s - c e l l ,
for
o
JxI
where
L×I
these sub-
has as k - c e l l s
a simplex of J
of
L,
of
and s u i t a b l e
are the k-1 c e l l s
of
its
boundary
s-ball
L.
This
proves
the lemma. Let the lemma.
R
d e n o t e the s - c e l l It
follows,
the e x i s t e n c e of
structure
tion), r:
the map
[IKI,
BPL(n)] + [ I K i ,
structure
on
~IQ~
we may e x t e n d the s - b l o c k
R + Qn
(extending
IQnl
(by an argument s i m i l a r
e x t e n d the u n d e r l y i n g b l o c k bundle of p u t any s - b l o c k
on
id
Qn)
Yn
(with
structure
classify
22
to
to
x I that
Qn]) ~
g u a r a n t e e d by which showed
that
over
R,
if
we
and i f
r e s p e c t to the t r i a n g u l a to a l l
this
of
R.
structure.
Let
we
1.13 NOW consider a map by a map
g: K + Q~,
f:IK[
+ [Qn[-
S i m p l i c i a l l y approximate
and then note t h a t
g*(~IQ~)
f
acquires an
s - b l o c k - b u n d l e s t r u c t u r e once s~me s u i t a b l e order on a s u b d i v i s i o n of K
has been chosen.
this
structure.
[KI ÷[Qn[. (i.e.,
Moreover,
But then
(rlQ~)°g
l(rlQ~)°g[
is
is
the c l a s s i f y i n g map f o r
homotopic to
f
as a map
So we have an s - b l o c k bundle over some s u b d i v i s i o n of
the s - b a l l s of the s - c e l l
complex are the s i m p l i c e s of the
s u b d i v i s i o n ) whose c l a s s i f y i n g map is
f
up to homotopy.
We then
may e a s i l y c o n s t r u c t by amalgamation an s - b l o c k - b u n d l e and over itself
(i.e.,
map is
homotopic to
[IKI, Yn
IQnl]
the s - b a l l s are the s i m p l i c e s of f
as w e l l .
i s onto.
must be i n j e c t i v e , The proof of
IQnl + BPL(n)
follows that
[IKI,
1.6 i t s e l f ,
we have seen how
TM
complex on
aM)).
a map
M
aM ~ ~,
Note that
M ÷ IQn{,
BPL(n)] ÷
exists classifying
BPL(n)] ÷ [ I K I ,
IQnl]
it
on the
PL
whose s - b a l l s are the c e l l s of
T*
T
T
manifold
acquires an s - b l o c k bundle s t r u c t u r e (over
d e f i n e d s - c e l l - c o m p l e x map
on
[[K[,
Qn"
angulated and ordered as subcomplexes of
point that for
whose c l a s s i f y i n g
immediately below, is q u i t e s t r a i g h t -
Given a l o c a l l y - o r d e r e d t r i a n g u l a t i o n
the s - c e l l
K
and the p r o p o s i t i o n i s proved.
f o r w a r d , given the c o n s t r u c t i o n of
Mn,
K)
This shows t h a t
Since a map
as a block bundle i t
K
T:T* ÷ Qn. is a c e l l
is well
T'.)
T*,
tri-
Thus we get a w e l l -
(Again, we note the t e c h n i c a l s t r u c t u r e on
Mo = M- ( c o l l a r
defined as a c e l l u l a r map, i . e . ,
as
is s p e c i f i e d p o i n t w i s e , given the l o c a l l y -
ordered t r i a n g u l a t i o n
T,
and not merely as a homotopy c l a s s .
we are j u s t i f i e d
in t h i n k i n g of
the geometry of
T),
T
Thus
as a "Gauss map" ( a p p r o p r i a t e to
r a t h e r than as a " c l a s s i f y i n g map" which merely
records tangent bundle i n f o r m a t i o n in the homotopy category. :low l e t
be 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 f o r PL i n-manifolds, i . e . c ~ H ( B ~ ( n ) ; G). T h e n C(yn) i s an element of i H (Qn; G)
c
and hence i s
represented by an i - d i m e n s i o n a l c e l l u l a r
23
1.14 (oriented) c(Mn).
But
defined
maps on
TI~
i F ~ C (Qn;
co-chain
= T
note
that the
T
G).
as a g l o b a l
cells
of
T*.
i T#? ~ H ( T * ; G )
Thus map,
That
is
is,
the
over
union o*,
of
locally-
the
is determined purely by the s - c l o c k bundle
is c l e a r t h a t , as an s - b l o c k bundle, t h i s
represents
map
TMI~*,
and i t
depends merely on the
o r d e r i n g of st(~,K). That i s , given an ordered co-dimension j n- j j -i n-star A *~ , we may r e g a r d i t as an o r d e r e d m a n i f o l d D, and hence
get
a classifying
An o r i e n t a t i o n
j-cell
A*
( zxn-i
i -1
,
*~
obviously
m
on
of
D
,~)
as
has
the
map Z
d:D*
is
dual to
hence
1.6
is
~
1.13
Remark.
ence
theorem.
= Poincare
to
an o r i e n t a t i o n
tantamount
d#r(z~*,O).
This
for
j
dual
= i,
of
~ *
~.
o
on t h e
we may define
locally-ordered
formula
property
represents
therefore
D*
n-j A , hence,
~(T) and
+ Qn,
the
= T#F
characteristic
class
c.
The p r o o f
of
complete. As n o t e d
before,
the
proof
of
1.6
is
purely
an e x i s t -
Construction would amount to c o n s t r u c t i o n of a
s p e c i f i c c o - c y c l e r e p r e s e n t a t i v e on the " u n i v e r s a l example"
Qn.
Before discussing other approaches to the problem of constructing local
formulae, l e t
us observe t h a t
1.6 and i t s
c o r o l l a r i e s may be extended to a somewhat wider c o n t e x t . we have r e s t r i c t e d our a t t e n t i o n to
PL
Heretofore,
manifolds and 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 s t h e r e o f , but note t h a t c h a r a c t e r i s t i c - c l a s s theory i s of i n t e r e s t as well
in studying homology man~folds (or r a t i o n a l
homology m a n i f o l d s ) . Recall t h a t an ( i n t e g r a l ) homology n - m a n i f o l d i s a s i m p l i c i a l complex such t h a t (i-I)-sphere.
~k(c n - i )
has the i n t e g r a l homology of an
Replacing the i n t e g e r s by some other u n i t a r y r i n g
of c o e f f i c i e n t s y i e l d s the d e f i n i t i o n of ( e . g . , r a t i o n a l homology m a n i f o l d s ) .
24
A
A
homology manifolds
The notion of
(A)-homology-
1.15 manifold with
boundary may a l s o r e a d i l y
be d e f i n e d i n
Given an A-homology m a n i f o l d s a dual generalizing That i s ,
the
ba *
~k(~).
since
its
of
b~
in
Of c o u r s e ,
boundary is
There i s manifolds.
the f i r s t
~
is
a natural
base space a r e " c e l l s "
in
merely a c e l l
we p o i n t
space
the c o n s t r u c t i o n
manifold.
is
defined
s u b d i v i s i o n of the
homology" sense
the " c e l l s "
"dual
cells"
this
kind of
in
AK
in
the
of an blocks
block-bundle
We use
t o denote the dimension
n.
an A-homology m a n i f o l d has a is
essentially
the same as f o r
one puts an A-homology b l o c k - b u n d l e s t r u c t u r e
r e g u l a r neighborhood of Mn
~
Bh (n) A A-homology b l o c k b u n d l e s of f i b e r
may a l s o be seen t h a t
manifold
PL
c o n d i t i o n holds f o r
[M-M].
Further,
i.e.,
out t h a t
and a s i m i l a r
space f o r
manifolds,
in
t h e same sense t h a t
classifying
tangent bundle;
barycentric
may then be seen t h a t
t h e o r y has a c l a s s i f y i n g
it
the c e l l
exists,
b l o c k bundle t h e o r y a s s o c i a t e d to A-homology
A-homology m a n i f o l d a r e , It
K,
of a
way.
a (A)-homology sphere.
Omitting details,
over cells.
structure
P o i n c a r e ' dual o f a t r i a n g u l a t i o n
g i v e n the A-homology m a n i f o l d
t o be the s t a r
"cell"
the usual
K x K.
c o m e s e q u i p p e d , as i t
PL on a
T h e r e f o r e , an A-homology
were,
with a classifying
map
TM: M ÷ Bh (n) and c h a r a c t e r i s t i c c l a s s e s f o r such m a n i f o l d s are A thus d e f i n e d i n the usual way s i m p l y by s e t t i n g , ( f o r any cohomology class
c ~ H (Bh ( n ) ; G ) ) c(M) = TM*c~ H {M;G). We n o t e , as an A example of s p e c i a l i n t e r e s t , that for A = G = Q, the r a t i o n a l P o n t r j a g i n classes
L-class
for
their
rational
p., or t h e r a t i o n a l I homology m a n i f o l d s s i n c e f o r
needs P o i n c a r e d u a l i t y suitable
transversality
(with
twisted rational
theorem in
~
may be d e f i n e d i d e f i n i t i o n one o n l y
co-efficients)
and a
the r ~ t i o n a l - h o m o l o g y - m a n i f o l d
category. I~ any case the n o t i o n of characteristic
class
briefly
the p r o c e d u r e .
sketch
is
readily
local
(ordered) formula for
extensible First
25
of a l l ,
a given
to A - m a n i f o l d s . Def.
1.3
(local
We
1.16 o r d e r i n g ) may be a p p l i e d unchanged to A - m a n i f o l d s . alter
the n o t i o n of codimension n-i i -1 i -i A "7 where ~ is
form
w i t h the A-homology type of
i
S i m i l a r l y we may
n - s t a r to mean a complex of
the
now merely an A-homology m a n i f o l d
the ( i - 1 ) - s p h e r e .
T h u s one may speak in
this
c o n t e x t of ordered n - s t a r s , as w e l l as o r i e n t e d ones, keeping in n-i i -i mind t h a t an o r i e n t a t i o n of I" *~ now means an o r i e n t a t i o n of i-1 Z f o r the c o - e f f i c i e n t groups A. E q u a l l y t r a n s p a r e n t is
the a d a p t a t i o n of
(ordered) formula, v i z . ,
a local
w i t h c o e f f i c i e n t s in
is
G
the n o t i o n of l o c a l
ordered formula f o r an i - c o - c h a i n ,
now taken to mean a f u n c t i o n d e f i n e d on
isomorphism classes of o r d e r e d , o r i e n t e d c o d i m e n s i o n - i n - s t a r s ,
(in
the newly m o d i f i e d sense) which takes values in the commutative group G
and which r e s p e c t s change of o r i e n t a t i o n .
b e f o r e , is Local
f o r m u l a , as
one which d i s r e g a r d s o r d e r i n g . ( o r d e r e d ) formulae o b v i o u s l y s p e c i f y G-valued i - c o - c h a i n s
on any ( l o c a l l y manifold.
A local
o r d e r e d ) t r i a n g u l a t e d n - d i m e n s i o n a l A-homology
Here, of course,
the c o - c h a i n s are d e f i n e d on a c e r t a i n
A-module chain complex not n e c e s s a r i l y having the i n t e g r a l homology of
M.
Thus,
it
makes sense to speak of a l o c a l ( o r d e r e d ) formula i as 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 class c ~ H (Bh ( n ) ; G ) . FurtherA more, i t is n a t u r a l to t r y to extend the r e s u l t of Theorem 1.6 to the l a r g e r c o n t e x t of A-homology m a n i f o l d s . 1.14
Corollary.
Given 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
A-homology n - m a n i f o l d s , t h e r e e x i s t s a l o c a l representing
for
o r d e r e d formula
c.
We c l a i m t h a t t h i s
extension is
e a s i l y achieved.
It
r e a d i l y be seen by the reader t h a t the basic d e f i n i t i o n s s-cell
c
complex, s - b l o c k b u n d l e , s - c e l l ,
etc.
will s-ball,
- may be mimicked in
c c n t e x t a p p r o p r i a t e to the study of A-homology m a n i f o l d s .
the
For
example, an s - b a l i must now be taken to mean an ordered s i m p l i c i a l
complex c o n s t i t u t i n g an A - h o m o l o g y - m a n i f o l d w i t h boundary having the
26
1.17 A-homology type of Qn, (an s - c e l l
Dj ,
Sj - 1
f o r some
j.
complex in the new sense)
The analogue is
qA(n) of
then constructed as i s ,
m u t a t i s mutandis, the a p p r o p r i a t e canonical A-homology s-block bundle y In)
over
qA(n).
One then r e a d i l y sees, j u s t as in the proof of 1.6, an A-homology m a n i f o l d obtain a " c e l l u l a r "
M
map
t h a t given
with l o c a l l y - o r d e r e d t r i a n g u l a t i o n
T:T* + qA(n) M
s t r u c t u r e on
of course, c a n o n i c a l , and, given any
T
is,
co-chain r e p r e s e n t a t i v e z#r
represents cM n-i that T#r((~ )*,0)
T*
i r ~ C (qA(n);G)
and, since
~
is
we
c l a s s i f y i n g the tangent
A-homology block bundle of Mn).
(here
T,
is
of
the dual A-homology c e l l
C(YA(n)) ~
Hi ( q A ( n ) ; G ) ,
determined l o c a l l y ,
we see
depends only on the o r d e r e d , o r i e n t e d class of n-i n-i n-i the c o - d i m e n s i o n - i n - s t a r st ~ = o *~k(o ) with i t s orientation
0.
I.e.
T#r
is
determined by a l o c a l ordered f o r m u l a ,
as r e q u i r e d . As a s p e c i a l a p p l i c a t i o n of the remarks above we take note of the f o l l o w i n g . 1.15
Corollary.
In the category of t r i a n g u l a t e d r a t i o n a l homology
n - m a n i f o l d s , there e x i s t l o c a l
(unordered) formulae f o r r a t i o n a l
P o n t r j a g i n classes and r a t i o n a l L - c l a s s e s . Proof of 1.15 comes by using the t r i c k
of C o r o l l a r y 1.7 on the
g e n e r a l i z a t i o n 1.14 of Theorem 1.6. We now turn our a t t e n t i o n to the question of o b t a i n i n g e x p l i c i t l o c a l formulae f o r manifolds.
c h a r a c t e r i s t i c classes of t r i a n g u l a t e d
PL
Of course, we began our discussion with Whitney's well
known c h a r a c t e r i z a t i o n of is
w by l o c a l f o r m u l a . Thus, as the reader i doubtless aware, a t t e n t i o n has, of l a t e , been concentrated on the
r a t i o n a l P o n t r j a g i n classes
p. 1
(and to a c e r t a i n e x t e n t on the
L-classes).
27
1.18
We b r i e f l y by
refer
Gabrielov,
this
paper
note
[Gel
Gelfand leaves
makes
it
Pontrjagin
slightly
the as
(1) of
leads
comes
choices
of
the
content
formula
of
this It
is
well
integral
class
over
the
7-skeleton).
a local
ordered
There
is In
,
any
St2~,
well
approach.
Patodi,
and
Riemannian
such
the
the
course,
known
is
p
further
appear
The
starting
in
[A-P-S]
on
is
the
not
over
all
in
any
clear
Thus
the
geometric
Pl
is
sense
a well-defined are
guarantees
integral
been
smoothable
the
existence
coefficients.
so
well
MacPherson
details work
point
procedure
PI'
[GGL].
of
[Ch 2 ]
construction
rational
PL m a n i f o l d s
has
papers
Cheeger's
Details
that
in
[GGL]
the
the
obscure.
with
a formula of
for
therefore
I
though
thus
(since
1.6
for
work
even
is
worth
approach.
formula.
of
expository
Singer
and
[Mac]
Stone
and
here.
on l o c a l that
the
clarified
formulas
we m e r e l y
well-known
for
the
real
sketch work
so-called
n-invariant
characterizing
this
of of
Atiyah, a
manifold.
Perhaps use o f
of
some p o i n t s
are
an a v e r a g i n g
quite
first
there
of
still
the
subsequent
by e n l a r g i n g
formula
Theorem
the
for
local
a local
achieved 1 terseness of
Gelfand's
particular
and
p
specified
clear,
manifolds
so we may o m i t
L-classes. the
is
formula
case, by
Less
makes
for
PL
this
data
known,
for
no h i n t
publicized [St
of
"hypersimplicial"
by
However
expense
al~orith~
of
[GGL].
for
Although
formula
indeed
a canonical
at
[GGL].
a local
can
note
a combinatorial
(2)
that
of
formula
somewhat obscure,
limitations
to
only
Lussik
class
As G e l f a n d ' s
[GGL]
this
the
combinatorial
matter
clear
results
to
the
and
the
rational
noting
to
the
easiest
way o f
1
form
is
by
~ f o r the r e a l L-class. i We remind the r e a d e r t h a t , g i v e n a smooth R i e ~ n n i a n m a n i f o l d Mn, 4i there is a canonical l o c a l l y - d e f i n e d 4 i - f o r m ~ (M) ~ R (M) with i Hl i d~. = 0 and [ ~ . ( M } ] : L (M) E (M;R). The n - i n v a r i a n t is I
the c a n o n i c a l Chern-Weil
invariant
i
28
1.19 defined for M
bounds o r i e n t a b l y , i . e .
q,
in
M = @W.
which case we may compute
assumption, l e t with
M × I
on a c o l l a r
not,
n(M)
as
q.M
bounds f o r
1 n(q.M).)
some
Under t h i s
neighborhood of
M.
the s i g n a t u r e of
the o r i e n t e d m a n i f o l d
There are then two
is merely the usual a l g e b r a i c - t o p o l o g i c a l s i g n a t u r e
of a m a n i f o l d - w i t h - b o u n d a r y i . e . form in
(If
We assume
W be equipped w i t h a Riemannian m e t r i c i s o m e t r i c
obvious " c a n d i d a t e s " f o r The f i r s t
4i-i M
o r i e n t e d Riemannian ( 4 i - 1 ) - m a n i f o l d s
2J-dimensional r a t i o n a l
the s i g n a t u r e of homology.
f
W. o(w)
the i n t e r s e c t i o n
The second i s
given by
~ (W) WI
(The two c o i n c i d e of course on m a n i f o l d s w i t h o u t b o u n d a r y . ) We d e f i n e
n(M)
as n(M)
It
~.(W) W1
o(W)
is an easy e x e r c i s e to show t h a t
(with i t s t i o n of
[
:
n(M)
is
an i n v a r i a n t of
M
m e t r i c and o r i e n t a t i o n ) and r e v e r s e s sign when the o r i e n t a M
is
reversed.
The r e l e v a n c e of the
n - i n v a r i a n t to the search f o r
f o r m u l a s may be suggested by the f o l l o w i n g " p l a u s i b i l i t y " (based on a d i s t i n c t l y
contrafactual hypothesis.)
local argument
We may 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 as a m e t r i c complex and thus somehow analagous to a smooth m a n i f o l d endowed w i t h a Riemannian m e t r i c . t h a t such m a n i f o l d s a c q u i r e d as w e l l a " d i f f e r e n t i a l determined l o c a l l y ,
Now suppose
4J-form,"
analogous to the Chern-Weil form
make p o s s i b l e the d e f i n i t i o n
the
4i-I
sphere, and i t
this
h y p o t h e t i c a l game, t h a t f o r an o r i e n t e d t r i a n g u l a t e d 4 i - m a n i f o l d
W,
the s i g n a t u r e of
is
of
c . This would i n - i n v a r i a n t of a t r i a n g u l a t e d
a r e a s o n a b l y s t r a i g h t f o r w a r d i n f e r e n c e , in
W could then be computed as
the sum being taken over a l l the t r i a n g u l a t e d spheres c o n s i s t e n t w i t h t h a t of
vertices
~k v W).
v
o(W) = ~n(~k v,O) v o+ the t r i a n g u l a t i o n (and
having a p p r o p r i a t e o r i e n t a t i o n s This
suggests,
29
in t u r n ,
that,
0
f o r an
1.20 a r b i t r a r y t r i a n g u l a t e d m a n i f o l d Mn n-i each o r i e n t e d dual c e l l (o )*, 0 L (M). i Returning to r e a l i t y ,
the l o c a l
formula sssigning to
the real number n(~k o, O)
would r e p r e s e n t
forms" f o r less,
t r i a n g u l a t e d manifolds hardly seems a c h i e v a b l e .
the program may s t i l l
a smooth Riemannian definable. n(M)
however, the c o n s t r u c t i o n of "Chern-Weil
4i-i
seem p l a u s i b l e since the
Nonethe-
n - i n v a r i a n t of
m a n i f o l d has been shown to be i n t r i n s i c a l l y
The c e l e b r a t e d r e s u l t of A t i y a h , Patodi and Singer is
can be d i r e c t l y a n a l y t i c a l l y defined from the spectrum of
Hodge o p e r a t o r of bounded by
M,
that
the
w i t h o u t reference to any presumed 4J-manifold
M.
Cheeger has subsequently shown t h a t the Hodge o p e r a t o r may also be defined on a c e r t a i n class of v a r i e t i e s which are smooth manifolds away from s i n g u l a r p o i n t s of a p r e s c r i b e d k i n d , and which are p r o vided with a c o n s i s t e n t f a m i l y of Riemannian m e t r i c s on w e l l - d e f i n e d smooth s t r a t a .
Without g e t t i n g i n t o other examples, q u i t e i m p o r t a n t
in t h e i r own r i g h t ,
we s i n g l e out f o r
s p e c i a l a t t e n t i o n , as an
example to which Cheeger's c o n s t r u c t i o n s a p p l y , t r i a n g u l a t e d manifolds.
PL
Here, the s t r a t a in question are the s i m p l i c e s
themselves, each metrized by the Riemannian s t r u c t u r e of the standard s i m p l e x , or indeed, by any c o n s i s t e n t set of c o n v e x - l i n e a r m e t r i c s . [We note in passing t h a t t r i a n g u l a t e d r a t i o n a l homology manifolds can also be t r e a t e d in much the same way.] Cheeger goes on to show t h a t , this
sort,
for oriented
v a r i e t i e s of
the A t i y a h - P a t o d i - S i n g e r c o n s t r u c t i o n goes through, r e s u l t -
ing in a w e l l - d e f i n e d n - i n v a r i a n t . local
4i-1
Moreover, there thus a r i s e s a
formula f o r the real c h a r a c t e r i s t i c class
,(j
n-4i
*
y4i-1
where the t r i a n g u l a t e d sphere m e t r i c on each s i m p l e x .
, O) = n(~
4i-1 ~
viz,
,0)
is m e t r i z e d with the standard
Note that this
30
4i-1
L., 1
is a l o c a l f o r m u l a , r a t h e r
1.21 than a l o c a l - o r d e r e d formula.
However, d e s p i t e i t s
attractive
c a n o n i c i t y , t h i s formula has not been made " c o m b i n a t o r i a l " in the sense t h a t a d i r e c t a l g o r i t h m f o r computing i t purely from the combi4i -I n a t o r i a l data of is not y e t known. Indeed, the t r a d i t i o n a l difficulties
in a c t u a l l y computing the ~ - i n v a r i a n t of a smooth
Riemannian m a n i f o l d c e r t a i n l y seem to p e r s i s t in t h i s c o n t e x t as well as f a r as is c u r r e n t l y known. An a d d i t i o n a l a t t e m p t to c o n s t r u c t l o c a l c o m b i n a t o r i a l formulae f o r r a t i o n a l classes is
to be found in the paper of Gabrielov [Gab]
and representa a c o n t i n u a t i o n of the methods of [GGL] whose aim is
to
extend the compass of those methods so as to handle the higher P o n t r j a g i n classes.
The e a r l i e r paper, in f a c t ,
these a d d i t i o n a l r e s u l t s .
We s h a l l
briefly
briefly
r e f e r s to
o u t l i n e G a b r i e l o v ' s work,
or r a t h e r , an e s s e n t i a l l y e q u i v a l e n t f o r m u l a t i o n .
Perhaps an i n i t i a l
apology ought to be made f o r the present t r e a t m e n t .
As presented in
[Gab], G a b r i e l o v ' s r e s u l t s seem to be i n o r d i n a t e l y hard to read or understand.
F i r s t of a l l
the paper is exceedingly t e r s e .
completely absent, even in o u t l i n e . fill
in the geometric and i n t u i t i v e
Moreover, there i s no a t t e m p t to background of the c o n s t r u c t i o n ,
which thus appears to a r i s e out of t h i n a i r .
F i n a l l y , and r a t h e r
u n f o r t u n a t e l y , there is a gap in the c o n s t r u c t i o n , f i l l e d i n c o n s p i c u o u s l y by an ad hoc h y p o t h e s i s , which v i t i a t e s t h a t the problem of f i n d i n g l o c a l formulae f o r solved by these methods. references to i t ,
Proofs are
Pk
somewhat
any claim
has been f u l l y
The terseness of the paper i t s e l f ,
and of
has obscured t h i s p o i n t .
Nonetheless, G a b r i e l o v ' s c o n s t r u c t i o n , when put in a c l e a r geometric c o n t e x t , has some i n t e r e s t i n g aspects. nates the e s s e n t i a l d i f f i c u l z y proach.
Thus,
At worst i t
illumi-
which l i e s at the heart of such an ap-
the present author f e e l s j u s t i f i e d
in the e x p o s i t i o n
below. F i r s t of a l l ,
we place a small r e s t r i c t i o n
31
on the kinds of
1.22 triangulated an
n-star
if
there
manifolds K : is
an
simplices. say t h a t for
o
n-i
*
tion
~
be
If
Mn
is
said
KC
Recall to
Rn
be
which
(see,
e.g.
a Brouwer is
[Wh])
star
convex
if
linear
that
and on
only
all
a combinatorially triangulated manifold,
the t r i a n g u l a t i o n o.
studied.
is
embedding
every simplex
lation
to i-i
It
is is
Brouwer i f a fact
st(o)
that
is
we
a Brouwer s t a r
every c o m b i n a t o r i a l t r i a n g u -
of
an n - m a n i f o l d has a Brouwer s u b d i v i s i o n .
T h u s the r e s t r i c -
that
the m a n i f o l d s we study are equipped w i t h
Brouwer t r i a n g u l a -
tions
is
seen to
be a r a t h e r
i n e s s e n t i a l one. n-i i-i K ~ A * ~ we l e t
Given a Brouwer n - s t a r c o n f i g u r a t i o n space of b a + O,
K,
modulo the a c t i o n of
t o p o l o g y on
C(K)
p : # v e r t i c e s of C(K)
K,
with
the space of
the general
topology viz,
and t h a t
embeddings
linear
a r i s e s when one n o t i c e s
embeddings has a n a t u r a l
thus
viz,
that
be the
C(K)
group
KC
GL(n;R).
acts
the q u o t i e n t t o p o l o g y i s
The
the space of
as an open subset of
GL(n;R)
Rn,
freely
on t h i s
a m a n i f o l d of
n.p R , space;
dimension
n.(p-n). Now suppose t h a t
is
a s i m p l e x of
st(A*T,K)
= KT
technical
d e v i c e , we now i n s i s t
linear
and
n+r-i+l A * a specific
4'
KT
~k(T,~
)
map
embedding
rT:
TO put t h i n g s
The complex
Thus
Brouwer s t a r s n-i A *T
which has a n a t u r a l
fT = f l K ~ ' - f ( b A ' ) : n-i n+v-i+l = a * T ~ a
restriction
that
f:K,
b n-i
KT,b ~ , + Rn,O
as w e l l .
with
with
inherited ordering.
+ Rn,O,
As a
be equipped w i t h
may be i d e n t i f i e d
may be c a n o n i c a l l y i d e n t i f i e d
i-1
linear
~.
then an n - s t a r and a Brouwer n - s t a r
o r d e r i n g s on v e r t i c e s .
n+r-i+l A
ing
is
J T
Given
we o b t a i n an embedd-
where
This o b v i o u s l y induces a c o n t i n u o u s
C(K) ÷ C(KT).
i n t o a c o n v e n i e n t u n i v e r s a l c o n t e x t , we take one
i-i ( t o be t h o u g h t of as c~ ) f o r each c o d i m e n ~ i o n - i K n-i o r d e r e d Brouwer n - s t a r A *~ = K (e K = p o i n t f o r i = 0). Ranging i-cell
e
over a l l
o r d e r e d Brouwer n - s t a r t
K,
32
we assemble these c e l l s
into a
1.23 C-W
complex
identify is
e(n)
as f o l l o w s :
n-i i-i K = ~ *~ , T ~
Given
e
w i t h the c e l l 3" c~ dual t o ~. The i d e n t i f i c a t i o n K T made i n a c a n o n i c a l way, i . e . t h e r e i s a c a n o n i c a l PL homeomor-
phism from
= c(Ek(T,~)) to T*. o(n) KT these i d e n t i f i c a t i o n . The u n i v e r s a l i t y of that is
i-i
e
given a l o c a l l y
cell
o*
gm:Mn ÷ e(n) to
the c e l l
We now c o n s t r u c t a c e r t a i n will
not
(unfortunately!)
( w i t h compact f i b r e )
simplex of
K, ~,
space
Thus
let
=
thereby giving
tion
rise
similar
minor d e t a i l s ,
ceptual
where
y
if
local of
is
projection
information,
we a l e r t
that
of
be made i n reason t h a t
s t a t u s of a f i b e r
any p o i n t
x E int
the space
PL/O
binatorial
manifolds.
M
B(n) it
were
formulae B(n)
a p o i n t of of
e It
compatible with 7:
~
as a
For each o r d e r e d
identification
B + e is K K to a g l o b a l
to
will
The i n t u i t i v e
explicit
modulo these i d e n t i f i c a t i o n s .
of a d d i t i o n a l quite
In f a c t ,
space.
there
each s i m p l e x
However, one m i g h t t h i n k
(y,
B(n)
Mn
B denote e × C(K). Moreover, if K K x ~ ~*, we i d e n t i f y (x , f ) ~ e K × C ( K )
and
U B K K t h e p r o j e c t i o n map
point
bundle.
the c o n s t r u c t i o n o f
r T f ) E eKT × C(K T ) = B KT sponding t o x under t h e n a t u r a l
with
e(n)
C 8(n). st(~,M) B(n) "over" B(n).
c o a r s e a p p r o x i m a t i o n of a f i b e r
Brouwer s t a r
~ e modulo K K l i e s in the f a c t
which f o r e
be a f i b e r
c o u l d be made to f o l l o w . certain
thus
ordered B r o u w e r - t r i a n g u l a t e d manifold
a c a n o n i c a l Gauss map
sends the dual
is
T
is BK
=
e
a
corre-
KT with
KT is c l e a r
T . that
identifications B(n)
+ e(n)
(As a
the reader t h a t a c o n s t r u c -
e ( n ) and B ( n ) ,
differing
only in
§8.) B(n)
has i n
bundle o v e r
e(n)
some r e s p e c t s the conis
that
the " f i b e r "
e
at
is C(K) which i s a "rough a p p r o x i m a t i o n " t o K w e l l known t o s t u d e n t s of smoothing t h e o r y f o r comHowever, we s h a l l
a n a l o g y any more p r e c i s e ; " c o n s t a n t " from one c e l l We remark t h a t
certainly
n o t a t t e m p t to make t h i s
"fibers"
a r e by no means
to an a d j a c e n t one.
the " u n i v e r s a l "
33
construction
e(n)
is
not
1.24 strictly
necessary.
"over"
M
That i s ,
for
We might merely have c o n s t r u c t e d a space
any l o c a l l y - o r d e r e d B r o u w e r - t r i a n g u l a t e d m a n i f o l d
over each c e l l
take the space
a*
of
the dual
Ba = a* × C(st a,M)
triangulation
of
ture.
a < 3,
We do t h i s
T C st
a
defined. with T :
x,r
by n o t i n g t h a t
and the r e s t r i c t i o n
Thus, If] P
if
x
if
T* < a * ,
map from
where
E T* × C(St T),
C(st
we i d e n t i f y
T* C O *
of
M,
P C Ek ~
cell
so struc-
thus to
x,[f]
we
Ba s
the dual
a)
Mn.
I
and glue the v a r i o u s
as to r e s p e c t the v a r i o u s i n c i d e n c e r e l a t i o n s
st
B(M)
C(st
T)
is
~ o* × C(st ~)
is
d e f i n e d by
~*P. Putting
6(n),
B(n),
consider a manifold
Mn PL
( ~ - n ) - d i m e n s i o n a l plane x e M that
if
for
and o n l y
B(M)
if
etc.
aside f o r
embedded in
PC R
there
is
is
R .
a neighborhood
y,z
d U w i t h y ~ z, y - z eP.
Now l e t
K :
A*Z
embedding
on s i m p l i c e s and w i t h linearly
be a Brouwer n - s t a r K
independent.
by which
the s e t of [Thus
is
vectors
E >> 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
L
I
~ ~ ~ e . {olJ=L~} L,~ L
K,L
49
F i n a l l y , we l e t
e
L
2.8 We proceed t o
the d e f i n i t i o n n+k + R We s h a l l
denote
G:
viewing
the
that
this
whereby will
n,k d o m a i n as
cell-by-cell the
be c l e a r
C~L,
simplicial
c~L~
that v
are
the
local
of
C , L
a barycenter
the t a u t o l o g i c a l
first
definition
various
Given a v e r t e x of
the
of
define
complex respects
note t h a t
of
CL.
the
amalgamated definition
b
G
to
on each
c~
, L then note
We s h a l l
identifications
form
~'
n,k a global
defines v
map, which we
.
Thus, map
it
G.
must be the cone p o i n t
some s i m p l e x
o
of
~L
or else
*
con-
U
tigous
to e i t h e r
a 1-simplex of ~(v)
= *.
Let
b ,
but not to
*
or some
C . L ~(v)
Now i f
*,
nor t o
= ~,
b , v
if
where " c o n t i g u o u s " means j o i n e d by
is
*
or c o n t i g o u s t o
*,
let
v = b , or i f v is c o n t i g o u s to o b f o r any p r o p e r f a c e T of ~. Let
0
T
P(*)
= {v]~(v)
together with disjoint
= *},
P(o)
the v a r i o u s
families.
= {vl~(v) P(o),
Now l e t
= o}.
partition
E(*)
(resp.
Note t h a t
P(*),
the v e r t i c e s E(o))
of
C into L be the subcomplex
of
C spanned by P(*) (resp., P(o)), i.e., the l a r g e s t subcomL p l e x c o n t a i n i n g no v e r t i c e s e x c e p t those i n P(*) (resp., P(~)).
E(*)
is
naturally
s u b d i v i s i o n of
isomorphic to
ZL
while
the cone on t h e second b a r y c e n t r i c
E(o),
in
t o t h e cone on the second b a r y c e n t r i c Thus,
for
the same f a s h i o n , s u b d i v i s i o n of
is
isomorphic
ck(°'ZL)
v ~ P(*)
(resp. P(o)), consider C~L (resp., n+k R , and l e t G(v) be t h e image o f v
as a subspace o f
~ ~L 0
C~L ) o under t h e
composition "
v ~ E(*)
C~
iso. (resp., Having d e f i n e d all each
of
CL. CL
G
But i n
respects
"o",
for
on v e r t i c e s so d o i n g ,
~ CZLC L homeo "*"
and
n +k R
"L "
for
of
"L")
C , we m e r e l y e x t e n d l i n e a r l y L we n o t e t h a t t h i s d e f i n i t i o n of G
the i d e n t i f i c a t i o n s
on
was formed as ,k the u n i o n , mod i d e n t i f i c a t i o n s , of c e l l s c~ = IC I . Therefore, L L t a k i n g t h e union of a l l t h e l o c a l d e f i n i t i o n s of G yields a global
50
made when V~n
to
2.9
map
G:
/•n
It
is
n+k + R
,k i m p o r t a n t to note t h a t
G(~ ) C_V for all L. This f a c t L L a l l o w s us t o o b t a i n a v e r y s i m p l e d e f i n i t i o n of y "locally." n,k where TV denotes That i s , f i r s t define Yn,kJeL as (GIeL)*TV L, L the t a n g e n t PL n - b u n d l e o f the PL m a n i f o l d V . We must m e r e l y L show t h a t these l o c a l d e f i n i t i o n s c o i n c i d e c a n o n i c a l l y on i n t e r s e c t -
ions of
the form
either
L < J
~
or
~ ~ . But n o t e t h a t ~ /~ ~ ~8.. implies that L J L J J < L. So assuming, say, t h a t L < J, note t h a t
G(~
~ ~ ) is c o n t a i n e d in L J VL ~ V j . Thus the bundle fined
the i n t e r i o r Yn,k
g l o b a l bundle over a l l
the non-empty
U Yn,k 'eL L
is,
in
n-manifold
fact,
a well
de-
of ~ n , k "
Having d e f i n e d the bundle triangulated
:
of
Yn,k ,
we must now n+k show t h a t ,
n - m a n i f o l d embedded or immersed i n
a
linearly with n r e s p e c t t o the t r i a n g u l a t i o n , the n a t u r a l Gauss map g: M + is o n,k n c o v e r e d i n an e q u a l l y n a t u r a l way by a b u n d l e map TM + y 0 npk n We may t h i n k o f TM as r e p r e s e n t e d by the a s s i g n m e n t t o each o n n point p ~ M of a neighborhood U of p in M so t h a t °n n P n n n n n L J { p } x U C M ×M is a r e g u l a r neighborhood of AM C_ M ×M ~ M xM . p p o n o o o o Now c o n s i d e r t h e g i v e n c e l l u l a r s t r u c t u r e on N , i.e., the c e l l s n
of
M ,
a r e t h e dual c e l l s
t o the s i m p l i c e s o f
M
not c o n t a i n e d in
0
aM . n
~
On
M n
R
for
We s h a l l
let
a
denote t h e dual c e l l 0
n
and then proceed to decompose
M
into
t o such a s i m p l e x subspaces
0
bears t h e same r e l a t i o n construction a
is
to
o f an a l t e r n a t e
,
subdivision
N
as
where 0
~
i
subcomplex o f
the f i r s t
barycentric
n
of
n
M ,
i s o m o r p h i c t o t h e cone on
For
hood i n
o c
a face of of
T,
the v e r t e x
we d e f i n e b ,
a
which i s
,
(~k(o,M))
then we may s u b d i v i d e once more~ o b t a i n i n g t h e t r i a n g u l a t i o n a .
, c
T
of
the o r i g i n a l
triangulation.
51
of
as the r e g u l a r n e i g h b o r the b a r y c e n t e r o f
the
T
simplex
of
did t o e i n our p r i o r L L decomposition of ~ n ,k That i s , i f
t h o u g h t of as a c e r t a i n
0
~
~
o
We l e t
a
be t h e
2.10
regular
neighborhood of
b
in
c .
o
Under t h i s ~L(~,Mn ).
definition,
g,
it
covering
Gog. n U of TM at P On+k standard R ,
will
r
the Gauss map
suffice
X . o = n-dim o.
a
T~O
C~)~ "
takes ~ to 0 n bundle maPn TMo + Yn,k
PL
This
q
~
clearly
maps
d e t e r m i n e s an a f f i n e
plane
X
in
U such t h a t P s e t c o n s t i t u t e s a PL
the v e c t o r
p-q
of
0
the unique such p l a n e c o n t a i n i n g
o. is
Con-
orthogo-
W of d i m e n s i o n p enough, we c l a i m t h a t Gog
U i s taken small P homeomorphically onto a PL
takes
g
to c o n s t r u c t
the s i m p l e x
those p o i n t s
to
= ~
TM I~ ÷ TV o o L(o,M n ) Consider, t h e r e f o r e , a p o i n t p ~ ~ , and the f i b e r n o p. Since M is t h o u g h t of as immersed in t h e
t h e same d i m e n s i o n , i . e . ,
nal
~ 0
In o r d e r t o c o n s t r u c t the
covering
sider
Then
0
If
W P Gog{p) ~ Z , so t h a t Z is P P d i m e n s i o n = dim ~ p a r a l l e l to
disc
Z , with P P L - t r a n s v e r s e t o the a f f i n e X
r-disc
and p a s s i n g t h r o u g h
p l a n e of
Gog(p).
G
Moreover, W ÷ Z P P Gog(p).
Z lies entirely P t o a homeomorphism
in
VL(~,Mn ).
We e x t e n d the map
U ÷ T where T is a neighborhood of P P P In p a r t i c u l a r , if U is viewed, ( w i t h o u t loss of g e n e r a l P i t y ) as W +d, for d a small r - d i s c p a r a l l e l to X , then T p a p may be viewed as Z +d. This s e t o f homeomorphisms may then be P n t a k e n t o d e f i n e a bundle map TM I~ ÷ ¥ I~ L Mn , c o v e r i n g the o o n,k (0, ) Gauss map g. It tent,
remains to o b s e r v e t h a t
this
family
of bundle maps i s
consis-
i.e.,
i t s e l e m e n t s p i e c e t o g e t h e r t o form a w e l l - d e f i n e d PL n b u n d l e map TM ÷ y covering g. D e t a i l s are l e f t to t h e r e a d e r o n,k as an e x e r c i s e . We remark t h a t
there is
an a l t e r n a t i v e
way o f
visualizing
the
n
bundle
and t h e bundle map TM + y which ,k o n,k c o v e r s the Gauss map. C o n s i d e r the spaces cZ which a r e t h e p r e L images o f the c e l l s e L of ~ , k . We may r e p l a c e C~L, f o r conceptual
Yn,k
over
p u r p o s e s , by
n+k
b C_ R L
where
52
b
L
is
the l i n e a r
complex i n
2.11 n+k R
whose v e r t i c e s are the o r i g i n
p l u s the v e r t i c e s of
ZL
and
whose s i m p l i c e s are the convex h u l l s of those sets of v e r t i c e s of CZL
which span s i m p l i c e s of
CZL.
That i s ,
bL
" i n s c r i b e d " polyhedron which corresponds to morphic to
CZL
as a complex, we may t h i n k
as the preimage of
is
ZL"
the cone on the
Since
of b L, n+k R , it
bL
is
iso-
r a t h e r than
CZL
e . As a subspace of is c l e a r , L b L C C~LC VL. Let EL = TV Ib • Next, c o n s i d e r a s i m p l e x ~ of L L b L (which corresponds to a s i m p l e x of ZL). Abusing our e s t a b l i s h e d n o t a t i o n m i n i m a l l y , we o b t a i n a face h(L,~):
b
L
of
L
and a map
which i d e n t i f i e s b w i t h the dual c e l l to o in L L b . h(L,o) extends to a map~ ~: V + Rn+k That i s , L L o n+k t h i n k of V as Q (~) X Then h ( L , o ) extends to ¢: Q + R L L L L + , ~R, by @: tp + ~ ( L , ~ ) ( O ) ~ t ( h ( L ~ o ) ( p ) - h ( L , o ) ( O ) ) . Here p E L thus tp i s a t y p i c a l element of Q Now extend ¢ l i n e a r l y to L L In f a ~ t
,
on
is a
+ b
0
QL + XL
by
,(q+x)
PL °embedding.
sees t h a t
V L of i d e n t i f y i n g
= @(q)+x, q
coincides with
TV L I b L
~(V
with EL
L
diagram (2) above w i l l
b L's
and the
,
Clearly,
x E XL
~
h's
by
) one L 0 Thus, we have a s t a n d a r d way
).
TVL~h(L,O)bL
In o t h e r words, we
+ BL
h(L,o).
same c o n s i s t e n c y c o n d i t i o n s ho~d f o r
by
QL
Moreover, in a neighborhood of ° h ( L , o ) ( b
o b t a i n bundle maps o B ( Le, ~ ) :
i.e.
c
covering the
e's
as f o r the
Moreover the
the
still
commute i f
CZL'S
e's.
Thus (now viewing ~ n'k
h's, be r e p l a c e d
as having
been pieced t o g e t h e r from
b 's r a t h e r than c~ ' s ) , we have a way L L of p i e c i n g t o g e t h e r s i m u l t a n e o u s l y a bundle x from the B 's. n n+k n,k L Given an immersed m a n i f o l d M in R and a c e l l ~ of n
M , i t is c l e a r t h a t , over o i : o* ÷ bL(o,Mn ) ; i clearly neighborhood of VL(o,M), i:
TMnlo*
~
in
n
M
.
o ,
the Gauss map a r i s e s from a map
extends to a homeomorphism to a neighborhood of
and we thus o b t a i n , l o c a l l y ,
÷
BL(~,Mn ) •
L(o,M)
from a in
a s t a n d a r d map
These maps, in t u r n ,
53
b
u
piece t o g e t h e r to form
2.12 n Mn TM I + Y o n,k We b r i e f l y note t h a t , in order to show t h a t the two proposed
the global map
d e f i n i t i o n s of
e s s e n t i a l l y coincide we use techniques which n,k n n i m i t a t e the c o n s t r u c t i o n of the covering map TM I M ÷ y of the n f~ o n,k Gauss map g: Mo + ,k' as t h a t c o n s t r u c t i o n was done using the
initial first ~L )
y
d e f i n i t i o n of
Yn,k"
YI
denote the
proposed version of and
Y2
¥ ( i . e . , l o c a l l y given by G*TV over n,k L the second ( i . e . , l o c a l l y TVLIbL). We may show t h a t
f o r any p a i r of formal l i n k s bK
More p r e c i s e l y , we l e t
of
~L ~ eK'
L < K,
if
there is a bundle map
U K,L
is the preimage in
TVKIUK,L ÷ TVL
covering
UK,LC ~L G--~->VL" (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~<~-+l,k'--
The d i f f e r e n t
triangulations
of
n M
with respect to
which the two immersions a r e l i n e a r may g i v e r i s e t o s l i g h t l y differn ent M 's: n o n e t h e l e s s , these may be i d e n t i f i e d t o a l l i n t e n t s and o purposes.) We s h a l l
not give a f u l l
depends e s s e n t i a l l y
p r o o f of
2.3.
We m e r e l y n o t e t h a t
on e x t e n d i n g the n o t i o n of
Gauss map t o
it
slightly
more c o m p l i c a t e d s i t u a t i o n s . For the moment, we s h a l l assume t h a t n M i s w i t h o u t b o u n d a r y ; we l e a v e t o the r e a d e r t h e minor t a s k o f e x t e n d i n g the c o n s t r u c t i o n
below to c o v e r the case where a n o n - v o i d
boundary i s
tion
present. n n Suppose M = @W , and suppose f u r t h e r n+l n n+k+1 n+k f: W , M + R+ , R Here
57
that
there is
an immer-
2.16 n+k+l n+k n+k n+k+l R+ = R x R+ R × R = R We assume t h a t -1 n+k n f (R ) = M . We wish to d e f i n e a Gauss map n n ~ ~ /<~ g: W ,M + n + l , k " n , k ' where is i d e n t i f i e d w i t h a subcomplex n,k o f ~<~ via ~. This map i s to have the p r o p e r t y t h a t n +l,k gIMn +~n is the Gauss map of flM n + Rn+k C l e a r l y , we may begin ,k n n+1 by d e f i n i n g g on W = U over those s i m p l i c e s ~ of W not 0
n
in
O*
M
This c o i n c i d e s w i t h the s t a n d i n g d e f i n i t i o n coming from r e n+l n+k+1 garding f as an immersion of W into R Ne a l s o d e f i n e n n n+k glM as the Gauss map of the immersion flM ÷ R Thus the p r o b n+l n lem reduces to e x t e n d i n g the d e f i n i t i o n to the c o l l a r on @W = M n n+1 between M and BW In e f f e c t , t h i s means s p e c i f y i n g g on the 0
half-cell
: dual-half-cell
to
~
in
W f o r each s i m p l e x
~
of
W
n
M ,
a
so as to agree w i t h = dual c e l l
to
~
g
as a l r e a d y d e f i n e d on ~* ~ W and W 0 n M . For each such ~, of dimension n - j ,
in
we d e f i n e a c e r t a i n formal L(~,W). usual,
Let let
u
link
of dimension
denote the v e c t o r
b
(n+l,k;j+1)
(0,0,0 ,.. .0,-1)
denote the b a r y c e n t e r of
~,
in
and l e t
denoted by Rn+k+1
c
As
= b +u.
Clearly,
the immersion ( i . e . , embedding) n+k+1 n+k f l ( s t ( o , W ) , s t ( o , M ) ) C_ R+ , R extends to an embedding
D
= st(~,W)
L~ c st(~,M) into st(~,M] c and e x t e n d i n g l i n e a r l y
p o i n t to L(~,W)
as
L(o,D )
under t h i s
that
L(o,M ) ^ [L[~,W))c
n
L(~,M ) : ding to
c ).
over
of
by sending the cone c st(~,M).
embedding.
It
is
We d e f i n e
c l e a r from i n s p e c -
^
no
tion
Rn+k÷l
f
is
i n c i d e n t to
(where
(Here, be i t
c
is
L(o,W),
i.e.
the v e r t e x of
~L~(o,W)
a l s o n o t e d , we are i d e n t i f y i n g
corresponthe
0
(n,k,j) Thus
link giG*
~{ c .Co,W) that of
+
L(~,M) factors
eZ(~,W)
gl~*w ~ Wo
with i t s
image under
through the diagram
C ~ --+1,k"
factors
~k(o,W) ~ W , L(T,W) o
~
as an
-L~(o,W)"
i n c i d e n t to
58
(n+l,k,j)-link.)
+ CZL(o,M) + Zt(o,W)L
At the same t i m e ,
through is
A
it
I.e., L(~,W).
is e q u a l l y c l e a r f o r each s i m p l e x Thus,
there is a
2.17 s t a n d a r d map
~* ~ W ÷ ZZ through which w o (o,w) , these r e s p e c t i v e maps o* ~Z
÷
and
g2
respectively.
disjoint g l U g2
j-discs
in
It
the
(o,w)'
factors.
oW FIw O ÷ ZC(o,
is c l e a r t h a t
j-sphere
g
im ~g l and
ZL ~,w)"
Denote A by g l
W)
im g2
are
Thus the homeomorphism
extends to a homeomorphism
g : o* onto > 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
MLn
i.e.,
M L ~ =
Let
transversality
MK ÷ i n t ( V L ~
VK )"
w i t h a s m a l l open
Now l e t n-disc
U M . e L cT~z L
71
MI
denote the p u n c t u r e d
removed from each component.
ML,
4.3 We now proceed i n d u c t i v e l y . been d e f o r m e d , k e e p i n g each int(V
~
VK),
to some map
Let us assume t h a t
is
an i m m e r s i o n .
dim
L ~ j-l,
{TML
÷ TV ) , L
has
Min V and each L L N(J_I) h so t h a t , on
L h
GoflM
Note t h a t
this
means, i n
M- ~ Min L K : L~ M , dim L < j - I particular, that for
Mhas been immersed w i t h c o d i m e n s i o n O i n t o the L n-manifold V . We may assume f u r t h e r t h a t the d e f o r m a t i o n t o h L has been c o v e r e d by a d e f o r m a t i o n of the f a m i l y of bundle maps
is
that
for
dim L < j - l ,
induced by the codimension-O
further will
and t h a t
t o deform
c o n t i n u e to Note i n
h
so t h a t
hold with
this
the b u n d l e map
immersion.
We w i s h ,
to
(j-l)
r e p l a c e d by
connection that
deform
him
for
j.
j-dimensional
to an immersion o f
L
t h e d e f o r m a t i o n may be made r e l in
this
fact
regard,
that
ML
that
the c o l l a r
the v a l i d i t y
of
of
L,
will
keep
~
this
as f o l l o w s .
VK)
that
h).
HL q
least,
PL
manifolds
V . L
In f a c t ,
1)
on
this
M ~ N( j - l )
MK
n.
L < K. with
L < K,
72
M , L
take
(which maps to
on t h e p u n c t u r e d v e r s i o n o f of
him L
We see
Deform the r e s p e c t i v e r e s t r i c t i o n s
We e x t e n d the d e f o r m a t i o n t o a l l
by
d e f o r m a t i o n of
to maps wi~ich a r e immersions on each component o f b o r h o o d s , or a t
Now
immersed.
M- ~~ N ( J We n o t e , L argument depends on the
in V ~ V for all K with K L K For each n-dimensional link K
under
we may assume
on
M-
a closed r e g u l a r neighborhood of int(V L~
into
M m a p s to L TM + TV , and L L of aM i s
d i m e n s i o n s m a l l e r than
F u r t h e r m o r e , we may even i n s u r e M[
this
M L
may be o b t a i n e d from the c o l l a r
adding successive handles, a l l
now,
the same c o n d i t i o n s as s t a t e d above
V under h, the map b e i n g c o v e r e d by a bundle map L {j-l) m o r e o v e r , the c o - d i m e n s i o n - O s u b m a n i f o l d M ~ N L immersed i n V . In f a c t , w i t h o u t l o s s o f g e n e r a l i t y , L t h a t a c o l l a r neighborhood of M~ N( j - l ) in M is L L-we may a p p l y the H i r s c h - P o e n a r u immersion theorem f o r [H-P]
TM + TV L L
rel
of
h
such r e g u l a r n e i g h such a component.
a neighborhood of
4.4 M- ~ L this
N( J -
1)
Since
L < J < K
implies
int(V
~ V ) ~int(V ~ V ), K L J d e f o r m a t i o n may be chosen to keep M ~ M in int(V ~ V ). L J L J Now c o n s i d e r (n-1)-dimensional links K with L < K. In what L
remains of
M a f t e r removal of the i n t e r i o r s of L n e i g h b o r h o o d s , p i c k a r e g u l a r neighborhood of
lar
the p r e v i o u s r e g u M ~
•
deform once more, at
rel
the
subspaces a l r e a d y immersed,
l e a s t modulo p u n c t u r i n g i n t o
all
V {~ V . L K
HI.
of
Proceeding in
this
M
and
L
Extend the d e f o r m a t i o n to
manner a dimension a t
stage where we have immersed, a t
to an immersion
a time,
we reach a
l e a s t modulo m u l t i p l e p u n c t u r i n g , a
~
-
neighborhood of
-
H ~( M ). We then immerse, r e l L L
regular
Strictly
s p e a k i n g , we have now immersed a m u l t i p l y - p u n c t u r e d
through a d e f o r m a t i o n of
h.
But t h i s
will
of
course,
M L
contain a
s m a l l e r copy of
M within itself, and we may now s u b s t i t u t e t h i s L s m a l l e r copy f o r the o r i g i n a l M . Note t h a t the d e f o r m a t i o n a t a l l L (j-l) stages i s made r e l N , and t h a t i t may be extended to a deform-
ation
of
h
on a l l
property that all
links
M J
J,K.
of
M . J
ej
goes to Finally,
This d e f o r m a t i o n p r e s e r v e s the
V and M F~M to i n t ( V f) V ) for J J K J K the d e f o r m a t i o n on each M , i s covered by ~J
a d e f o r m a t i o n of
the bundle maps
dim L ~ j
which are
is
the
t h a t of
immersed
in
immersion.
Proceeding w i t h
the i n d u c t i o n on
-
each
in
M L n+k R
immersed
in
But note t h a t
c o u n t a b l e number of d i s c s hemeomorph immerses,
j,
we reach a stage where
( I
is
[
TM ÷ TV , and, on those M , J J the r e s p e c t i v e V , the bundle map L
V L (n) N
and thus
is
merely
removed.
n M
-
ek~L C
/7~z
n M
Therefore
(n)
M = N L with, (n) N
is
immersed
a t most,
a
contains a
n M )
M of itsFlt. Thus M (i.e., certainly 1 1 n+k , via r e s t r i c t i o n , in R Now set M = M ~M . Without L L 1
73
4.5 lOSS
of
generality,
submanifold
of
linear,
so
a
of
image is
and the
M' may be assumed t o be a c o d i m e n s i o n - O L Now t r i a n g u l a t e M , so t h a t t h e i m m e r s i o n i s i
M . i that
the
V . L
with
V L
M is L i t must
immersion
some i n t e r i o r
X C L
I
each
triangulation,
under
that
each
In
this
in
point
case,
and hence,
is
since
lie
the of
it
a subcomplex. in
one
Thus,
such
maps,
immediate
}~L i s
under that
assumed
V
to
a simplex
M and t h u s i t s L V . One p o s s i b i l i t y L the immersion, into
corresponding o
given
i
coincides
L(o,M 1 )
be a g e o m e t r i c sub-
complex~ e L ( ~ , M
1) On the
other
hand,
suppose no i n t e r i o r
point
of
~
goes i n t o
X . Let T be a s i m p l e x of ~ and l e t E d e n o t e the subspace of L L T V c o n s i s t i n g of n+k vectors w of the form w = x+st where L 0 x ~ X , t ~ T and s > 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
We wish to extend t h i s
deforming
h(L ) r e l
(L)
=
(L)
r a i n e d by e q u i v a r i e n t l y
C(L ).
ob-
(L)
The problem,
M C does not n e c e s s a r i l y s a t i s f y the r e l a (L)' (L) A = ~" - C and l e t Bierstone c o n d i t i o n . Therefore, let
though, tive
is
that
(L)
A
denote f o r
(L),i minimal elements of a c t i o n of
A
~,
(~C
(L),i handles.
n
[I.e. A
Assuming
the s t r a t i f i c a t i o n
such t h a t
A
A
will
(L),i A (L),i
A
the v a r i o u s
a s s o c i a t e d w i t h the
(L)
will
A
be a
B-component of some A(¢) K-component of f o r any
be a m a n i f o l d of
from
(el A(L ) , ¢ > 8; r,
some d i m e n s i o n , say
C involves r-handles.] (L},i (L) each such A(L), i , p i c k some o r b i t ,
e < ~; thus
and o b t a i n i n g
A
Now, f o r
and remove from Clearly,
of
(L)
i ~[7,
cannot be o b t a i n e d from a c o l l a r on (L),i by handle a t t a c h i n g w i t h o u t the use of t o p - d i m e n s i o n a l
(L),i c o n t a i n s no
(L},i
(L)
with
some i n d e x i n g s e t ~ ,
A
Rx ,
i
x EA
i
(L),
i'
a small e q u i v a r i a n t t u b u l a r n e i g h b o r h o o d .
(L)
such t u b u l a r neighborhoods may be assumed m u t u a l l y d i s j o i n t .
The removal of
these neighborhoods y i e l d s a m u l t i p l y - p u n c t u r e d
A
(L)
and hence a m u l t i p l y - p u n c t u r e d M- , C (L} (L) deform h(L
satisfies
M which we denote M. (L) (L) 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 , hence we may
) rel C(L )
Covering t h i s TM(L ) + TV(L ) ,
to a map
I
h(L )
which is an immersion on
d e f o r m a t i o n by a d e f o r m a t i o n of
h
(L) ~-bundle map which, over
we o b t a i n a
to
: (L) M(L ) is
M~L).
that
l
induced by the immersion
h(L)IM~L ) ÷ V(L ).
Moreover, we c l a i m t h a t the d e f o r m a t i o n of be choser so as to keep latter
as a subspace of
M ~ M in V(L ) /'~ (L) (K) K with V ) for all
(L)
98
h
(L)
V(K )
to
h
I
(L)
(regarding
L < K.
The
may
the
5.13 argument here e s s e n t i a l l y of 4 . 2 ,
replicates
the analogous one in the p r o o f
the presence of a group a c t i o n
in t h i s
case n o t w i t h s t a n d i n g .
We t h e r e f o r e o m i t d e t a i l s . Thus,
if
we p e r f o r m the d e f o r m a t i o n of
orbit
(L),
well)
may be s u i t a b l y
all
K
it
is
h
(L) h (K)
immediately apparent t h a t
of d ~ e n s i o n
deformed ( t o ~ j,
h
so t h a t
h
(K)'
t(Kl)
o
(K)
to
h' f o r each (L)/~ (and h as (K) respectively) for
I
h(Kl)
:
t(K2)
o
I
h(K2)
on
M ~ M K , K of a r b i t r a r y d i m e n s i o n . Here, i t is under(K 1 ) (K2)' i 2 I stood t h a t h = h for dim K < j - 1 . Moreover, i t is under(K) (K) I stood t h a t the d e f o r m a t i o n s of h to h s a t i s f y the same com(K) (K) patibility c o n d i t i o n s as the h or h' themselves. (The de(K) (K) f o r m a t i o n is the t r i v i a l one f o r dim K < j - 1 . ) T h e r e f o r e we have d e f i n e d , g l o b a l l y , rel is
M( j - 2 )
It
is
an immersion on
i n d u c t i v e step, H(O)
clear that M( j - l )
h',
defined locally
= M( j - 2 ) ~
o b t a i n i n g hypothesis
h
a d e f o r m a t i o n of
(K)
o
I
h ,
h'
(K)' This competes our
( J M-
(L) (L)
H(j)
as t
tO
H(j-I).
from
Since
was known, we may a s s e r t
H(n+l). In consequence, we may n n+k a s s e r t the e x i s t e n c e of a map h: M + R , d e f i n e d l o c a l l y by (n) t(L) o L)' which, moreover, is an immersion on M ~ M. We s h a l l
h(
immerse a l l Clearly
of
M( n )
M, is
or r a t h e r ,
a
~-homeomorphic copy, as f o l l o w s .
a multiply-punctured
M,
that
o b t a i n e d by removing some t u b u l a r neighborhoods of For the sake of b r e v i t y ,
let
n o n - v o i d boundary ( t h e case when n B i e r s t o n e c o n d i t i o n holds on M , P
of any
M
(e)
Theorem 3.1 in is
itself
must s a t i s f y [Bi].
us assume t h a t n M is open i s it
is
a manifold-with-boundary.
o r b i t - n e i g h b o r h o o d s removed to o b t a i n borhood P
T
of
the o r b i t
~.x.
~.x
d e s c r i b e d above and the image of
99
has been
n-orbits. n M
is
compact w i t h
similar).
f o l l o w s t h a t any
P ~ aMn ~ k ~
Moreover, i t
M( n )
is,
Since the
~-component
(See the analogous
readily
seen t h a t
P* = P/E
Now suppose one of the
M( n ) lies
f r om
in one of
T F~ P
in
P*
M
is
the is
the n e i g h -
n-components k a disc D ,
5.14 (k
= dim P)
about
y = image ~ . x ~ P*. k p from D to @P*
so we p i c k an a r c int
P*.
The i n v e r s e image o f
paths from
@T (] P
to
@P.
t u b u l a r neighborhood of each o r b i t form
~.x,
M(n)),
claim,
in
from
M
No e s s e n t i a l This as
M
arcs disc
Doing t h i s
last
the
whose i n t e r i o r is
s t a y s in
a disjoint
family
M(n)
p
of
an e q u i v a r i a n t
procedure s u c c e s s i v e l y f o r
~-manifold
H ~ o M.
that
i n v o l v e d the d e l e t i o n
loss
M(n) C-M.
We
M is /~-homeomorphic t o o under t h e s i m p l i f y i n g a s s u m p t i o n t h a t o b t a i n i n g
of
generality
is
the e q u i v a r i a n t
of
just
us n o t e t h a t
neighborhood of
deleted. (I.e., extend k D , t o produce ~. If
p
in ~
one o r b i t - n e i g h b o r h o o d .
created.
assumption u n d e r s t o o d , l e t
with
P
P*(~ at411~ ~ O,
(whose n e i g h b o r h o o d has p r e v i o u s l y been removed t o
We show t h i s (n) M
in
@P* :
We then e x c i s e from
p.
we o b t a i n a t
fact,
p
Now
P*
is
a
M may be viewed o ~ - i n v a r i a n t f a m i l y of
by the " d i a m e t e r " o f
the f a m i l y o f arcs
the
covering
M is M l e s s an open ~ - n e i g h b o r h o o d o f u-(inside endpoints). o k-1 But t h e n , l e t S be a p r o d u c t n e i g h b o r h o o d S = ~ x D of o P*. fiber
Recalling that PC M , n-k e D (a b l o c k bundle,
there will more
be a
precisely)
O-bundle over
S,
~
~,
in
with
such
that
B = E x ~ is ~-isomorphic to the equivariant tube about ~ (whose e interior was e x c i s e d to produce M . Thus M may be d e c o m p o s e d as o ~ M UB. where B = [Ix E U ]Tx ( ( l endpolnts ~). Now i t is easy to find o B e 0 an e x p l i c i t ff-homeomorphism M ~ M so t h a t M ~ M C__ M i s o o ~ - h o m o t o p i c t o the i d e n t i t y . n+k Thus, changing n o t a t i o n , we r e g a r d him ÷ R as an immersion n+k o h: M ÷ R Taking a ~ - e q u i v a r i a n t t r i a n g u l a t i o n o f M so t h a t h
is
linear
g(h). as in result
~mbedding on s i m p l i c e s ,
Th~.t
im g(h)~_~ ~ /
the p r o o f o f of
this
4.2.
we o b t a i n an e q u i v a r i a n t
follows This
by the g e o m e t r i c i t y
complete~ t h e p r o o f o f
section.
100
of
5.2,
Gauss map 2~7/, j u s t the main
6.1 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 ( w i t h R. M l a d i n e o )
This c h a p t e r , and the succeeding one as w e l l , are based on the d o c t o r a l t h e s i s of my s t u d e n t , work is
Regina Mladineo [ M l ] .
to extend the immersion r e s u l t s of
e x t e n s i o n s of
§5) to the case of
ulated manifold.
The idea of
§4 (and t h e i r e q u i v a r i a n t
immersions i n t o an a r b i t r a r y
Triangulated manifolds, it
stands,
in
relation
m a n i f o l d to i t s
triang-
should be u n d e r st o o d ,
are g e o m e t r i c , r a t h e r than merely t o p o l o g i c a l o b j e c t s . speaking, a p a r t i c u l a r
Roughly
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 of a m a n i f o l d
to the u n d e r l y i n g
PL
underlying differentiable
s t r u c t u r e , as a Riemannian structure.
In p a r t i c u l a r ,
by p u t t i n g the standard m e t r i c on each s i m p l e x , s u c h n o t i o n s as c u r v a t u r e , in
this
PL
the sense of D. Stone, become d e f i n e d [ S t 3 ] , [ S t 4 ] .
Given a t r i a n g u l a t e d m a n i f o l d of dimension, i t ask whether i t
seems n a t u r a l to
supports an " a s s o c i a t e d Grassmann b u n d l e . "
way of background, t h a t any v e c t o r bundle
C
of f i b e r
R e c a l l , by
dimension
G (5) wherein the f i b e r n,k the base space of ~ c o n s i s t s of
n+k
has an a s s o c i a t e d Grassmann bundle
G (x) over any p o i n t x in n,k space of l i n e a r n-planes in the f i b e r
v e c t o r space
G (~) supports a n a t u r a l n - v i c t o r bundle n,k over the p o i n t in G (~) r e p r e s e n t i n g the njk
itself. in
Recall
terms of
this
~(x)
~ (~) n,k n - p la n e
over
the x.
whose f i b e r Y~
x
is
Y
that Hirsch's
immersion theorem thus may be r e s t a t e d , n n+k construction: Let M , W be smooth m a n i f o l d s
and
TM, TW t h e i r r e s p e c t i v e t a n g e n t v e c t o r bundles. Then according n n+k to H i r s c h , f: M ÷ W is homotopic to an immersion i f and o n l y i f
n , g: M ÷ G (T ) so t h a t g (T W) = T . (Of course, n,k W n,k M n we must assume [a~ u s u a l ] t h a t M has no t o p - d i m e n s i o n a l c e l l i f f
lifts
to
k=O.) Moreover, pushing th~s r e s u l t Phillips set,
in
the d i r e c t i o n of
theorem, we may a s s e r t t h a t i f
and i f
f
lifts
to
g
with
101
U C__G (T) n,k W im g C U, then f
the Gromovis
some open
deforms to an
6.2 immersion from
i
with
im h C U, where h denotes the n a t u r a l l i f t i i (W) determined by the immersion.
W to
G n,k Our aim in t h i s
result.
The f i r s t
section i s
to demonstrate an analog to t h i s
t h e r e f o r e , w i l l be to c o n s t r u c t , at l e a s t n+k for a t r i a n g u l a t e d manifold W , the analogue to the Grassman
bundle and i t s
task,
canonical
n-plane bundle.
This spece w i l l
be denoted
(W), and the canonical bundle by y (W). We w i l l then study n,k n n+k n n,k immersions f : M + W , with M a PL m a n i f o l d such t h a t (I) of
f
is
in general p o s i t i o n with r e s p e c t to the t r i a n g u l a t i o n
W
(2)
n M
is
t r i a n g u l a t e d so t h a t each s t a r i s embedded in
with each simplex l i n e a r l y embedded in some simplex of
W.
For such an immersion there is a n a t u r a l Gauss map, i . e . g: Mn +~n,k(W)
covered by a n a t u r a l bundle map
w i t h i n a corresponding sub-complex s u l t of t h i s
section i s
that i f
X
a map
TMn ÷ Yn,k(W)"
Moreover, geometric c o n d i t i o n s on such an immersion i n t e r p r e t e d as a requirement t h a t the Gauss map
W
g
f
has i t s
may be image
XC~_ ( W ) . The p r i n c i p a l r e n,k is a s u i t a b l e subcomplex of
(W), t h a t i s , to extend our previous t e r m i n o l o g y , a geometric n,k n n subcomplex, then any bundle map M , TM + X, y ( W ) I X implies that n,k n there i s an immersion f : M + W whose Gauss map g has image conrained in
X.
We now proceed to the c o n s t r u c t i o n of ~Y
n,k complexes
(W)
~
(W). n,k to be the union, mod c e r t a i n i d e n t i f i c a t i o n s ,
is
(
j+k
where
), )
defined in §2 above.
where
~
j+k
is a
i s a copy of the It
~
j+k-simplex PL
of
W,
Grassmannian
is unCerstood, of course, t h a t f o r
of
and k
j
as
< O,
~j
= ~. The i d e n t i f i c a t i o n s in question are modeled on the s i m p l i ,k ~ j+k i+k c i a l complex W; t h a t i s , i f ~ < ~ are s i m p l i c e s of W, we shall
identify
~.
(a j+k ) j,k This is done as f o l l o w s :
with a c e r t a i n subcomplex of ~
102
i,k
(B I"+k ).
6.3 For each dimension
we choose, once and f o r a l l , a j+k f i x e d i s o m e t r i c copy of the standard simplex A embedaedl i n e a r l y in
Rj+k
j+k
n+k
barycenter at the o r i g i n . ( T h e choice of the parj÷k~ n+k t i c u l a r i s o m e t r i c embedding A R w i l l not a f f e c t the d e f i n i -
t i o n of
with i t s
~
(W).) For each simplex j + k of W pick an isometry n,k j+k j+k in e f f e c t , a 1 - i correspondence of v e r t i c e s , @: ~ ~ a
that is, j+k C R It
j+k i+k ~ < B C_ W,
is then c l e a r t h a t i f
defined i n c l u s i o n i+k subspace of R
t
RJ+k
: ~,~ That i s ,
+
i+k R
there is a w e l l -
j+k R
embedding
the composition
j+k ~-i A ~
~B i+k ~ < B -----~ A C
i+k R
extends uniquely to an a f f i n e embedding
RJ+k ~
i+k R
shift
where
b
of
t h i s embedding by the vector
~6 (m),
inclusion
we obtain
t ~ B.
-b,
: ~ . k C - / ~ i , . To see t h i s , ~,B J, k formal l i n k of dimension ( j , k ; r ) . Let :
face
an i s o m o r p h i s m
relations,
*a,B : 4
D e f i n i t i o n . /<~;fn
n,k
(W) ,k is a copy of
f i e d with i t s j+k i+k
=
1
o
Y,6
induces an
L : (UL, ~L )
(L) = L'
to
specify
image
:
(W)
is
then immediate:
is
t h e complex
be a
be the
an i n c l u s i o n
I
L~ ~ j,~.j,
and where
~,k (eL) ~ / ~ i , (B) ~,6 j , k k
I t must be noted, of course, t h a t i f ajy
t ~6
map
-g
- ~ j , k ( ~ j+k)
I
the barycenter
U = t (U) and Z = the admissiL' ~,B L L' t (Z )" Then we may think of as ~,~ L' :a,B e L ~ eL,, This is clearly compatible with
and may be t a k e n
,k ,k The d e f i n i t i o n of
6.1
we then
determined by
bly t r i a n g u l a t e d sphere defining
let
If
is
Furthermore, the map
:
(i,k;r)-l
as a l i n e a r
•
~,6
103
(j+k) k . k(~ j+k )
where is i d e n t i -
f o rJ' each face r e l a t i o n
<
g
< y,
then
6.4 In g e n e r a l i z i n g ~
to ~ (W) we also f i n d i t convenient n,k ~,n'k n+k to g e n e r a l i z e the canonical map ~ + R to a canonical map - n ,k be a simplex of W and / ~ . (m) C_Gw:~n k(W) ÷ W. Let j + k , j,k ,k
(W)
for V~B
L :
the corresponding copy of a
(j,k;r)
formal l i n k . (3
a i+k
¢-1(V~ B ~,B(L)
V~Biq = V~ . Now, l e t L L PL n - d i s c e m b e d d e d i n
).
,k
We set
V~ = ,EI(V /3 a L= L
j+k i+k ~ < B
Let ~B
Thus
.
V L
~
= ~ / V ~B . L ~
and l e t
is a
j+k - disc in
It
seen t h e n
is
B,
that
with
V~ L
is
a
With these c o n s t r u c t i o n s in hand, we proceed to the d e f i n i t i o n of
Gw. Let
-~j+k-1
denote the sphere in
Rj+k centered at the o r i g i n j +k ~j +k - 1 and circumscribed by the chosen copy of A , i.e. is conj+k j+k tained in A and t a n g e n t t o t h e m a x i m a l p r o p e r f a c e s o f A at their
respective
subdivision
berycenters.
where,
subcomplex
for
any
(j,k;r)
whose s i m p l i c e s
complex
(see
§2,
Recall
are
p.2.9).
that
~.
link
the
images
Thus,
L, of
has a s i m p l i c i a l
j,k the the
cell
(W) by l e t t i n g the simplices be those of each n,k ~ i gives a s i m p l i c i a l s t r u c t u r e to Vn~'~ k(W) since l,k
(B)
~,B
of
a
the
triangulate
(j+k).
This
J~
j,k
(~)
is a s i m p l i c i a l i n c l u s i o n .
Hence, to specify the canonical map s u f f i c e to define i t t3
is
L
simplices
we may s i m p l i c i a l l y
,
e
.
v~ ~ i , k ( ~ ) , ~
GW:~n,k(W)__ ÷ W,
it
will
on v e r t i c e s , provided
G ( v ) E ~ whenever W f o r in t h i s case convex l i n e a r extension w i l l be w e l l -
defined. This in mind, l e t
v
(j+k), and l e t ~, j,k Thus, v moreover, be the smallest simplex f o r which t h i s holds. corresponds to some vertex t a u t o l o g i c a l map t i o n on
~j+k-1
be a vertex of ~ .
Recall the v of the "standard" .J~",k'. °j+k ÷ R Let p(x) ~enote r a d i a : pro iec-
G:~. j,k if x ~ O,
0
if
104
x = O.
So, in p a r t i c u l a r ,
6.5 pG(v) ~ a
j+k
= ~-IpG(v).
Definition.
6.2
G :~ (W) . W is W "-n ,k defined on t h e v e r t i c e s
as a l r e a d y
G
W
With
the
the space
but
e
pair
~ ~ e y ~ L'
L y,
L
(W).
of
for
W and
i
L
L
c_
(L)
i
a
of_~
n,k recall
(W).
{j,k;r)-link
such t h a t
y < ~
L
i s an i+k (8 )
i,k
the convex l i n e a r e x t e n s i o n of
G in mind, the W L a formal (j,k;r)-link.
f o r any
where
we have e ,k
definition
~L c _ ~ j , k
a s i m p l e x of
A~n
G (v) W
Let
(i.e.
(h,k;r)
link
for all
is
~ ) ~.
y
Let
of
j+k m
be
e C /~Z/ (m) L -j, k i ~ L f o r any
not
and
definition
h+k
Let
j+h < m ). Thus -~ -8 e = U el L ~
l~B
We make the f o l l o w i n g o b s e r v a t i o n l e a v i n g the e l e m e n t a r y task of verification
to the r e a d e r : ~
6.3
Proposition.
G (e ~) C V ~ C W. W
L
L
We may now s p e c i f y the
n-dimensional ~C{
locally
by s p e c i f y i n g t h a t
obtain a global
L
c o n s i s t e n c y of PL
PL ~~
bundle ~
y
n,k
(W)
--CI
¥n'k(W)]eeaspaces (Gw]eL)}/TVL"
Noving t h a t the v a r i o u s merely show l o c a l
:
n-bundle
cover ~ n . k ( W )
these v a r i o u s l o c a l
we need
definitions
to
y
(W). To t h i s end, we note t h a t n,k e (~ ~ xQ, o n l y i f ~ < B (or B < ~) and I L < K L K ~,B ( o r ~ K < L). In t h i s case -~B i s a codimension-O s u b m a n i f o l d of K
B,~
V and i t is L and (G -B
~
thus c l e a r t h a t the two l o c a l d e f i n i t i o n s (GwIe)*T~ ~c~ -B through a c a n o n i c a l i d e n t i f i c a e ~ e L K
WleK)*TVB K agree on
tion. Having d e f i n e d ~ map
GW, n
map
the bundle
(W) and, w i t h the a i d of the t a u t o l o g i c a l n,k Yn,k(W), i t remains f o r us to sec how a Gauss
°
sion
M , TM + (W), y (W) a r i s e s from a g e n e r i c PL immer,k n,k n M . W. In p a r t i c u l a r , we s h a l l assume t h a t the immersion f
is
general p o s i t i o n w i t h r e s p e c t to each s i m p l e x of
in
105
W.
Thus i f
6.6 is
a
(j+k)-simplex
manifold forth
of
M
that
o M ~ c~ (not
M
with is
of
W,
M
boundary
:
~M
triangulated
f
-1
~
= f
is
a
(@~).
so t h a t
f
We s h a l l
~
o M be d e f i n e d by M = ~ T * where c~ cz TI~M c~ c~ c o n t a i n e d in the boundary) ~ i t s dual
M ).
As u s u a l ,
g
to
is
yielding
o M = M -
be d e f i n e d
as
(collar
o
a Gauss map
g : M
¢ o f
+
.
sub-
assume h e n c e -
is
a subcomplex.
T
is
cell
on b d y . ) .
follows:
j-dimensional
a simplex of (in
o M ,
On
immerses
Let
~.l c( the m a n i f o l d
the
Gauss map
M
in
Rj + k ,
the
latter
Identifying
space
°
with
(~)C (W) we g e t glM + (W). We now wish t o e x j,k ,k m n,k O this definition to all of M (or to M = M - (collar on @M)
tend if
,
Let sion
~
n+k
be a s i m p l e x o f in
W.
Let
~ = a
My,
= ¢ (f(st (~',M))/q ¥), y face of ~ having ~ ~ 8.
Given
8 < A
a = 8"8. through
let
Let b
o('B) ,
a certain
"~
Finally,
complex
denote the denote the
¥
is
K
of
maximal
~ = ~
o f(~'),
Y where
B
( d e p e n d i n g on
complementary face reflection
(in
dimen
is
a
~,¥).
of
A,
i.e.
n+k R )
of
"~
i.e.
o(~)
NOW l e t
where
= ¢ (y), and s e t Y X (~,y) = 8 ~ X(~,y) 8
~(~',y)
We now c o n s t r u c t
~M
n+k
a
:
{y
designate
:
the
b
-
(x-b)ix
smallest
~
face
~}
of
a
such
that
o ~
which
take
~.
set K
Here,
it
the
form o f
K.
Note t h a t
is
joins K
x(~,~)u
:
understood that are is
naturally an
a s i m p l e x and t h e r e f o r e , spondingly
eL(~,K)
L~ (o(~)*xB(~,y)) the
terms
in
embedded i n
n-manifold we have t h e
may be t h o u g h t
106
(with
union as
is
the
boundary) containing
formal of
this n+k R ,
link
as a c e l l
L(~,K). o f "n,KA~Y)"
LomDlex o
Corra-
as
6.7 Recall
that
for
each
6 < ¥,
we have a l r e a d y d e f i n e d
M , and im ga y)C (W). I t is o b v i o u s by i n s p e c 6 /x~ , ,k that im g l i e s , in f a c t , i n the c e l l e with im g 6 L(~,K) moreover i n the boundary o f t h i s c e l l . We d e f i n e g on a ~ y by g~ tion
linear
extension; that
which
g
y,
through g
CZL(o,K ) , It
if
we l e t
ZL(o,K)C
may e a s i l y
CZL(a,K ) .
remains to o b s e r v e t h a t
so t h a t
the d e f i n i t i o n
is
this
noting that
in
the e x t e n s i o n o f
t o map to a c e l l which a r i s e
of y ~ (a)
details
t h e r e f o r e on a l l
of
map t o a
by c o n s t r u c t i n g on
then t h i s
the r e m a i n d e r of
i.e.
o ~
for
o C 6 < y,y
Roughly, we see t h i s
above,
which i s
on
gl~f~* ~ a
common t o a l l
y.
each ,
by
may be chosen
the c e l l s
(n+k)-simplices
e
L(a,K) We
y.
to the r e a d e r .
We have thus d e f i n e d
is
6.
from the v a r i o u s c h o i c e s o f
leave further
tend t h i s
~ g
X
e x t e n s i o n may be chosen f o r
consistent, a
g ,
a ~
Using t h e c o n t r a c t i b i l i t y
be e x t e n d e d t o
the two e x t e n s i o n s c o i n c i d e on
e
be t h a t p a r t of
o
has a l r e a d y been d e s c r i b e d by the v a r i o u s
map f a c t o r s of
is,
g
on a l l
M where M o o PL bundle map M
a cellular
of
M n M x o as usual i s ~
for ~
each c*.
y
and
We must e x -
a¢~M
TM ÷ y (W). This is done o n,k d e c o m p o s i t i o n {e } of M, where
a r e g u l a r neighborhood ( i n
g:
M )
of
the b a r y c e n t e r
b
of a
O
given simplex this
is
~
of
perfectly
M.
We o m i t d e t a i l s ,
analogous to
merely p o i n t i n g out t h a t
the d e c o m p o s i t i o n
{eL}
of
/~
.
{eL}
of
defined L
i
immersion ~,
~L n,k g(e ) a where
(W).
n,k
or
So,
in p a r t i c u l a r , under the Gauss map g as 6 lie in e where 6' < 6 and L(a, M ) = L' M ) denotes the f o r m a l ] i n k o f a under the
will L(o,
. o(flM ). ~6 ~
Thus,
Gogle
~
will
be an e m b e d d i n g
of
e
a
in
V ' and hence t h e r e i s a bundle map c o v e r i n g t h i s embedding L 6' .~a' 6' TMIe = Te ÷ TV . But, l o c a l l y , y (W)]e is G*TVL, and so a a L' n,k L A t h i s bundle map on Te may be c o n s t r u e d as a map g to y (W) a n,k c o v e r i n g g. I t remains t o n o t e t h a t i f x E e ~ e , the two d e f i T
nitions
co-lncide.
Again, d e t a i l s
are l e f t
107
to t h e r e a d e r .
6.8 In
the
spirit
of
§2,
we c o n s i d e r g e o m e t r i c subcomplexes of
n,k (W)" 6.4
Definition.
iff
given
Theorem,
Let
of
B
and
(W) i s s a i d to be g e o m e t i c n,k 6 = V ~ a ~ ' B ~ f o r some maximal simL
As u s u a l ,
~
C L W
BC~
V ~ L' then e C B. L' we s h a l l now assume t h a t
plex
e
A subcomplex
n M
has no c l o s e d components. Q~
6.5
a fixed
B
be a g e o m e t r i c subcomplex o f
n,k
(W)
(W w i t h
triangulation).
Assume t h e r e
is
a b u n d l e map
f
TM ~ y
n,k
(W)IB
S,S M -.-0 B
Then t h e r e
exists
Gauss map
g
F
Proof:
an immersion
F:
h a v i n g image i n
B.
Consider
B =
~
M + B
e .
with
F
G o f,
As we have p r e v i o u s l y
and the
remarked,
e c B L
in
the
fence,
case of
L subcomplexes o f ~
and we may thus
By the
usual
regard
general
s i m u l t a n e o u s l y t r a n s v e r s e to
e
: e
N
...n
e
for
f
a finite
,
BC B
is
as a map i n t o
position
is
L1
v n ,k
arguments,
all
"strata"
B.
we may assume t h a t of
collection
a homotopy e q u i v a -
of
B
h a v i n g the
formal
f
form
links
Lr
= {L
n
M
..... L } ( o r d e r e d by the face r e l a t i o n ) . 1 r -iv Let M = f (e ) ; thus M is a codimension-O submanifold of L L L The i d e a of the p r o o f i s to o b t a i n a c o d i m e n s i o n - O immersion
M + ~ L L' set
of
with
formal
M ~ L links
compact s u b s e t o f
M + ~ ~ ~ . K L K
Let
o r d e r e d by the
face
int
( ~ ~ ), Le.P L
which
108
~
= {L I . . . . .
relation. is
a
L } r
Then
be a f i n i t e G(ej)
is
PL-homeomorph of
R
a
6.9
Let
M~
G(e~)
denote
in
int(
NOW n o t e We work the
dim ~ =
j,
and
V
be a r e g u l a r neighborhood of
eA# ~ 8 ~
J
must have s i z e
M L
~ # ). Le~ L that for
inductively,
increasing
starting
dimension
j
with
d
as a b o v e ) .
(GIB)o
F
,I ~
of
j
: O; i . e . ,
the manifold
Our i n d u c t i v e
has been d e f o r m e d t o
n+1-j,
the
M#~
F'
so t h a t
induction
(Note
assumption
0 < j
is
~ n.
is
on
that that
F'IMA~
is
an i m m e r -
%
sion
into
~
dim < d. out
for
all~]~
of
We assume t h a t
(--~ o f
arbitrary
length
this
~ n+l
deformation
length).
d,
i.e.
keeps
Moreover,
on a l l
M~
in
MA~ o f
V~ t h r o u g h -
we assume t h a t
on each
M~
#, we have a
PL
bundle
map
the deformation
of
of
bundle
the
original
i
TMIMJ - - ~ maps i s
that
(w)l
y
this
to
under
family
of
covering
covered
restriction
F,
from
deformations for
inclusion
F' I ~
been c o v e r e d
We assume t h a t
Finally
sub-bundle
-=~ T ~
having
map w h i c h
restrictions.
the
F
I
+ T~.
~pk consistent
Moreover respect
F
TMIM~
of
+ ~,
by a d e f o r m a t i o n
viz. this
M~
family to
bundle
maps i s
dim M~O < d,
TM~ C ~* T~p
Mr,
of
bundle
~/c
~.
also
taken
we s h a l l
coming f r o m
assume the
immerF
sion
~'
coincides
These of
with
inductive
dimension
the
sub-bundle
assumptions
d + I.
If
in
we t a k e
inclusion
mind,
d + i
TM~ CTMIM~
we c o n s i d e r
< n,
to
I
~
T~ ,
a typical
we see t h a t ,
in
~p view of I
the
relative
be d e f o r m e d constant
version
property
of
that
the
immersion
the
trivial
of
!
~i~
to
the Hirsch-Poenaru
t o an i m m e r s i o n
on each
deformation
of
Mj,
bundle
~,, ITMj
M~ - - - +
~
is
~ : Mj + ~ .
M~ C @M_~
and,
maps f r o m merely
~.
It
is
109
Finally, the
will
that
inclusion that
restriction
will
latter arising
this
since
F~
be c o v e r e d
where t h e
understood
of
[H-P]
deformation
moreover,
the bundle
M~ c 3 ~ .
a deformation
This
to
is
one Gver each II
II
theorem
may be by a
has t h e from
deformation
is
each d e f : ~ r m a t i o n of
the
global
map
6.10 F:
M ÷ W,
agree)
and
since
on a l l
the
Mj,
~
various
of
deformations
length
> n+l
I
is
a global
that
the
deformation
deformation
ii
-
:
are
d,
it
of
F
to
f
keeps
Mr
in
VTC
for
the
respective
follows
M ÷ W, (~
constant
with
(and that
the
arbitrary),
hence
there
property where
n
F " I M~
:
F~,
(and
similarly
^
wise
the
deformation
each
M ,
TV~,
which,
at
the
see
end
that
of
of
bundle
> 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
7.17 dimension
r.
Again,
we note t h a t as i t
has been d e f i n e d on the
c l o s u r e of
X-X , u e x t e n d s , in a r a t h e r n a t u r a l way, to o u: X ÷ X i n d u c i n g a homeomorphism i n t X + i n t X. For a s u i t a b l e o o p a r a m e t e r i z a t i o n of MIX) as a f i b e r bundle over X , i t is seen o that u on M(X)IZ i s the same as uox, where Z i s t h a t p a r t of BX meeting L~ M(Y) and o dim Y
obtain
u
on a l l
The u s e f u l n e s s of
x
is
the bundle p r o j e c t i o n map.
uo~.
Proceeding i n d u c t i v e l y ,
we
n M
of
the map
u
subsequent s e c t i o n §7.3 where i t
will is
become c l e a r in the
critical
to the d e f i n i t i o n
Gauss map f o r a c e r t a i n c l a s s of p i e c e w i s e - d i f f e r e n t i a b l e Prior
So
to b e g i n n i n g t h a t d i s c u s s i o n , however, we s h a l l
of a
immersions.
digress briefly
to c o n s i d e r the problem of c l a s s i f y i n g L S - s t r a t i f i c a t i o n s on a given n PL m a n i f o l d M , up to concordance. We s h a l l c o n s t r u c t a c e r t a i n space B such t h a t concordance c l a s s e s of L S - s t r a t i f i c a t i o n s of LS n n M correspond to bomotopy c l a s s e s of maps M + B . Our c o n s t r u c LS t i o n of B is l a r g e l y in the s p i r i t of our p r e v i o u s c o n s t r u c t i o n s . LS First, for fixed j, c o n s i d e r the set of a l l a d m i s s i b l y - t r i angulated (i.e •
Rj+k
~ C sJ+k-Ic If
Zv
(j-1)-subspheres
Z
is
~
for
of
the standard E u c l i d e a n sphere
sufficiently
such a sphere and
v
is
large
the segment from
0
to
v
in
Rj+k
k).
a v e r t e x of
the f o l l o w i n g a d m i s s i b l y t r i a n g u l a t e d
~,
we denote by
(j-2)-sphere:
and the
S
Consider
(j+k-1)-plane
U !
o r t h o g o n a l to t h i s be a small ~'
= S' ~
phic to S'
segment and p a s s i n g through the m i d p o i n t .
( j + k - 2 ) - s p h e r e in c Z.
Note t h a t
~k(v,~).
onto a
alized just
(j+k-2)-sphere
subsphere of as in
is
By the ,isual
thus o b t a i n i n g the image triangulated
Z'
U
S" Z~v
centered at t h i s a triangulated
translation of
of
radius ~'.
sJ+k-Ic
§2 so t h a t f o r any
132
S~
Let
m i d p o i n t , and set
( j - 2 ) - s p h e r e isomor-
f o l l o w e d by d i l a t i o n 1
S
map
c e n t e r e d a t the o r i g i n ,
Clearly
Zv
is an a d m i s s i b l y -
Fbe procedure may be g e n e r r-slmplex
~
of
~
we o b t a i n
7.18 the
admissibly-triangulated
(j-r-2)-sphere
~ .
We s h a l l
write
a
T < ~
whenever
T = ~
f o r some simplex
~
of
~.
N o t e t h a t as
0
as in our c o n s t r u c t i o n of ~ h(~,a):
c~
in
n,k (where o*
÷ ~*C_~
§2,
there are natural maps
is the dual c e l l
to
~).
These
0
maps have the usual consistency p r o p e r t i e s and thus we may form a cell
complex
B
with one
j-cell
e
~ image
cZ
f o r each
&
(j-1)-sphere
~,
by i d e n t i f y i n g
c~
with
~*
under
h(Z,~).
Note
0
t h a t t h i s complex has a s i n g l e "embedding"
of
S-1 =~8, in
O-cell corresponding to the unique
S~. ^
Note t h a t there is a n a t u r a l f o r g e t f u l map a r i s e s from f o r g e t f u l l y assigning to the formal a d m i s s a b l y - t r i a n g u l a t e d sphere In f a c t ,
Z,
,k + B
s: link
(U, Z)
which the
regarded as a subsphere of
S .
t h i s f o r g e t f u l map is c o n s i s t e n t with the n a t u r a l maps
~/
~+~ and ~ B + ~n so t h a t we obtaln a natural n,k n,k+l ,k +l,k f o r g e t f u l map s: l i m / ~ + B. n,k n,k We now r e t o p o l o g i z e B ( j u s t a s ~ was r e t o p o l o g i z e d to obn,k r a i n ~ n k.) That i s , i f I 'I are a d m i s s i b l y - t r i a n g u l a t e d ( j - I ) , of subspheres sidered
S~
(i.e.
c-close i f f
such t h a t
h(v)
T
for
large
k)
l e t them be con-
there is a s i m p l i c i a l isomorphism
is within
Euclidean m e t r i c ) f o r a l l that i f
sJ+k
of
~
of
v
vertices
g: ~1
÷~
2
( w i t h respect to the usual v
of
ZI"
(In passing, note sj - i ,
is an equivalence class of t r i a n g u l a t i o n s of
this
procedure puts a topology on In the s p i r i t
B = the set of a l l Z with Z ~ T.) T of the r e t o p o l o g i z a t i o n of ,k t o obtain n,k'
we may now r e t o p o l o g i z e cial
space.
I.e.,
x ~ e i , y ~ ez2 to cZ1
y
B
as the geometric r e a l i z a t i o n of a s i m p l i -
x,y ~ B and
Z1
are now to be considered
is
~-c~ose to
in Euclidean space where ~,~ and
cZ2
rmspectively.
while
~
are pre-images of
This r e t o p o l o g i z a t i o a of
space which we s h a l l designate f l e d w i t h a subspace of
~2
~-close i f is x B
~-close any in is the
B . Note that B may be I d e n t l LS T B L S , namely, the union of the cone points
133
7.19 of all
such t h a t
Z~
t
T.
/
M o r e o v e r , the n a t u r a l map s : ~ ÷ B n,k c passes o v e r t o a c o n t i n u o u s map s : ~ ÷ B . In f a c t , p a s s i n g t o n,k LS the l i m i t , we o b t a i n s: l i m ~ _ + B . n,k n,k LS I t remains f o r us to j u s t i f y the c l a i m t h a t B i s the u n i v e r LS sal c l a s s i f y i n g f o r LS-stratified s t r u c t u r e s on PL m a n i f o l d s . We
first
cZ
d e t e r m i n e some a d d i t i o n a l
tlon
of
the
(j-1)-sphere.
i.e.
a specific,
facts
Given
T,
concrete triangulated
concerning pick
BT,
T
a triangula-
a representative
sphere r e p r e s e n t i n g
K, T.
Let
E be t h e space o f a l l p i e c e w i s e - g e o d e s i c embeddings o f K into S T so t h a t t h e image i s an a d m i s s i b l e - t r i a n g u l a t e d s u b s p h e r e . Clearly, all
such embeddings a r e d e t e r m i n e d by t h e i r
K,
so we may t o p o l o g i z e
number o f
factors
7.2 Lemma.
E T
Proof:
is
is
Let
v a l u e s on t h e v e r t i c e s
E as a s u b s e t of S x...xS T the number o f v e r t i c e s o f K.
of
where t h e
weakly c o n t r a c t i b l e .
~:
i S
+ E . We may t h i n k o f ~ as p a r a m e t e r i z e d T i f a m i l y o f a d m i s s i b l e embeddings ~ : K C S , x E S We must show x that ~ is c o n t r a c t i b l e over E . W i t h o u t l o s s o f g e n e r a l i t y we may N T assume t h a t ¢ : K ÷ S f o r some l a r g e f i x e d N independent of x. x
Let
p
d e n o t e t h e number o f
arbitrarily
as
be the v e c t o r
vertices
v I . . . . . VD~+lConsider (O,q
) ~ RxR
p
where
of
K, and o r d e r t h o s e v e r t i c e s
N+p+l N+I Rp R = R x q.
1
v e c t o r of
is
t h e i th
and l e t
ni
standard basis
1
The a s s i g n m e n t v + n d e f i n e s a unique a d m i s s i b l e i i N+p embedding ¢: K C S . We now d e f i n e a d e f o r m a t i o n ¢ , 0 < t < 1 x,t such t h a t ~ = @, ¢ = ¢. I t s u f f i c e s t o s p e c i f y how ¢ x,O x x,1 x,t behaves on t h e v e r t i c e s of K. ( I t w i l l be c l e a r t h a t t h i s s p e c i f i cation will
Rp.
definej
The f o r m u l a i s : ,
~
in
for
each
Cx,t (vi)
t,
an a d m i s ~ i b ! e embedding o f
= (1-t)$x(Vi)+t.ni/~t
t h e d e n o K i n a t o r d e n o t e s t e e usual
numerator.
7.2
is
thus e s t a b l i s h e d .
134
l;
K).
where t h e symbol
E , c ~ i d e a n norm o f
the
7.20 Now l e t Clearly,
AT
A be the group of s i m p l i c i a l automorphisms of K. T -i acts f r e e l y on ET by a.¢ = ¢oa Thus ET/AT is
the c l a s s i f y i n g space
B
f o r the f i n i t e
group
A .
AT parent t h a t any element of
ET/AT
I.e.,
is ap-
is c h a r a c t e r i z e d p r e c i s e l y by
s p e c i f y i n g 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 ~ T.
But i t
T
there i s a n a t u r a l i d e n t i f i c a t i o n
of
Z
in
B
S
with
= E /A
AT
T
with
T
BTC-BLs" Now, l e t
us consider
D = U c~ T ZeTDj This is p r e c i s e l y the u n i v e r s a l
with the a p p r o p r i a t e t o p o l o g y .
B with s t r u c T t u r e group A . N o t e t h a t , as B has been c o n s t r u c t e d , D is T LS T the pre-image of L.J e . U n d e r the i d e n t i f i c a t i o n map, D -S goes 2ET X T T homeomorphically onto i t s image, where S denotes the boundary T sphere bundle of D . T n n Let M , M be two L S - s t r a t i f i e d s t r u c t u r e s on the same undero 1 n lying PL m a n i f o l d M 7.3
Definition.
MxI
so t h a t
7.4
Lemma.
n
M and o f l e d s t r u c t u r e s on M i f f M i
bundle over
n
M are said to be concordant L S - s t r a t i 1 there i s an L S - s t r a t i f i e d s t r u c t u r e on
c o i n c i d e s with t h a t induced on
Mx{i},
i
= 0,1.
B i s the u n i v e r s a l c l a s s i f y i n g space f o r LS-stratiLS n f l e d s t r u c t u r e s on PL m a n i f o l d s . I . e . , f o r any PL m a n i f o l d M , concordance classes of correspondence with The proof is
set i t
First
n
M
are in
1-1
[Mn,BLs ] .
rather straightforward.
of the argument w i l l we s h a l l
L S - s t r a t i f i e d s t r u c t u r e s on
However, inasmuch as p a r t CT-
be needed f o r the proof of the main theorem ~ 5 ,
out in some d e t a i l a t t h i s p o i n t .
of a l l ,
we s h a l l
"stratify"
B i n t o ,ion-manifold s t r a t a LS one f o r each equivalence class of t r i a n g u l a t i o n s of a f i n i t e T dimensional sphere. Each X i s to have an open n~Ighbornood I o o k T X ,
ing l i k e an open d i s c bundle w i t h s t r u c t u r e group proceed i n d u c t i v e l y .
At the
0 th
A . As usual, we T stage, we take the unique O - c e l l ,
135
7.21
superscript stage, "I"
context.
over the it
o X1.
by
the
X . This i s t o be p a r t o f o d e n o t e s the " t r i a n g u l a t i o n " of
we l o o k a t t h e unique t r i a n g u l a t i o n
in
O
O
and d e n o t e t h i s
X , where the o in -1 =~,. At t h e f i r s t S O
S ,
of
of B LS image c o n t a i n s
0cZ-*,
C o n s i d e r the image i n
O-spheres in S®. 1 We l e t X1 = B1
This
which we d e n o t e
X
~
ranging
and we d e n o t e
O
be the union o f a l l
t h e cone p o i n t s of
~. Now suppose we have reached the
jth
stage,
i.e.
we have
stratified
LJ im D , T a t r i a n g u l a t i o n o f a sphere o f dimendim T < j - 1 T T sion <j-l. The s t r a t a a t t h i s stage a r e X . We then a d j o i n a l l J D with dim T = j by the a p p r o p r i a t e a t t a c h i n g maps on t h e c o r r e T T sponding S and we d e f i n e X = B for dim T = j . For T T TJ+I T dim T < j - 1 we l e t X = X. U U i m ( c ~ - * ) where ~ ranges o v e r j+l 3 the simplices of j-spheres ~CS such t h a t ~k(o,Z) ~ T.
The f i n a l T X =
result
of
In t h e i n t e r e s t
inductive
of c l a r i t y ,
deformation retract. how the cone (o
this
procedure
is
to
have
T L.) X.o j>dim T J
cZ
a simplex of
decomposition different
It
is
further
useful
was decomposed i n t o ~)
{M(X))
in
at
this
contains
B as a T to r e c a l l
point
codimension-0 cells
the p r e l i m i n a r y
o f an
T X
we n o t e t h a t
N , N i
t o our s p e c i f i c a t i o n
LS-stratified
manifold
spheres a r e i n v o l v e d we use the n o t a t i o n
M. N~~J
of
the
When N~
etc.,
and
O
note t h a t A , Z c T. T with
t h e d e c o m p o s i t i o n may be t a k e n t o be i n v a r i a n t M o r e o v e r , we must remark t h a t
the m a p s
h(Z,a):
c ~k(~,~)
+ o .
the choice is
I.e.
if
we l e t
we have h - i N Z = N~ O
*
h - I N ~ = N~'! T
and
where
o < T
T
136
and
T = T'*~
under consistent Z
m
= ~k(°,Z)
7.22 with
T
a simplex of
~k(o(,Z).
(Of course
h-lN ? '~ =~Q, i f
~ ~ T.)
T
Thus t h e image o f This
{-J NI~ i n B ~ET * LS l e a d s us to a n o t h e r d e c o m p o s i t i o n
is
a disc-bundle over
{MT)
of
BLS,
B . T d e f i n e d as
follows: Let B( j ) denote 'L-) im D C__ B . Assuming t h a t B(j) dim T<j T LS M (~ has a l r e a d y been s p e c i f i e d f o r dim T ( j - 1 we l e t T MT ~ B( j + l ) = (MT~ B( j ) ) L)W where W = image(~iU_NZ).o For dim T = j
we l e t
M C~ B( j + l )
=
U im N!.
At the c o n c l u s i o n o f
this
T M~ B( ~T T i n d u c t i o n we thus have M = L] J) Now s e t X = X /l M . ^T TT j T T We c l a i m t h a t BI.C_ X C X and t h a t the d e f o r m a t i o n r e t r a c t i o n o f T T X to B may be f a c t o r e d as a d e f o r m a t i o n r e t r a c t i o n o f X onto T ^T AT X , f o l l o w e d by a d e f o r m a t i o n r e t r a c t i o n o f X onto B . Clearly,
T~ T B = U M . Moreover M i s a PL d i s c b u n d l e o v e r X onto with LS T T T s t r u c t u r e group A . T These p r e l i m i n a r i e s taken care o f , we a r e now ready t o i n d i c a t e n n how a map f : M + B , M a PL m a n i f o l d , i n d u c e s an LS-stratified LS n s t r u c t u r e on M F i r s t o f a l l we may assume t h a t , modulo d e f o r m a -i tion, f has the p r o p e r t y t h a t f ( M ) =~8.. f o r dim T > n. This T i s c l e a r as a m a t t e r of e l e m e n t a r y g e n e r a l - p o s i t l o n c o n s i d e r a t i o n s . We n e x t c o n s i d e r a l l
of d i m e n s i o n n-1 and make f PL T -I T transverse-regular to X , so t h a t , in g e n e r a l , f X is a set of n T T i s o l a t e d p o i n t s of M . Let Z denote t h i s s e t and n o t e t h a t Z -1 has a t u b u l a r n e i g h b o r h o o d K = f M . In f a c t , K + M has the T T T T T T s t r u c t u r e o f a b u n d l e map o v e r Z + T Let Q = LJ K, T o dim T=n-1 T and n o t e t h a t fIQ is t r a n s v e r s e to a l l X f o r dim T a r b i t r a r y . o Now assume, i n d u c t i v e l y t h a t f o r a l l T of dimension ) n-i: -I n a) f ( M ) is a codimenslon-O submanlfold of M such t h a t T ^T -1 flf-l(MT ) is t r a n s v e r s e to X , and such t h a t f:f (M) + M is a oi T) aT T T b u n d l e map c o v e r i n g f (X + X .
b)
T
~ f-I(MT) dim T>n-t
is a codtmension-O
137
submanifold
Q of Mn. 1-1
7.23
It
follows
that
flQ
i-i Assuming ( a ) and ( b ) , l e t of
course,
-1 XT) f ( ~
f
is
is
t r a n s v e r s e to T
be o f
a l r e a d y t r a n s v e r s e to
n
- I ^T Q. ) = f (X). I-IT t o be t r a n s v e r s e t o ~ . In f a c t (M - i n t
T
X
for
all
T.
dimension n - i - 1 . T X . In f a c t
On
BQ., I
We then deform f rel Q so as T -1 T i-1 Z = f (X) acquires a neigh-
borhood K = f - 1 ( M ) so t h a t K + M is T T T T T T Z ÷ X M o r e o v e r , we may do t h i s f o r a l l
a bundle map c o v e r i n g n-i-1
dimensional
T
s i m u l t a n e o u s l y , and s e t t i n g reached a s i t u a t i o n
Q = Q U ~ K we have i i-1 dim T = n - i - 1 T where s t a t e m e n t s (a) and (b) a r e t r u e w i t h n-i
r e p l a c e d by n-i-l. The i n d u c t i o n t e r m i n a t e s when i = n, with n Q = M . At t h i s t e r m i n a l s t a g e , f is s i m u l t a n e o u s l y t r a n s v e r s e to n T all X , and i s , in f a c t , L S - s t r a t i f i e d , w i t h s t r a t a g i v e n by t h e -i T connected components o f the v a r i o u s f (X). -1 T stratum with X ~ f (X), then T = link X.
If
XC M
is
a
n {M(X)} of M may be s p e c i f i e d -1 t o be t h e component o f f (M) such t h a t X is a -1 T T f (X) and M(X) ~ X ~ ~ Then X = X n M(X) is O
M o r e o v e r , the d e c o m p o s i t i o n taking
M(X)
component o f -1 AT f (X).
I t i s c l e a r t h a t t h i s argument may be r e l a t i v i z e d ; if n n f: M ÷ B i s a map such t h a t flBM has a l l t h e p r o p e r t i e s LS o b t a i n e d i n t h e t e r m i n a l stages of t h e above p r o c e d u r e , then f be d e f o r m e d , properties.
rel It
LS-stratified homotopy c l a s s from
[Mn,B
structures
LS on
]
@M,
so t h a t
the r e s u l t i n g
cal
may
map a l s o has these
therefore
f o l l o w s t h a t , up t o c o n c o r d a n c e , the n s t r u c t u r e on M o b t a i n e d from f depends o n l y on t h e n of f in [M ,B ]. Thus, we have e x h i b i t e d a map LS t o t h e s e t o f concordance c l a s s e s o f LS-stratified n M .
[We ask the r e a d e r t o n o t e t h a t which w~ s h a l l
by
allude
t h e argument above i s
s u b s e q u e n t l y , in
t o an i m p o r t a n t s t e p i n
that
the p r o o f of
To c o n c l u d e the p r o o f o f
the c u r r e n t
138
it
Is
ir
one t o
essence i d e r t l -
subsequent r e s u l t s . ] lemma, we must show t h a t
7.24 n
any
L S - s t r a t l f i e d s t r u c t u r e on M a r i s e s , up to concordance at n l e a s t , from a map M ÷ B and that concordant L S - s t r a t l f i c a t i o n s LS' a r i s e from homotopic maps. Let {X} be the s t r a t a of the s t r a t i f i e d n n manifold M and {M(X)} the corresponding decomposition of M , as e a r l i e r described.
map
Now for a l l
O-strata
X,
there is c l e a r l y a
f : M(X), X ÷ M~
x
,B~ . Moreover, t h i s is a map of ~(x) spaces in the sense t h a t for any stratum Y with
,~(x)
stratified
#
X < Y, Y ~ M(X) goes to
X7(Y)
~My(x) .
T h u s , on the subspace
M(X) of Mn we have a stratum-preserving map f =~f dim X=O (j-l) o x Suppose, now, i n d u c t i v e l y that on M = L_) M(X) we M(J_I) dim X(j-1 J(X) have defined a map f : + B such that f (X) C X j -t LS j -1 n and f j _ I ( M ( X ) ) c ~M"(x) for a l l s t r a t a X of M having dimension j-l.
Suppose, moreover, that each map
A~
- d i s c bundle map over f IX (X) j-I o l a r , that for every Y of dimension f j _ l ( Y ~ M(X)) C x~(Y)~
M~(X);
f
j-11
Y ~ M(X)
(Y)
^~(Y)
o
Therefore, l e t
~ j
with
furthermore i t
f j _ I ( M ( Y ) ~ M(X)) C M~(y) yq ~ ( X ) ' f j _ I ] ( M ( Y ) ) ~ MIX)) ~ +
f IM(X) ÷ M, is an (x)j-I ~ (X) ÷ X This means in p a r t i c u X < Y, implies that
and, in f a c t ,
is a~(y)
that
A~(y)-disc bundle map over
M~(X~
Y
be a
j-dimensional stratum and r e c a l l the l
A~(y)-disc bundle homotopy type of
Y . Recall too that X4{Y) is of the o ~(Y) _B~(Y) and thus X i s , e s s e n t i a l l y , the M ( Y ) over
universal c l a s s i f y i n g space for the group _A~(Y) while the associated canonical disc bundle.
M~(y)
is
Thus,
fj_llY
~ M( j - l ) + x~(Y) extends to a map gy : Yo ÷ x~(Y) while (j-l) fj_IlM(Y) ~ M extends to f y : M(Y) ÷ ..M~(Y) with the property o
that
fyIY,
= gy and o We thus may define f j and, c l e a i l y , be replaced by
f
j j.
f
is an
~AF; -disc bundle map over gy" (Y) (j) : f _1U L_~ f on M = L_~ M(X) J dim Y=j Y dim X<j has a l l the stated p r o p e r t i e s of f if j-1 j-t Finally, let f = f . I t i s clear t h a t n
139
7.25 n M + B produces on LS w i t h which we began, i . e .
f:
n M
LS-stratified
structure
for all X, f being n t r a n s v e r s e to a l l X . T h u s the map from [M ,B ] t o c l a s s e s of T LS n L S - s t r a t i f i e d s t r u c t u r e s on M is s u r j e c t i v e . F u r t h e r m o r e , the argument above i s n show t h a t i f M
f
p r e c i s e l y the -I
(Xy(x))~ .-~X
quite easily relativized, has two
LS
stratifications
n maps f ; f : M + B such t h a t M ,M o 1 LS o 1 then t h e r e i s a map F: Mxl + 8 with LS duces t h i s
concordance.
cordance c l a s s e s of
the
proof
of
Hence,
so t h a t one may q u i c k l y
M ,M a r i s i n g from o 1 are a b s t r a c t l y c o n c o r d a n t ,
F = f , F = f which i n o o 1 1 goes b i j e c t i v e l y onto con[Mn,BLs]
LS-stratified
s t r u c t u r e s on
7.4.
140
n M .
This completes
7.26 7.3.
P i e c e w i s e - D i f f e r e n t i a b l e Immersions and T h e i r Gauss Maps; The Main Theorem Our aim in
this
section is
to c o n s t r u c t a Gauss map
g(f):
M ÷ ~c where f i s a p i e c e w i s e - d i f f e r e n t i a b l e immersion { i n n,k n a sense soon to be m o r e p r e c i s e l y d e f i n e d ) o f the PL m a n i f o l d M n+k in R Before d i s c u s s i n g the p r e c i s e class of mind, we must f i r s t i n t r o d u c e d in
r e f i n e somewhat the n o t i o n of
the p r e v i o u s s e c t i o n .
duce the concept of Recall f i r s t §7.2.
immersions we have in stratification
In p a r t i c u l a r ,
we s h a l l
intro-
smooth s t r a t i f i c a t i o n .
the d e f i n i t i o n
of
strictly
stratified
m a n i f o l d from
Consider the a d d i t i o n a l r e q u i r e m e n t t h a t the s t r a t i f i c a t i o n
be
smooth i n the f o l l o w i n g sense: Given a s t r i c t l y - s t r a t i f i e d manifold n M , we s h a l l c a l l i t smoothly s t r a t i f i e d as w e l l i f and o n l y i f each stratum is
smooth.
By t h i s
we mean simply t h a t each st r a t u m
be t h o u g h t of as a co-dimension-O submanifold of X'
( w i t h o u t boundary).
(X'
is
X
some smooth m a n i f o l d
merely to be t h o u g h t of as i n d u c i n g ,
the smoothness of
X.
We do n o t view
X < Y,
is
as a subspace of
X
in
M .)
t o g e t h e r smoothly in
then the i n c l u s i o n
embedding of a neighborhood of It
n
X
Moreover, we r e q u i r e t h a t the s t r a t a f i t sense t h a t i f
!
X
XC Y into
the
extends to a smooth e
Y .
now easy to d e f i n e a s m o o t h l y - s t r a t i f i e d m a n i f o l d in
more general case where the s t r a t i f i c a t i o n cular, a stratified
may
is
not s t r i c t .
m a n i f o l d becomes smoothly s t r a t i f i e d
the
In p a r t i when p r o -
v i d e d w i t h an a t l a s of c h a r t s smoothly and s t r i c t l y , the smooth, s t r i c t
stratifications
respectively coincide. smoothly s t r a t i f i e d
{U ) s u c h t h a t each U is s t r a t i f i e d 1 i and s u c h t h a t on o v e r l a p s of the form U. ~ Uj, I
It
i n h e r i t e d from
U
i
f o l l o w s t h a t the i n t e r i o r s
and
U
j
of s t r a t a of
m a n i f o l d s have s p e c i f i c smooth s t r u c t u r e s .
Let u~ now narrow the c a t e g o r y under c o n s i d e r a t i o n to smoothly stratified This
is
m a n i f o l d s whose u n d e r l y i n g
PL
stratification
is
LS.
the kind of geometric o b j e c t whose immersions i n t o Euclidean
141
7.27 space we s h a l l
be s t u o y i n g .
Accordingly,
let
us s p e c i f y
a smoothly
LS-stratified
map.
note t h a t
First
precisely n manifold M in
what k i n d o f immersion of n+k R g e n e r a t e s a Gauss
g i v e n such a m a n i f o l d ,
we a c q u i r e a c e r t a i n
a d d i t i o n a l s t r u c t u r e on i t s t a n g e n t b u n d l e . Assume, m o m e n t a r i l y , n that M is s t r i c t l y - s t r a t i f i e d . For any s t r a t u m X, let X be t h e smooth, b o u n d a r y l e s s m a n i f o l d c o n t a i n i n g s u b m a n i f o l d , as p o s i t e d in fied
manifolds.
certain with [p]
Let
subset of
p(O) of
p
= p, at
p
the d e f i n i t i o n
be an a r b i t r a r y
T (X'): Let p P and p ( [ 0 , ~ ] ) C_ X
0
is,
essentially,
of
X
as a codimension-O
smoothness f o r
point
in
X.
strati-
We c o n s i d e r a
be a smooth t r a j e c t o r y
p:
for
The l - j e t
some small
~ > O.
a tangent v e c t o r to
X'
R + X'
at
p.
We l e t
T (X) be t h e union o f a l l such v e c t o r s . In g e n e r a l , T (X) P P i s a E u c l i d e a n cone b u t n o t n e c e s s a r i l y a s u b v e c t o r - s p a c e o f T (X). n p 1 We may then v i e w T (M) as LJ T ( X ) . The t o p o l o g y a r i s e s P p~X P from n o t i n g t h a t i f p ~ X < X then T (X) i s n a t u r a l l y viewed I 2 p 1 as a sub-cone o f T (X), i n d e p e n d e n t o f the p a r t i c u l a r c h o i c e o f p 2 I I X and X . Thus, the union must r e s p e c t t h i s i d e n t i f i c a t i o n , and I 2 t a k e s on t h e o b v i o u s weak t o p o l o g y of u n i o n . n Now we drop the a s s u m p t i o n t h a t M be s t r i c t l y stratified and n note t h a t the decomposition of T (M) as L) T (X) may s t i l l be P p~X P made s i n c e a s t r a t u m X containing p is, locally, t h e union o f strata
of a s t r i c t l y
stratified
We a r e now i n a p o s i t i o n
neighborhood of
p.
t o c h a r a c t e r i z e t h e immersions t h a t
we
r e a l l y want t o c o n s i d e r . F i r s t of a l l , we demand t h a t an immersion n n+k f: M ÷ R be smooth on each s t r a t u m . (For a s t r i c t l y - s t r a t i f i e d manifold,
this
means t h a t
for
each s t r a t u m
X,
fiX
extends to a n
smooth map on course,
X .
In : h e case o f
non-strictly-stratified
M ,
we m e r e l y impos~ the c o n d i t i o n above on each s t r i c t l y -
stratified chart.)
142
of
7.28 This c o n d i t i o n , that,
of i t s e l f ,
is
not enough f o r
our purposes.
Note
in
i t s presence, we o b t a i n a w e l l - d e f i n e d " d i f f e r e n t i a l " n n+k df : T M ÷ T (R ). To see how t h i s i s d e f i n e d assume, f o r P P f(P) n the moment, t h a t M is s t r i c t l y - s t r a t i f i e d ; then we may d e f i n e ,
for
p { X, df
on T iX) as df IT iX) where f is a smooth e x t e n p p i p 1 sion of f l X to a neighborhood of X in X' Clearly, df is P w e l l - d e f i n e d in t h i s way, i . e . i t does not depend on which T iX) we P r e g a r d as c o n t a i n i n g a s p e c i f i c element of T(M). Equally c l e a r l y , P the d e f i n i t i o n of df may be made to hold f o r n o n - s t r i c t l y P stratified m a n i f o l d s as w e l l .
Of c o u r s e , we do n o t c l a i m t h a t df = ~ df is a continuous n pEM P map on the t a n g e n t bundle of M . In f a c t , t h e r e i s no " n a t u r a l " t o p o l o g y on the
L,) T (Mn), i . e . no c a n o n i c a l way of i d e n t i f y i n g P P n t a n g e n t bundle of the u n d e r l y i n g PL m a n i f o l d M .
PL
it
with
However
o
when we r e s t r i c t
to the i n t e r i o r
X
of a s i n g l e s t r a t u m
X
(i.e.
o
X
=
U Y)
X-
we f i n d
that
T ( M n)
Y<X
= ~ o T ( M n)
X
p~X
t o p o l o g y , w i t h r e s p e c t to which
dfITxM
Now in the case of an immersion n M , f
n
n+k
÷ TR
is
continuous.
of a smoothly
LS-stratified
smooth on each s t r a t u m as above, we may e s t a b l i s h the f u r t h e r
requirement that its
f
does have a n a t u r a l
P
: T ( M n) + T (R n+k) be a homeomorphism onto P P f(P) Moreover, we note t h a t i f t h i s r e q u i r e m e n t is f u l f i l l e d ,
image.
then the cone
df
(T (Mn))c_ T may be r e p r e s e n t e d as a (R n+k) P P f(P)n c e r t a i n d i r e c t sum, i . e . df (T M )) = P (~)Q where P is the subo P P o vector-space d(flX)(T iX)) where p ~ X, and where P Q = df (T ( M n ) ) ~ P~ . This f a c t is e a s i l y seen since the cone P P n , , T ( M ) may be decomposed as T iX) ~) Q f o r some cone Q . Since p P df is a map of cones, i t f o l l o w s t h a t Q may be taken to be the P image o f Now
df
P i 0 ,
df
(Q)
a~d hence
combinatorial object. infinite
cone
(Rn+k ) + Pf(P) are n a t u r a l ~ y a s s o c i a t e d w i t h ~ c e r t a i n
under p r o j e c t i o n
c iX)
O.
That i s ,
Q'
T
may be t h o u g h t of as the
on the s i m p l i c i a l
143
complex
%i X ) .
The same
,
7.29 p i c t u r e holds f o r unit
sphere in
Q,
therefore.
That i s ,
the sub-space
U = P ~ T P f(P) is a t o p o l o g i c a l sphere, and i t
Q ~ S(U ) = ~ P o v e r , where the s t r a t a are of the form over the s t r a t a
such t h a t
that,
p,
for
each
~
X < Y.
To summarize, t h e n :
e)
f
B) df
Y
df
is
be the
stratified,
more-
df
with
X < Y.
Here
We s h a l l
be i n t e r e s t e d in smoothly LSn n+k and immersions f : M ~ R such t h a t
n M
is p i e c e w i s e smooth, i . e .
strictly-stratified
is
S(U ) P then
is admissibly t r i a n g u l a t e d with simplices
(T Y) ~ S(U ) f o r each P P P dim a(Y) = dim Y-dim X - I .
manifolds
we l e t n+k (R )
(T Y) (~ S(U ), Y ranging P P P Our l a s t r e q u i r e m e n t of df is
o(Y) = df
stratified
if
local
smooth on each s t r a t u m ( o f any
chart).
nowhere s i n g u l a r ,
i.e.
for all
p,
Rn+k
P
n : T (M ) ÷ T ( ) i s a homeomorphism onto i t s image. P F(P) o y) f o r any p ~ X (X a s t r a t u m of dimension n - j ) , the o
natural where
stratification
(T M) i s of the form df (T X) ~ c ~ P P P P P is 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 in t h e ~ u n i t
~p
sphere of
of
df
X)], ~ P P Now note t h a t when f
where
X
[df(T
i s an
formal l i n k
being s i m p l i c i a l l y
equivalent to~(X). o is an immersion as above, and p ~ X,
( n - j ) - d i m e n s i o n a l s t r a t u m , we o b t a i n a c e r t a i n
of dimension
j.
That i s ,
have: of
L ( p , f ) = (U ,~p)._ (U n+k P P R in the obvious way.)
is
with
identified
We now a s s e r t t h a t given a smoothly
U = [df(T X)], we P P w i t h a v e c t o r subspace
LS-stratified
manifold
n M
and s u f f i c i e n t l y large k, t h e r e e x i s t s an immersion ( i n f a c t an n n+k embedding) M + R satisfying ~, B, y. Moreover, t h i s w i l l be unique up to i s o t o p y .
We leave t h i s
The Gauss :nap g ( f ) : satisfying
~, B, y
Mn +~C~n __,k
to
mean t h a t
of an immersion
may now be d e f i n e d ,
a r a t h e r u n i m p o r t a n t way, canonical
to the r e a d e r .
g(f)
g(f) is
fails
n
n+k ÷ R
We note in advance t h a t ,
to be c a n o n i c a l i f
determined
144
f:M
exactly
i,
we take
on e a c h
point
of
7.30 n
M
by the geometric data of the immersion. However, up to an n isotopy of M , the Gauss map w i l l be w e l l - d e f i n e d so t h a t t h i s f a i l u r e is of no more than passing c u r i o s i t y . n n+k Given f : M ÷ R , r e c a l l the decomposition
We begin be d e f i n i n g X.
g(f)
on
This is e a s i l y done since
s p e c i f i c way, where
Z
a c e r t a i n formal l i n k ZL(x,f ) = Z(x,f) then
is
M I X ) for a l l MIX)
is
N~
of dimension
n,
where
{x}
isomorphic, in a s p e c i f i c way, with~
M I X ) with
cZ(X,f)
in a
At the same time, we have
since there is a standard way of i d e n t i f y i n g
may i d e n t i f y
n M.
of
0-dimensional s t r a t a
i d e n t i f i e d with
is the l i n k ~ ( X ) . L(x,f)
{MIX)}
and thus map
NL
MIX)
= X,
and
~.
But
with
c Z,
to ~ c
we by
C ~ cn'k composing t h i s i d e n t i f i c a t i o n map with Doing t h i s f o r a l l on
M(0)
0-dimensional
X
L~ MIX). dim X=O Suppose now, i n d u c t i v e l y , t h a t
(j-l) M
c~(x,f) ÷ eL(x,f)
n,k y i e l d s the d e f i n i t i o n of
g(f)
:
g(f)
has been defined on
=
~ MIX). We wish to define g ( f ) on MIX) f o r a l l X dim X<j of dimension j . Recall t h a t MIX) is parameterized as a twisted product
X x N~, OT
NL
as
cZ
where
Z =~(X).
Again, we may re-parameterize
*
and thus we have a way of t h i n k i n g of
MIX)
as
X x cZ. OT
Moreover• f o r any
x ~ X,
we may i d e n t i f y
x × c ~ C X× c Z
with
T
cZ(x,f).
Hence, we have a continuous map
x + L(u(x),f),
where
the decomposition to cover
ulint
X
u
is the map
{MIX)}
M
of
( i n t X )x c~ ÷ P o ~
Z
described in 7.1. we r e : e i v e a map
is s t r a i g h t f o r w a r d
by a bundle map from the normal bundle of
int X
•
to the normal bundle of
x ÷ L(u(x),f)
+~P given by --?. O described e a r l i e r when
X + X o was s p e c i f i e d . Now i t
O
in
int X
O
X.
Consequently, the map
X into ~ . may be covered by a bundle map o where P is the natural c~ bundle over ~ L
But 9X
P from
is tile pre-image of int
a l r e a d y , by assumption, defined
X × cZ ~M(X)
OT
g(f)
145
NZC~Z~n,k and thus tO
N .
Z
!~9 have
on the remainder of
MIX)
7.31 (i.e.
M(X)-M( j - l ) M( j - l ) g(f)l
with
M(J-1)Lj
M(X).
M(J) : M ( J ' I ) u
= ( i n t X x c~)). We claim t h a t gx is c o n s i s t e n t o T (j-l) i.e. g(f)JM ~ gx is a continuous map on
In other words, dimL'~x:jM(X)
we may define
by
g(f)IM j - I U
g(f)
on
dlm"V gJx ' : "
We s h a l l leave to the reader the v e r i f i c a t i o n of the claim of continuity. Note t h a t the c o n s t r u c t i o n of the decomposition
{M(X)}
g(f)
depends on the choice of
and on the p a r a m e t e r i z a t i o n of e a c h M(X).
However, since the choices are unique up to a stratum-preservlng n isotopy of M , the Gauss map is unique in p r e c i s e l y the same sense• Of course, we are also obliged to define the bundle map n c TM + y which covers the Gauss map g ( f ) of an immersion• n,k (j-l) w i l l s u f f i c e to c h a r a c t e r i z e t h i s bundle map on TMJM(X)-M all
j-strata
X x c~,
X,
all
y~ = ~ ( X ) .
o T
(uo~I*TX ~)~*n p r o j e c t i o n is
where
C
is the
with
T
P, [
i.e.
D = C)T L (. u ( x~) , f
in
is merely
naturally
as
isomorphic
to
PL ( n - j ) - b u n d l e
T X
x o
is
M(X)
for
X × c~ whose o T More s p e c i f i c a l l y , l e t x ~ int X and consider o In a natural way TMJC = T ~)~ where r. is the
7.
n
TMJM(X)
bundle with constant f i b e r
identify of
Once more, parameterize
We see t h a t
xxc Z C M(X)-M ( j - l ) trivial
j.
It
TD (E) B
u(x)
where
8
.
T
Now l e t
u(x)
X.
H e r e we are using
u
to
D
denote the image f i b e r disc c T h e n Yn,k p u l l e d back to D
is the t r i v i a l
bundle with constant f i b e r
(U
) = (U(u(x),f)) Since, on C, g ( f ) f a c t o r s through a L(u(x),f) homeomorphism C + D and dfJT X identified T X with u(x)
U(u(x),f)) pullback to it
,
u(x)
we have a n a t u r a l i d e n t i f i c a t i o n of TMIC with the c g(f) as C, via D, of Yn,k • Thus the covering of
r e s t r i c t s to
C,
is defined, and since every p o i n t of
M
is in
w
TM + y covering n,k (Again, v e r l f i c a t i o l J t h a t t h i s n,a? i s , f i r s t of a l l , c o n t i a u -
p r e c i s e l y one such g(f)•
C,
we get a bundle map
ous, and, in a d d i t i o n , a bundle map is l e f t
146
to the r e a d e r . )
7.32 We now specify what is to be meant by a "geometric subspace" of ~n
i . e . , those spaces which are to play a r o l e analogous to t h a t k" of geometric subcomplexes of --/~-,k" In t h i s regard, r e c a l l t h a t i f
T
sJ-i,
is an equivalence class of t r i a n g u l a t i o n s of
the set of formal
j-dimensional l i n k s
L = (U,Z)
admits a natural map to the standard Grassmannian, defined by
(U,Z) + U
t h a t we t h i n k of hood
;
by 7.1,
this
then
with
Z ~ T
e T : ~ T ÷ Gn_j,j+ k
is a f i b r a t i o n .
as embedded in
~T =
Recall also
with a c e r t a i n neighbor-
N . n,k T Consider subspaces H C ~ C~ with the f o l l o w i n g property " n ,k (I) If H /~ ( i n t NT) = ' B ~ then H ~ ( i n t NT) is an open disc A
bundle ( i . e .
with f i b e r
open in ~ T
and
~T IH ~ T
~
cT-T)
BT(H~ ~PT)
over
HflO~T.
is open in
Moreover
Gn_j,j+ k.
H(~ ~ T
is
Finally,
is a f i b r a t i o n .
We c a l l
such subspaces " g e o m e t r i c . "
The obvious examples of
where ~-/~is some c o l l e c t i o n U N T~ T of equivalence-classes of t r i a n g u l a t i o n s having the property t h a t i f such subspaces are of the form
Z s T ~~
and
v
is a vertex of
Z,
then
[~k(v,Z)] ~.
The main theorem of t h i s section may now be stated: c n 7.5 Theorem. Let H be a geometric subspace of ~ . Let M be n,k a PL manifold with no closed components. Suppose t h a t the map n n c h: M ÷ H is covered by a bundle map TM + y IH. Then: n n,k M admits 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 m a n i f o l d , so that,
with respect to t h i s s t r a t i f i c a t i o n , there is a piecewisen n+k d i f f e r e n t i a b l e immersion f:M + R ( s a t i s f y i n g ~, 6, y) so n c g ( f ) : M +~/ has i t s image in H and, moreover, g ( f ) is n,k homotopic to h in H. There i s ,
as w e l l , a r e l a t i v e version of t h i s r e s u l t : n
be a smoothly L S - - s t r a t l f i e d manifold and C o r o l l a r y . Let V p n M be obtained from V by adding handles of dimension < n. let n n+k is a p i e c e w i s e - d i f f e r e n t i a b l eimmersion, with Suppose f IV + R o 7.6
147
7.33 r e s p e c t to t h i s
stratification
so t h a t
g(f
n
):
+ H (H o ,k g e o m e t r i c ) extends to h: M + H while the bundle map Tvn + yC IH n c n,k covering g ( f ) extends to a bundle map h: TM + y IH covering o n,k h. Then: There is a smooth L S - s t r a t i f i c a t i o n of n V
n and
an
immersion
f:
V
n
M
M
÷
R
extending
f
•
d i f f e r e n t l a b l e with respect to t h i s g(f): rel
lish
Mn + , ~n
,k
has i t s
), in H. o We s h a l l b r i e f l y
extending t h a t of
n+k
image in
,
stratification. H
and
and
f
piecewise
0
g(f)
is
Moreover, homotopic to
h,
g(f
postpone the proof of 7.5,
some consequences f i r s t .
in f a c t ,
in order to e s t a b -
We take note of the f a c t t h a t 7.5 i s ,
somewhat s t r o n g e r than the analogous r e s u l t 4.2 r e l a t i n g
p i e c e w i s e - l i n e a r immersions to the geometric subcomplexes of
o
- ' n ,k We may use t h i s a d d i t i o n a l s t r e n g t h to good e f f e c t in a n a l y z i n g the homotopy type of
X c
n,k
.
Note, f i r s t
,k
of a l l ,
t h a t the double sequence
+l,k
n,k+l
+l,k+l
passes over, under r e t o p o l o g i z a t i o n , to a s i m i l a r double sequence
148
7•34
c
~
c
• " Y ~ n , k +/~n+l,k
•~c
~
c
n , k + l + ~ n + l , k+l
which maps to
BPL
s t a b i l i z a t i o n of
by the map which, at each stage, c l a s s i f i e s the y
c
. This suggests the c o n j e c t u r e t h a t n,k is a homotopy e q u i v a l e n c e , which c o n j e c t u r e , however,
lim~" + BPL n,k n,k i s not q u i t e t r u e .
There i s a more or less obvious o b s t r u c t i o n to
the t r u t h of the c o n j e c t u r e which, moreover, is
the only one.
I n f o r m a l l y , t h i s o b s t r u c t i o n a r i s e s as f o l l o w s :
Consider
LS-strati-
f l e d m a n i f o l d s , i g n o r i n g f o r the moment the s m o o t h a b i l i t y of the stratification•
M o r e s p e c i f i c a l l y consider an
j - s p h e r e ; we ask whether t h i s LS-stratified
(j+l)-disc.
the s t r a t i f i c a t i o n
sphere is
j-sphere.
will
However, i t
be s t r a t i f i e d ,
have f o r i t s
case, s i m p l i c i a l .
"yes" when, e . g . ,
of the sphere is 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 ,
not work f o r more general the cone w i l l
the boundary of an
Obviously the answer i s
f o r then we merely t r i a n g u l a t e the on the
LS-stratified
link
(j+l)-disc
as the s i m p l i c i a l cone
is c l e a r t h a t t h i s
construction w i l l
L S - s t r a t i f i c a t i o n s of the sphere, f o r then but the
the o r i g i n a l
O-stratum a t the cone p o i n t j - s p h e r e , which i s not,
Hence the s t r a t i f i c a t i o n
in t h i s
of the disc f a i l s
to be
LS. We may, however, note the f o l l o w i n g about the n a t u r a l map _~:~
c
: llm n,k
7.7 Lemma. jective;
÷ BPL. ,k For any f i n i t e
complex
K,
[ K ~ c]
moreover, there Is a n a t u r a l s p l i t t i n g
149
+ [K,BPL] t:
is
[K,BPL] .
sur[ K , ~c ]
7.35 of
¢,.
Proof: by the n.
PL
Given the map n - p l a n e bundle
I n d e e d , by
K ÷ BPL,
represent its
~
over
for
structure
we may f i n d
K,
homotopy c l a s s
some s u f f i c i e n t l y
some m a n i f o l d
large
of the s i m p l e n homotopy t y p e o f K such t h a t ~ is, essentially TW . Then, f o r n n+k l a r g e enough k, we s h a l l have a PL-embedding f : W ÷ R , n n+k that is, W i s a p i e c e w i s e - l i n e a r subspace of R , with respect n t o some t r i a n g u l a t i o n . Ipso f a c t o , W i s g i v e n a smooth LS stratified
[Wa],
a:
Wn
which the embedding f is p i e c e w i s e - d i f f e r n c e n t i a b l e and o b v i o u s l y g ( f ) : W +z[/" has the p r o p e r t y t h a t Cog(f) -~n ,k classifies a. Hence the s u r j e c t i v i t y c l a i m e d by the Lemma i s e s t a b n fished. But, i n a d d i t i o n , the m a n i f o l d W , the t r i a n g u l a t i o n thereof,
and t h e
dance ( i f
n,k
PL
[a].
embedding are u n i q u e l y d e t e r m i n e d up t o c o n c o r -
be l a r g e e n o u g h ) .
unique element i f of
for
[K, 4
]
Hence t h i s
and hence i n
Thus the c o n s t r u c t i o n
defines
[K
c o n s t r u c t i o n produces a /~]
as the p r e - i m a g e
the s p l i t t i n g
map
t
as
required. We may say something more a b o u t homotopy group l e v e l . the kernel
e
of
0*:
i xiBPL ~ e i . e
(In
fact,
That i s , x ~c i it
is
@:~c +
we s h a l l
÷ x BPL. i
BPL,
at
least
on t h e
characterize geometrically
Of course
~ ~fc i
splits
p o s s i b l e to d e s c r i b e a space
BSL S
with
afield.) i of S , and c a l l two i i such e q u i v a l e n t i f t h e y a r e c o n c o r d a n t , i . e . S and S are o 1 i e q u i v a l e n t when t h e r e i s a smooth LS s t r a t i f i c a t i o n T of S xI i i whose boundary i s S _L~ s . Let e d e n o t e t h e s e t of e q u i v a l e n c e o 1 i classes. C l e a r l y , there is a d i s t i n g u i s h e d element of B., namely i t h a t r e p r e s e n t e d by the s t r a t i f i c a t i o n w i t h o n l y one s t r a t u m , i . e . i a l l of S ( w i t h i t s usual s~ooth s t r u c t u r e ) . I t is e a s i l y seen i that a stratification of S f a l l s w i t h i n t h i s c l a s s i f and o n l y i f i
it
= ~ B b u t t h i s would t a k e us t o o f a r i SLS C o n s i d e r a l l smooth LS-stratifications
as
is
t h e boundary o f a s m o o t h l y
LS-stratified
150
(i+l)-disc.
7.36 0 i s , in i e v e r , note t h a t technicalities define it
$
fact,
there is
a n a t u r a l map
c o n c e r n i n g the r o l e
by n o t i n g t h a t
by a s m o o t h l y
denote
a g r o u p , as w i l l
of
be seen s h o r t l y . e ~ ÷ ~ ~c . i i the base p o i n t ,
First,
how-
We o m i t some and s i m p l y
g i v e n an e l e m e n t in
o., we may r e p r e s e n t I i - s p h e r e which, f o r c l a r i t y , we
LS-stratified
T,
and t h e n , we c l a i m , proceed to embed T i n some i+k E u c l i d e a n space R , the embedding f being piecewise d i f f e r e n t i -
a b l e and s a t i s f y i n g
g(f): T of
of
~,
B, y.
We thus o b t a i n a Gauss map
Si
÷~ c , which we may c o n s i d e r t o r e p r e s e n t an e l e m e n t i,k We c l a i m t h a t t h i s e l e m e n t depends o n l y on t h e e q u i v a l e n c e
_:
~cJ-. i
class
conditions
T,
7.8 Lemma.
and n o t on 0 + e
i
+ ~
T
split
nor t h e embedding
f.
¢*
#c
÷ ~ BPL + 0 i
i
We have a l r e a d y seen, sequence i s
itself,
in essence,
surjective.
is
that
We n o t e t h a t
¢,
exact.
in
the above
im ~ C k e r ¢ ,
merely
because, f o r t h e Gauss map o f any p i e c e w i s e d i f f e r e n t i a b l e immersion n+k f: M ÷ R , ~og(f) classifies the s t a b l e t a n g e n t bundle o f M; i thus i f M = S , Cog(f) is h o m o t o p i c a l l y t r i v i a l , hence ¢*o~ is trivial. which i s i n v e r s e t o ~. d e f i n e a map ~: ker ¢ , + e i c @,[u] = 0 i n ~ B P L . Think of u as a Suppose u: ÷x~ with i i i map S +~ , n and k both l a r g e . We may then r e p l a c e S by i n-i n,k c W = S xD and t h i n k of u as a map W + ~ c o v e r e d by a p a r t i n,k c c u l a r b u n d l e map o f the ( t r i v i a l ) t a n g e n t bundle of W to Yn,k" If We s h a l l
i S c
we now a p p l y the main r e s u l t n k be l a r g e enough) W in some smooth
LS-stratification
7 . 5 , we may immerse ( i n f a c t , embed i f n+k R via f so t h a t , w i t h r e s p e c t t o of
W,
and t h e Gauss map Now t h i r k small
isotopy,
transversally.
f
has p r o p e r t i e s
~,
B, y
g(f) i s homotopic t o u. i i i n-i n+k of S as S × { 0 } C S xD : W C_R After a i we may assume t h a t S meets the s t r a t a o f W i I.e., S is s t r a t i f i e d so t h a t a t y p i c a l
151
7.37 codimension-j stratum codimension Y
stratum
has a t r i v i a l
inherit is
j
Y
is X
i S ~
a component o f of
W.
X
Moreover, since
normal b u n d l e in
X,
it
follows
for X
that
some is
smooth and
Y
must
smoothability.
More p r e c i s e l y , the embedding o f W i c o n c o r d a n t t o one i n which S ~ W is a " s u b m a n i f o l d " of
smooth
LS-stratified
manifold
d e f i n e s an e l e m e n t o f ~([u])
is
e
well-defined,
advice that
the r e l a t i v e
W.
This
stratification
which we d e n o t e by
i
details
being l e f t
v e r s i o n 7.6 o f
~([u]).
of
n+k R
in the i S
We c l a i m t h a t
to t h e r e a d e r w i t h
the main theorem i s
the
a key
ingredient. So• t o o , i s the f u r t h e r f a c t t h a t t h e s t a n d a r d maps c c c c ~:~ ÷~
immersion
then
~og(f)
n+k+l R , while Bog(f) n f n+k C n+k+1 M + R R It
is
clear
that
~ g(fxtd)
,
= g(f
~o~
)
where
is
where
f
,
the i d e n t i t y
is
fxid:
n n+k M xR + R xR =
the c o m p o s i t i o n
e.. We must show I that ~ is i n j e c t i v e . C o n s i d e r once more the s i t u a t i o n o u t l i n e d i i n-i above, i . e . S sitting in W ~ S xD as a s u b m a n i f o l d ( i n the i sense o f s m o o t h l y LS s t r a t i f i e d manifolds.) Note t h a t S has a I
i
a
on
¢
n e i g h b o r h o o d o f the form W = S ×D where D i s a s m a l l e r copy of n-i D and where t h e smooth LS s t r a t i f i c a t i o n on W i n h e r i t e d from i W i s the p r o d u c t of the s t r a t i f i c a t i o n on S with a trivial stratification x i x~-- k '
of
g(f)
l
D .
g(flW').
It
is
clear that,
Now deform
flW'
r e g a r d e d as an e l e m e n t o f = f',
through piecewise
n+k
•
smooth embeddings, to f : W' + R so t h a t f i s o f the form 1 1 i i+k exi where e: S + R and i i s the s t a n d a r d i n j e c t i o n n-i n-i D' ~ D C_R We l e a v e i t t o t h e r e a d e r t o v e r i f y t h a t such a deformation is
p o s s i b l e when
k
g(fl }
But n o t e t h a t
g(f
g(f).
the i t e r a t i o n that
g(e),
of
is I ~
large. It follows that i n-i n-i )IS = ~ g(e) where ~
s t a n d a r d maps, : + . n-i ,k ,k hence ~ og(e) represents ~(~([u])).
152
i~
Also r e c a l l But
g(e)
7.38 represents
7.4
[u]
hence
~om = i d .
This completes the p r o o f .
Proof of Theorem 7.5 As the f i r s t
step in p r o v i n g 7.5 we must a n a l y z e the g e o m e t r i c
s t r u c t u r e of a t y p i c a l
g e o m e t r i c subspace
far,
has a s t r a t i f i e d
one sees t h a t
H
H
of ~ / c . In p a r t i c u njk structure essentially
analogous to t h a t of the space B of §8.2. LS c Given H C~" , let T be an e q u i v a l e n c e c l a s s of t r i a n g u l a ,k t i o n s of the i-sphere, -I < i ( n-l, so t h a t ~ ~ H is n o n - v o i d . T c is We s h a l l use the n o t a t i o n ~ T ~ H = . Recall t h a t NT C n,k the image of a d i s c bundle PT over - ~ T ; t h e r e f o r e , we use ~T to denote the image of
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
(Thus,
in p a r -
ticular
A is H~ ~ = H~ G .) On A we d e f i n e a s t r a t i f i c a o o r,i n,k i r t i o n w i t h open s t r a t a XT, , one f o r each component ~ T with ~r,i ~r,i+l ~ n dim T ~ i - l . We s h a l l have X ~ ~ ... C ^r, and T T T or,n or r or X = X . X i s the c l o s u r e of X T T Tt h or,o T s i m p l y be the components of ~ o " At the 0 stage, l e t X o Suppose, now, i n d u c t i v e l y , t h a t we have s t r a t i f i e d Ai_ I , with or,i-1 _r one s t r a t u m XT f o r each component ~ T of ~ T ' dim T < i - 2 . Consider any p a i r formal
link
s i m p l e x of Furthermore of
~T"
(L,o)
such t h a t Z-
Then
[ZL]
e , L L c~ C~ o T T ~r
where = S
L = (UL,ZL) with
L~S
is ¢ O,
an
i-dimenslonal and where
o
the image of for
cZ l i e s in ~ C__ H. L S and some component ~ r T = [~L- ] ,
is a
Keep In mind t~e subspace
i m ( ~ ~*) ~ imc~L C ~S.
T
Let
Z = ~im(~-*) where ~ ranges over a l l such o as above i . e . o T _r a s i m p l e x of _ZL, L~ -b~ f o r some S and L ~ ~ T f o r the given T
153
7.39 and A i
r.
or,i X T
Let
~r,i-1 =
T
c o r r e s p o n d i n g to
i-i,
we s i m p l y
T
of
~r,i X T
let
r Z . T
~
This
dimension
t a k e s care of < i-2.
For
all
T
of
strata
dimension
_r c o r r e s p o n d i n g component ~ T
be the
of
of
~T C Ai-Ai_l. n
Recall a map
is,
h
just
h
7.5;
we have a n o n - c l o s e d m a n i f o l d
t r a n s v e r s e to
the p r o o f o f
n M
on
the
Lemma 7.4 o f
may be taken t o
LS-stratification
c o u r s e , we may s t i l l n c TM + y IH since n,k b u n d l e maps.
n M
as f i x e d .
7.9
Lemma.
stratification §7.2,
We may now
of
after
M ,
H.
suitable
That defor-
be a s t r a t u m - p r e s e r v i n g map from an
into
H
(the
r t h e one j u s t d e s c r i b e d ) . Thus, i f X -1 r T h (X) has, as i t s components, s t r a t a T
of
n c TM + ¥ IH. n,k
c o v e r e d by a b u n d l e map
to make i t as in
mation,
hypothesis of
n M ÷ H
h:
define
the
assume t h a t
h
is
stratification is
of
H
being
a stratum of H, then n M whose l i n k i s T.
of
Of
c o v e r e d by a b u n d l e map
the d e f o r m a t i o n may be c o v e r e d by a d e f o r m a t i o n
H e n c e f o r t h , we s h a l l
regard this
stratification
of
n
smoothed i n
The s t r a t i f i c a t i o n a natural
Proof: stratified T
be the
a typical
smooth the smoothing
link
of
interior Y
Y
Y
stratum
itself.
We n o t e ,
open n e i g h b o r h o o d in
so
of
r + X T but t h i s
Y,
(this
as above can be
n
M
is for is
is
in
fact
some
r.
n M .
We s h a l l
essentially ~r X T
may be seen d i r e c t l y to
strictly
a submanifold of
the open s t r a t u m
leaving details H
Y
hlY
Note t h a t
as a d e f o r m a t i o n r e t r a c t struction).
arising
the moment t h a t
o
Let
M
way.
Assume f o r so t h a t
of
_r c o n t a i n s -~T
from the con-
the r e a d e r ,
which may be i d e n t i f i e d
e q u i v a l e n t to
with
that
or X T
has an
the
or or open-PL-disc bundle over X associated to the pullback to X T T _r _r (under the deformation retraction) of the PL b u n d l e P I;~T . ~ T J ~T" By way o f
clarification,
it
s h o u l d be u n d e r s t o o d t h a t
154
_r
o v e r ~-~T
7.40 _r the neighborhood looks l i k e int N . For n o t a t i o n a l s i m p l i ' T r r c i t y , l e t the bundle P -'T-~ be denoted by ~T and the r e t r a c t i o n by T r or _r n PT: XT ~ ~'T" Let v(Y,M) denote the normal bundle of ~ in M . itself
Then,
by
the
"transversality"
of
h,
we h a v e :
o r* r v(Y,M) = h*o T ~T But
recall,
in
light
of
§7.1,
that
Ycn , k I - ~
:
r r ~T~nT
r
where
nT
is
a certain vector bundle canonically i n d u c e d f r o m t h e map _r n r r r OTI~T * Gn-j k+j (dim T = j - l ) . T h u s we have: TN IY : h*PT ~T~)nT ). '
~
Hence, at l e a s t s t a b l y , we have,
r
r
TY = h*p T n~. s I . e . the PL tangent bundle of Y reduces to a vector bundle and, o by c l a s s i c a l smoothing theory (see, e . g . [ H i - M ] ) Y (and thus Y) i s smoothable, and, in f a c t ,
by c a n c e l l a t i o n :
endowed with a s p e c i f i c smoothing.
(That
is,
the s t a b l e bundle map [TV] + [n r ] may be chosen so t h a t i t s s Ts o r o r r sum with v(Y,M) ÷ ~T i s e q u i v a l e n t to the g i v e n TMIY + n T ~ T . ) It
is
not hard to show t h a t the r e d u c t i o n of tangent bundles of
s t r a t a as obtained above i s c o n s i s t e n t with incidence r e l a t i o n s . That i s ,
if
YP < Yq are i n c i d e n t s t r a t a , then there i s a canonical 1 2 way of i d e n t i f y i n g T~21~1 with T~I ~ ~q-p With regard to t h i s identification,
the s t a b l e r e d u c t i o n of
obtained above induces a r e d u c t i o n of
TY to a v e c t o r bundle as 2 TY1. We c l a i m t h a t t h i s i s
i d e n t i c a l to the r e d u c t i o n of It
TY t h a t has a l r e a d y been produced. I f o l l o w s , once more by c l a s s i c a l smoothing t h e o r y , t h a t a l l the
s t r a t a may be s i m u l t a n e o u s l y smoothed so t h a t ,
with
YI < Y2" Y1 ~
Y2
i s smooth, and thus the s t r a t i f i c a t i o n
i s smooth as w e l l . n We may now remove the ad hoc assumption t h a t M was s t r i c t l y -
stratified
by d i n t of the usual o b s e r v a t i o n t h a t a s t r a t i f i e d
fold is,
locally,
strictly-stratified.
strictly
stratified
That i s ,
mani-
we may smooth
c o - o r d i n a t e patches by the method o u t l i n e d abnve
so t h a t , on o v e r l a p s , the smoothness s t r u c t u r e s of the r e s p e c t i v e strata coincide.
155
7.41 We n o t e , slightly
however, t h a t
for
stronger properties
our purposes we s h a l l ,
for
the map
h
in
fact,
need
than have a l r e a d y been
shown.
We c h a r a c t e r i z e t h i s as f o l l o w s : R e c a l l t h a t f o r an open °r c r r* r r* r stratum XT o f H, Y n , k l X T = PT ~T (~9 PT nT" D e s i g n a t e these summands
by
~
!
and
q
J
respectively.
The p r o p e r t y we wish to have, t h e n , n r is: If Y is a stratum of M w i t h l i n k Y = T and h(Y) ~ X f n ~ c or then the b u n d l e map TM IY ÷ Y iX covering hl~ splits into a n,k T d i r e c t sum o f b u n d l e maps: O
I
i)
z:
v(Y,M)
ii)
y:
T(Y)
where
z
verse to
0
is
+
+ n
the n a t u r a l map a r i s i n g
the s t r a t i f i c a t i o n
ment on the s m o o t h a b i l i t y for
h,
of
T~
of
H. Y
(We n o t e t h a t
that
h
is
trans-
the p r e v i o u s argu-
does n o t q u i t e y i e l d
this
property
inasmuch as we m e r e l y o b t a i n e d a map from t h e s t a b i l i z a t i o n to
the s t a b i l i z a t i o n
I
of
n .)
L e t us denote the p r o p e r t y d e f i n e d above as p r o p e r t y
A.
We wish to show t h a t
may be
assumed t o of
of
from the f a c t
have p r o p e r t y
the h y p o t h e s e s o f
the assumption t h a t 7.10 Lemma. Suppose
under the hypotheses o f A.
This
7.5,
demonstration will
h call
upon one
7.5 which has h e r e t o f o r e been i g n o r e d , namely, n M has no c l o s e d components. n
a map t r a n s v e r s e t o t h e s t r a t i f i n c c a t i o n of H and i s c o v e r e d by a PL bundle map TM + y IH. Let njk n Y be a s t r a t u m o f the induced LS s t r a t i f i c a t i o n of M , and l e t hd Y to
h: M
÷ H
is
d e n o t e the homotopy d i m e n s i o n o f
Y.
Suppose t h a t
for
all
Y
of
Y
modulo the i n c i d e n t
dim > O,
hd Y < dim Y.
Then t h e b u n d l e map deforms to one h a v i n g t h e p r o p e r t y The p r o o f o f t h e argument o f
the lemma i s
straightforward.
7.9 which shewed t h a t
be {mposed on t h e s t a b i l i z a t i o n O
quite
cf
A.
We a d v e r t to
a v e c t o r bundle s t r u c t u r e
T~.
strata
could
We o b t a i n e d a s t a b l e map
l
(TY)
÷ n where t h e s in s u b s c r i p t denotes s t a b i l i z a t i o n . Hows s ever, if hd Y < dim Y = dim n ' , then t h i s s t a b l e map desuspends t o
156
7.42 an
unstable
structure
map by o
on
TY
vector-bundle 7.10
is
is
in
this
on
T~
proof
of
theory,
induced
to
the to
the
stratification However,
achieve this
the
arising
7.5.
from
as f o l l o w s :
of
7.10 n
M
remove an open
n e i g h b o r h o o d o f a small
Int
(All
Y.
an a r c ,
with
the Thus
in
has
of dimension
discrete
general position
n-disc
open d i s c
is n
points).
if
n ) 3;
and
n+l,
7.9,
been
is
case to
of
of
that
the
be s m o o t h -
demonstrated.
Y
of
We
is n M
t h e form o f a small
d i m e n s i o n = dim Y
strata
the former in
in
(i.e.
it
subarcs,
meets the
We may assume the a r c s a r e d i s j o i n t with
some s l i g h t
neighborhoods of
removed t o
via
t r a n s v e r s e to a l l
tubular
M
which
For each s t r a t u m
where some care may be needed i n
puncture
in
far
( o r a component t h e r e o f )
n = 2,
n
not
thus
are d i s j o i n t . ) Order these d i s c s and th i t o the boundary of the (i+1)st by
the
such t h a t each a r c
only strata
these a r c s .
additional
the o r d e r i n g ) .
The e f f e c t
D
stratification
and
M , h
a s m a l l e r copy o f
may be r e p l a c e d by
t h e h y p o t h e s i s of 7 . 9 . n M has v o i d boundary or i s
h'
n
M ,
(by
argument i f Remove s m a l l
thus f a r
is
to
once. Now c o n n e c t the boundary o f what has been n @M by an a r c and remove a t u b u l a r n e i g h b o r h o o d o f
Thus we o b t a i n
If
bundle
the removed d i s c s
c o n n e c t t h e boundary of
satisfy
vector
smoothing.
Our a s s u m p t i o n
Suppose
tubular
arc.
the
stratification of H, n of M has b e e n s h o w n ,
hypothesis
hd Y = dim Y,
latter
th~
way may be i d e n t i f i e d
compact and has a n o n - v o i d b o u n d a r y . with
and
proved.
transverse
able.
obstruction
obtained
structure
We r e t u r n h
standard
the
w i t h an i n h e r i t e d
= hiM',
which does
non-compact, a s l i g h t l y
e l a b o r a t e argument may be made a l o n g the same l i n e s ;
this
is
more left
to
that
the
t h e r e a d e r as an e x e r c i s e . Therefore, it map
h
satisfies
smoothability
of
may now be assumed: property
A.
in
In v i e w o f
the s t r a t i f i c a t i o n
of
157
it
the proof of
7.5,
the argument above on the may now be assumed n o t o n l y
7.43 t h a t the s t r a t i f i c a t i o n
is
smooth b u t , f u r t h e r m o r e , t h a t f o r each c stratum Y, the bundle map TM + y JH r e s t r i c t s , on T~ to a map n,k r r of v e c t o r bundles T~ + nT, where h(Y) C XT. ( I n o t h e r words, the
v e c t o r - b u n d l e r e d u c t i o n of from t h i s
from which the smoothing a r i s e s comes
bundle map.)
We are f i n a l l y entiable
T~
immersion
in a p o s i t i o n t o c o n s t r u c t the p i e c e w i s e - d i f f e r n n+k M + R whose e x i s t e n c e is the c o n c l u s i o n of
7.5. We proceed, as is sion of
strata.
For
customary, by an i n d u c t i o n based on the dimenY
of dimension
embed, an open neighborhood of (a s i n g l e p o i n t ) L
in o~PT
to the o r i g i n
and we may i d e n t i f y
O,
M(Y)
in n+k R
of
M(Y)
we s h a l l immerse, in f a c t n+k R as f o l l o w s : send Y The image
h(Y)
is
w i t h the s t a n d a r d u n i t
a point n-disc
D in a s t a n d a r d way, and by e x t e n d i n g r a d i a l l y s l i g h t l y , we embed UL n+k a s l i g h t l y l a r g e r neighborhood M(Y) in R O b v i o u s l y , the Gauss map on
M(Y)
(which is a smoothly
LS-stratified
manifold) is
h
n v i a a s t r a t u m - p r e s e r v i n g i s o t o p y of M ). n+k Now assume t h a t t h e r e is an immersion f o : B ÷ R of a n e i g h -
(up to a d e f o r m a t i o n of
borhood
B
of
(i)
g(fo)
Let
Y
L~ M(Z) dim Z
be an
B C_B
is
on
= M( i - I )
so t h a t
B.
i-stratum.
We s h a l l
extend the immersion to
f
so t h a t h ~ g ( f ) r e l B' where (I-i) F i r s t , we note t h a t i f a s m a l l e r neighborhood of M
on a neighborhood of t
h
B' ~ M ( Y )
!
h(Y) C_ XT' r
,
_r
we may deform h on Y-B so t h a t h(Y-B ) c ~ T ' r i s smoothly immersed v i a fo h ( M ( Y ) - B ' ) C__ N . Note t h a t Y F~ ( B - B ) T n+k t h e r e i s an (n+k-i)-dimenin R so t h a t f o r y ~ Y ~ (B-B) n+k whicll i s the normal space to s i o n a l a f f J n e subspace of U of R Y foY" By d e f i n i t i o n , thls coincides with f o ( y r~ ( B - B ' ) ) at _r i s to be t h o u g h t of as a formal U where g ( f o ) ( y ) ~ Y T C ~ T g(fo)(Y)
linko
158
7.44 Obviously, the t r a n s l a t e to the o r i g i n of is
the subspace
hood of
U (via x ~ x-roY) Y Now consider a small t u b u l a r neighbor-
U . g(fo)(Y)
fo(Y ~ B - B ' ) ) ,
disc of radius
~
in
viewed as U . Y
U D y ~,Y
where
We may i d e n t i f y
D c,Y
D ~,Y
g(f
the
with Ug(fo)(y )
in a standard way. On the other hand, we have a map r whose image l i e s in ~ D C. g ( f o } : M(Y) ÷ M T y Ug(fo)(y ) p o s i t e of
is
Mr. T
The com-
)
with the map given by the union of i d e n t i f i c a t i o n s n+k D ( > ~ immerses M{Y) ~ (B-B') in R In f a c t we E,y Ug(fo)(y ) (I-I) claim t h a t , up to a s l i g h t r e g u l a r homotopy of fo r e l M , which 0
o
does not a l t e r fo" B
Now l e t and
B'
g(fo), B"
t h i s immersion of
be a neighborhood of
(i.e.
~'c
B", B" C B).
preserving way, r e l ~ " , Y ~ (B-B')
M( i - I )
coincides with
i n t e r m e d i a t e between
We may deform _r
so t h a t
the composite
M(Y) ~ B
h, in a stratum-
r
h(Y-B )C ~T__ C XT.
Y ~ (B-B')
h _r • ~T
standard Gauss map of a smooth immersion.
BT
N o t e t h a t on
, Gi,n+k_ i
is the
Since
i
i
e oh: Y-B ÷ G is covered by a vector-bundle map T(Y-B ) ÷ ~, T i,n-i where ~ denotes the canonical i - v e c t o r bundle over G it i,n+k-i' f o l l o w s from the Hirsch immersion theorem [H] t h a t there is a smooth immersion Y ~ (~"-B') eToh.
whose Gauss map is
In f a c t ,
open set,
¢: Y-B' ÷ Rn+k,
if
we l e t
c o i n c i d i n g with
homotopic,
G : eT(~)
then the Gauss map
¢
of
rel
which i s , ¢
Y~
fo
on
(~"-B'}
to
by assumption, an
has image in
G and is homo-
t o p i c to call
e oh in G ( r e l the same subspace). To see t h i s , we reT the extension of Hirsch's immersion r e s u l t via Gromov-Phillips
theory [ P ] .
Now by the d e f i n i n g p r o p e r t i e s of geometric sub-
, i t f o l l o w s t h a t the deformation of e to spaces of ~C Toh -v n ,k may be covered by a deformation of hI(Y-B') to some map h, m
(Y-B)
~r
÷~rT
again
rel
Y ~ ~".
This deformation may be extended,
159
7.45 t e l ~"
to a deformation of
c a t i o n of
M
U , Y
I.e.
Y
h
M(Y) h
on B".
f o r each
f
of
Obviously, t h i s
ing the o r i g i n a l
M(Y)
h
is now c l e a r t h a t
M ( Y ) via the same argument
we have a normal
D Uh(y )
with
D ~,Y'
n+k-i
M(Y)
(so t h a t
B"u
g(f)
on
= h
plane
when composed fo
on
extended to a M(Y)
i s open).
B" U M(Y) (h
has been m o d i f i e d to t h i s
mean-
p o i n t by deform-
F i n a l l y , we note t h a t t h i s procedure may be c a r r i e d out i-dimensional
d i f f e r e n t i a b l e immersion Gauss map
It
immersion be " r a d i a l l y "
of
as i t
simultaneously for a l l
g(f):
C ÷ H
sion of the o r i g i n a l it,
y ~ Y-B ,
be t h i s e x t e n s i o n ; c l e a r l y
ations).
which preserves the s t r a t i f i -
M ( Y ) produces an immersion, c o i n c i d i n g with
small neighborhood Let
M
may be extended to
and the i d e n t i f i c a t i o n
with
on
and the t r a n s v e r s a l i t y c o n d i t i o n .
the immersion on as b e f o r e .
h
f
to produce a p i e c e w i s e -
on a neighborhood
c o i n c i d e s with
h).
Y,
h
(i.e.
C
of
(i)
M
whose
with a deformed v e r -
The i n d u c t i v e step is complete, and w i t h
the proof of the main theorem 7.5.
160
8.1 8. In t h i s
Some a p p l i c a t i o n s
to smoothing t h e o r y
s e c t i o n ~e s h a l l
examine an i n t e r e s t i n g c r i t e r i o n f o r n manifold M , whose p r o o f i s based on the
t h e smoothing of
a
c o n s t r u c t i o n s of
the p r e v i o u s s e c t i o n s .
First,
PL
c o n s i d e r the s e t o f e q u i v a l e n c e c l a s s e s o f
ordered 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 s
of
the
linearly
(i-1)-sphere
(equivalence
means o r d e r p r e s e r v i n g s i m p l i c i a l e q u i v a l e n c e ) . Given such an o b j e c t i-1 r e p r e s e n t e d by ~ , we have d e r i v e d o b j e c t s , o r f a c e s , one f o r each s i m p l e x
~
the i n h e r i t e d
o r d e r i n g and n o t e t h a t
classes.
of
~i-i
We see t h a t
That i s ,
Oim ~
we may now c o n s t r u c t a
we d e f i n e this
~
as
~k(o,~)
with
passes o v e r t o e q u i v a l e n c e
= dim Z-dim o.
complex i n t i l e b y - n o w - f a m i l i a r i-1 fashion: For each e q u i v a l e n c e c l a s s [Z ] we have one i-cell to i-1 be t h o u g h t of as c~ (There i s one z e r o c e l l f o r the - l - d i m e n sional
null
set.)
homeomorphisms dual
cell
Identifications
h : cZ
~* C ~
CW
in
c a t i o n s we o b t a i n a
a r e made, as u s u a l ,
+ ~ c c~
which i d e n t i f i e d
the s t a n d a r d way. CW
by the face
c~
with
the
After
making these i d e n t i f i ord complex which we l a b e l A
Note t h a t as in § i , a l o c a l l y - o r d e r e d 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 n n ord manifold M a d m i t s a c a n o n i c a l Gauss map g: M ÷ A (More p r e cisely, of
g
n
d e f i n e d o n l y on
M
= ~*,
n
o ~ ~M
being a s i m p l e x
0
n
M .)
t h e dual as
is
The d e f i n i t i o n cell
q*
to
of
g
the c e l l
is of
the usual one; b r i e f l y , we send ord A which i s v i e w e d , a b s t r a c t l y ,
n
c ~k(a,M )
class
of
(i.e., the c e l l which c o r r e s p o n d s t o t h e e q u i v a l e n c e n ~k(~,M ) as an o r d e r e d t r i a n g u l a t e d s p h e r e ) .
Now c o n s i d e r o r d e r e d t r i a n g u l a t e d linear
(on s i m p l i c e s ) embeddings i-1 p(v) C S for v a v e r t e x of (~1,01),
(~2,p2)
equivalent if
preserving simplicial
isomorphism
p: Z.
(i-1)-spheres together with i c Z , * ÷ R ,0 such t h a t We s h a l l
and o n l y ¢:
161
if
-~1 + ~
call
two such p a i r s
there is 2
such
an o r d e r -
that
8.2
Pl : a.(P2o ¢)
is an element of the orthogonal group O ( i ) . i-1 As b e f o r e , face o p e r a t i o n s are d e f i n e d . Given [(Z ,p)], and
the simplex
o
the sphere follows:
where
U
Let
For
v
x ~ b U
subspace of with
p',
(~ ,p )
by t a k i n g
Z
to be
as an ordered complex, and by s p e c i f y i n g
p
as
be the a f f i n e space of dimension i - d i m ~ -1 i p(co) c R and passing through the barycenter b a v e r t e x of Ov
u n i t sphere of
identify
we o b t a i n
U
with the ray
v i~ v",
~i-i
~k(o, Z)
orthogonal to p(C~).
of
m
U
and
Ek(,,),
v"
let
v'
be the i n t e r s e c t i o n of
the r a d i a l p r o j e c t i o n of
centered a t
b.
of
v'
on the
Extend convexly the assignment
to a l i n e a r embedding
p'
of
c ~k(~,~)
in
U.
We
i s o m e t r i c a l l y with an ( i - d i m o - 1 ) - d i m e n s i o n a l vector i R by the usual t r a n s l a t i o n -b and thus, by composing
we o b t a i n an embedding
o : cZ ~ 0
RJ,
J = i-dim o - i .
Of
0
course, there i s an a m b i g u i t y here which a r i s e s from the i n d e t e r m i n acy of the i d e n t i f i c a t i o n
of
U-b
with
i s w e l l - d e f i n e d , up to the a c t i o n of equivalence class
[(Z
,P ) ]
c l e a r l y depends only on
Rj .
But since t h i s
O(j),
it
is w e l l - d e f i n e d .
[(Z,p)]
and not on
is
isometry
c l e a r t h a t the
Moreover, t h i s class (Z,p)
itself.
ord in complete analogy to our previous c o n s t r u c t i o n of A , 8r we o b t a i n a C-W complex A which has one i - c e l l ( t o be thought Thus,
of as
c Ti-1 )
f o r each equivalence class of ( i - 1 ) - d i m e n s i o n a l
ordered Brouwer spheres. N a t u r a l l y , there i s a f o r g e t f u l map Br ord i-I, i-1 A ÷ A induced by the assignment (~ p) ÷ ~ , which i s o b v i o u s l y c o n s i s t e n t with equivalence r e l a t i o n s . that this torially
f o r g e t f u l map i s not onto,
We note in passing
inasmuch as there are combina-
t r i a n g u l a t e d spheres which are not Brouwer spheres in t h a t
the r e s p e c t i v e cones de not embed l i n e a r l y codimension
in Euclidean space w i t h
O.
We may r e t o p o l o g i z e
Br A
to o b t a i n a new space
analogy to the r e t o p o l o g i z a t i o n of x~ , n,k
162
cBr A
which produced ~ f
in c. n,k
That
8.3 is,
g i v e n two c l a s s e s
[(~l,Pl )]
o r d e r e d Brouwer s p h e r e s , p r e s e r v i n g isomorphism P2oh
in
the usual
new m e t r i c ,
if
[(~2,P2 )]
c o n s i d e r them h:
sense.
of
(i-1)-dimensional
~-close if
t h e r e is
Z + Z such t h a t p is 1 2 1 Br C a l l two p o i n t s x,y E A
t h e y have p r e - i m a g e s
~, ~
E-close to G-close i n
cZ1 , c~
in
an o r d e r
the
respectively
~2
for e of
(~l,Pl), (Z2,P2),
two o r d e r e d Brouwer spheres close to p2~
(~2,P2)
via
h,
as above, and i f
if
pl ~
in
is Br A
E u c l i d e a n space. With t h i s new m e t r i c cSr smaller topology A and we o b t a i n a f o r g e t f u l map which i s
the i d e n t i t y
naturally j-simplex [(~,p)]
pointwise.
Again,
the g e o m e t r i c r e a l i z a t i o n (~([(~,p)],T)
for
and each j - s i m p l e x
by t o p o l o g i z i n g
the s e t o f
each c l a s s T
of
we n o t e t h a t
of a s i m p l i c i a l
c~.
of
within
is e
acquires a Br cBr A + A Br A is
s e t h a v i n g one
o f o r d e r e d Brouwer spheres cBr Br A i s o b t a i n e d from A
j-simplices,
and t a k i n g the g e o m e t r i c r e a l i z a t i o n
(~l,pl)
all
j,
in
the o b v i o u s way
the c o r r e s p o n d i n g s i m p l i c i a l
space. Now n o t e f u r t h e r Br cBr f ord A + A ~ A , Our smoothing r e s u l t
that
the f o r g e t f u l
where t h e f i r s t concerns l i f t i n g
map
map i s of
A
Br
ord ÷ A
f a c t o r s as
the p o i n t w i s e - i d e n t i t y .
the d i a g r a m
cBr A
,f n g ord M ---@ A where
n
M
is a
triangulation
and
PL
manifold with a locally-ordered ord g the n a t u r a l map t o A
combinatorial
8.1
Theorem. In the diagram a b o v e , t h e r e e x i s t s a homotopy l i f t i n g n cBr h: M + A i f and o n l y i f t h e r e i s a smoothing o f the PL n s t r u c t u r e on M . In f a c t , a homotopy c l a s s o f such l i f t i n g s h
d e t e r m i n e s a h o m o t o p y - c l a s s o f v e c t o r b u n d l e r e d u c t i o n s o f the s t a b l e n t a n g e n t bundle of M , and thus a concordance c l a s s o f smoothings of
163
8.4 n
M
Furthermore,
each
smoothing
n
of
a r i s e s from at l e a s t one
M
such l i f t i n g . n
8.2
n
Corollary. With M as above, n Br lifting k: M ÷ A in the diagram
M
is smoothable i f
there i s a
8r
A
If n g ord M ----) A We s t a r t our proof of 8.1 by i n v e s t i g a t i n g a c e r t a i n geometric Consider a formal l i n k L of dimension ( n , k ; j ) . subspace of ~z~ . ,k a c t u a l l y l i e s in some Say t h a t L is s t r a i g h t i f CZL c-U L j - d i m e n s i o n a l sub-vector space of UL. C l e a r l y , s t r a i g h t n e s s of l i n k s is
x~
preserved under the face o p e r a t i o n s , thus
t ~ = e ,k L straight L
is a subcomplex of ~ 1"n,k"
Equally c l e a r l y
this
i s a geometric subcomplex. When we r e t o p o l o g i z e ~ to o b t a i n ~ _ , we f i n d t h a t the " n ,k s~ n'k ~ c image of ~ t k is a geometric subspace ~_n,K of n,k" Moreover, we f i n d t h a t the r e s t r i c t i o n
I~s t ,
~ n,k
an n - v e c t o r bundle. over WL
x is
That i s ,
may be i d e n t i f i e d ,
if
has a canonical s t r u c t u r e as
,k
xe e L
c
,k , then the f i b e r of y n,k
as a vector space, with
U (~)W L L
where
the unique
j - p l a n e c o n t a i n i n g CZL (dim L = j ) . cst Note t h a t contains the canonical image of G in n,k ,~, n,k c (as the r e t o p o l o g i z e d O-skeleton of ~ ) and t h a t the n,k n,k is the r e s t r i c t i o n of the canonical n-plane bundle over G n,k cst I~/cst The f o l l o w i n g n a t u r a l vector bundle s t r u c t u r e on ~n,k = Yn,k ~ n . k " i s obvious:
~
8.3
Lemma.
The
PL
manifold
M
is
there i s a piecewise-smooth immersion
164
smoothable iT and only i f n n+k f: M ÷ R such t h a t
8.5 g(f)(M
)C
k To o b t a i n a proof of 8.1,
it
is useful
to r e v i s e some of our
p r i n c i p a l c o n s t r u c t i o n s to accommodate o r d e r i n g s . ordered formal l i n k L = (UL,~L)
(of dimension
(n,k;j))
F i r s t of a l l ,
i s a formal l i n k
t o g e t h e r with a l i n e a r o r d e r i n g on the v e r t i c e s of
Henceforth, the s p e c i f i c o r d e r i n g s h a l l be i m p l i c i t l y and we s h a l l
an
the n o t a t i o n
ZL,
L = (UL,ZL).
AS b e f o r e , we o b t a i n , f o r each simplex
~.
subsumed under
speak of the ordered l i n k a
of
ZL,
the
d e r i v e d ordered l i n k L . The usual c o n s t r u c t i o n o b t a i n s f o r us a ord o L of CW c o m p l e x ~ with one j - c e l l e f o r each ordered l i n k n dimension j . In passing, we note tha is the n a t u r a l t a r g e t of the Gauss map f o r l i n e a r immersions of t r i a n g u l a t e d , l o c a l l y ordered m a n i f o l d s . rd
~n
÷A
Moreover, there is a n a t u r a l f o r g e t f u l map
ord
,k
~ord Now we may r e t o p o l o Q i z e /~Yn,k' j u s t as was done with ~ n , k ' to ~cord ~cord obtain Again, we note b r i e f l y how f i t s i n t o the n,k n,k scheme of things as the n a t u r a l t a r g e t of a Gauss map. We consider n smoothly L S - s t r a t i f i e d m a n i f o l d s M s u c h t h a t , f o r each stratum X n of M , the t r i a n g u l a t e d sphere ck(X) i s ordered and, moreover, the o r d e r i n g s are c o n s i s t e n t with incidence r e l a t i o n s . Y < X, It of
the n a t u r a l i n c l u s i o n
~k(X) ~ ~k(Y)
i s understood t h a t o r d e r i n g of X,
i.e.
if
~k(Z)
That i s ,
if
i s order p r e s e r v i n g .
l a b e l s the l o c a l
incidences
M(X),
X = M(X) n X are in in §7, then we demand o t h a t f o r each component Y of Y ~ M(X), with dim Y = dim X + 1, r X < Y, there i s a s p e c i f i c v e r t e x v(Y ) of ck(X). Thus, viewing r M(X) as a c~k(X) bundle over X , we see t h a t i t is t r i v i a l i z e d o in a s p e c i f i c w a y . Denote such manifolds as smoothly-LOS s t r a t i f i e d manifolds (LOS = l i n k w i s e ordered s i m p l i c i a l ) .
Given a p i e c e w i s e n smooth immersion of the smoothly L O S - s t r a t i f l e d m e n i f o l d M in n+k R ( s a t i s f y i n g ~, ~, ~ of §7.3 as usual) we o b t a i n a Gauss map
165
8.6 n . cord M +~n,k " The analogues of theorems 4.2 and 7•5 may e a s i l y be obtained by simple m o d i f i c a t i o n s of the respective d e f i n i t i o n s and proofs, but t h i s is not our major concern. Consider now the analog of complex
L~/ e L straight L
, t h a t i s , the geometric subn,k w h e r e the union is taken over the set of
ordered l i n k s whose underlying unordered l i n k s are s t r a i g h t in the sense j u s t described.~
Denote t h i s complex by ~ o ~s~C"J"st
Its
image in
~LfcOrd is denoted /~f os and, as in the case of ~ , we see t h a t v~ n,k ~t-n,k n,k cord the natural PL bundle y over has a r e s t r i c t i o n n,k n,k cos cordj~nOS, r which admits a natural n-dimensional vector bundle Yn,k : Y n , k __,k structure• ( T h e existence of natural PL n-plane bundles over ~ord 2ord k' etc. is seen by obvious extensions of the conn, ~n,k c s t r u c t i o n of Xn,k, Yn,k in §2 and §7 r e s p e c t i v e l y . ) ord Now we note t h a t the f o r g e t f u l map ''/~X "°rd + A prolongs to "~n,k + A Moreover, we see t h a t there is a natural map 1~Tn,k
.~,cord
.}.~ord
"~n
os ,k
L,
ord
Br ÷ A
We see t h i s on the c e l l
l e v e l by assigning to the j - l i n k
the class of orderea Brouwer spheres
abstractly, embedded in
~,
is
ZL " as an ordered s i m p l i c i a l complex, and where
c~
RJ
[(~,p)]
by taking the v e r t i c e s of
ZL,
where
plus the o r i g i n ,
and convexly extending to obtain a l i n e a r embedding of UL,
cZ ~ CZL
in
which embedding, by d e f i n i t i o n of s t r a i g h t n e s s , f a c t o r s as
c Z C W ~U embedding Rj ,
is
L
where W is a j-dimensional subspace. p: c Z + Rj
and the class
This gives us an
when we pick any l i n e a r isometry of
[(Z,P)]
W with
is unaffected by t h i s choice. Hence we Br e L of "'/<~os to the c e l l of A corresponding may send the c e l l n,k to [(Z,p)]. This assignment e s s e n t i a l l y defines the map ~os Br ÷ A ( I t is a simple exercise to show t h a t t h i s ass]gnment n,k of c e l l s is c o n s i s t e n t with i n c l u s i o n of faces.)
166
8.7 We make the f u r t h e r r e t o p o l i z a . t i o n of
o b s e r v a t i o n t h a t the map above r e s p e c t s the
/~os
as
~nOS
n,k diagram
ducing the n a t u r a l
,k
It
thus p r o -
n,k
7vn,k
Br
cBr + A
i s a p p r o p r i a t e now to p o i n t out t h a t t h e r e are n a t u r a l
stabilizing X
cBr A ,
as
+SOS
OS
A
Br A
and of
maps in
is any of
n
and
n,k ÷
k
,k ÷
+1,k ,
,k+l
where
the s u p e r s c r i p t s t h a t have been i n t r o d u c e d to t h i s
point.
The d e f i n i t i o n
f o l l o w s the example of and ,k n,k .Y w i t h o u t e s s e n t i a l change. We use ~ f " to denote the d i r e c t l i m i t
lim~ n
•
n,k
~v ord A ° r d z_zc~°rd A°rd We note t h a t the n a t u r a l maps / 7 + , + cBr " ~ n,k "~n,k ~J os + ABr , ~nnOS + A c o m m u t ew i t h the s t a b i l i z i n g maps in n and n, k ,k ~)rd orcI . c . o r c I ord k. T h u s we have, in f a c t , n a t u r a l maps ~C~ + A , ~ ÷ A ~os Br j~cos cBr + A , + A To prove 8 . 1 , we s h a l l study the diagram cos
2] The key
idea
is
to
group
look ~os
0 = l i m O(m) on m a r i s e s in the f o l l o w i n g way: ordered l i n k transforms then
~L
L L
is
of dimension
cord
at
cBr
.A
+A
ord
the
action cord and ~<~
given
n
(n,k;j).
of
the
infinite
orthogonal
respectively.
and
k,
we c o n s i d e r an
An element
in the f o l l o w i n g way (as in
§5):
The a c t i o n
If
~
of
O(n+k)
L = (U L , ZL ),
s p e c i f i e d by
(~U ,~Z ) where, in the f i r s t instance L L ~U denotes the image of U v i a the n a t u r a l O(n+k) a c t i o n on L L of ~L Gj+k,n_ j and in the second ~ZL means merely the image n+k under the a c t i o n of
~
a c t i n g as an i s o m e t r y on
167
R
We assume of
8.8 course t h a t the v e r t i c e s of o r d e r i n g of each
ZL"
a c e l l u l a r homeomorphism ~: ~ n r d +~ n rd , ,k ,k
§5, we do not have an O ( n + k ) - a c t i o n on/~{ /~n rd O(n+k)x/~nrd ,k ÷ ,k"
sense of a continuous map a c t i o n of
are ordered corresponding to the
This a c t i o n on the set of formal l i n k s d e f i n e s , f o r
~ ~ O(n+k),
we observed in
~ZL
but,
as
ord
in the n,k' Rathe r , we have an
O(n+k)
with the d i s c r e t e t o p o l o g y . Our f i r s t ord o b s e r v a t i o n , however, is t h a t w h e n / ~ i s r e t o p o l o g i z e d to o b t a i n n,k cord n,k ' the O(n+k) a c t i o n does become continuous in the usual
x
sense. left
V e r i f i c a t i o n of t h i s
f a c t is s t r a i g h t f o r w a r d and d e t a i l s are
to the r e a d e r .
Furthermore, t h i s O(n+k) a c t i o n preserves s t r a i g h t n e s s of /~os /~os l i n k s , thus and are i n v a r i a n t under the a c t i o n . Next, n,k n,k we see t h a t under the n a t u r a l i n c l u s i o n O(n+k) ÷ O(n+k+l) the a c t i o n of /~
n,k
+ If
O(n+k) , X
commutes w i t h the s t a b i l i z i n g m a p s ~ COS,
the t r i v i a l
O(n+k)-maps i f
action.
we make
in
c°S/o(n+k) n,k
u
and
~
cBr + A
c ° r d / o ( n + k ) ÷ A°rd n,k
n
and
k,
we have the diagram
_~+ AcBr
< • c o/0 r- -dv+ Lemma.
ord Br A , A O(n+k)-spaces via
~
~C°Slo
8.4
we f i n d t h a t
Thus we have maps
~
Passing to the l i m i t
+1,k
cord.
n,k+l cord ord ~.cos cBr we look at the maps 5Y + A ~ ÷ A , n,k ) n,k
these maps are
x.2
n,k
ord A
are homotopy e q u i v a l e n c e s .
168
8.9 Proof:
To deal with
u
first
we claim t h a t i t
is,
in f a c t ,
a
homeomorphism. C l e a r l y , i t is onto, since, given a l i n e a r embedding i -1 i i-i i -1 p of c~ ,* + R ,0 taking v e r t i c e s to S , were ~ is t
ordered, we may take ~. to be the corresponding geodesic t r i a n g u l a i-1 t i o n of S , thus o b t a i n i n g the formal ordered l i n k ( o b v i o u s l y i , straight) L = (R ,Z ) of dimension ( i , O ; i ) . Obviously u(e L) is Br Br the c e l l of A , (and thus the subspace of A ) corresponding to the equivalence class of the ordered Brouwer sphere (Z,P). x~/c°s Now suppose x , y are points in /0 with ~(x) = u(Y). ~,~
denote preimages of
s u i t a b l y large interior
n
and
x
and
k.
Then
y
in % c°s
x,y
are,
~:Z~ c°s and thus in ~n,k r e s p e c t i v e l y , in the
Let for
e ,e where L and K are of the same dimension L K T h u s we may t h i n k of ~ and as points of C~L , C~K respect-
j.
cells
' ' ' r e s p e c t i v e l y , the isomorphic s i m p l i c i a l ~L' ~K n+k complexes l i n e a r l y embedded in R by t a k i n g , e.g. a t y p i c a l ively.
Now denote by
i
simplex of
ZL
simplex of
ZL.
~L ~
aL
to be the convex h u l l of the v e r t i c e s of a t y p i c a l There are standard homeomorphisms
ZK<~aK >
I
ZL and
of the respective cones• to
~
and
that
L
~
and
in K
t
ZK Let
which extend to standard homeomorphisms ~
and
~
be the corresponding points
I
cZ', c Z r e s p e c t i v e l y . Recall t h a t the c o n d i t i o n L K be s t r a i g h t l i n k s means, in the f i r s t place t h a t I
C~L
(resp.
C~K),
and t h e r e f o r e
n e c e s s a r i l y unique j - p l a n e s the image of points of sition
; ~ C~L
by i d e n t i f y i n g
aL
WL, WK
CZL'
(resp• CZK) l i e
respectively.
(resp • cZ') K
cos c o s +~ ) C~L ÷ eLc ~z~ n,k W (resp. W ) with Rj L K
@L thus o b t a i n i n g retopologization.
in /0 ~
Now i f
~
and
Br A ~
AcBr
Recall also t h a t under the compo-
cBr A
is determined
~L' ~LIC~L
which maps i n t o
and thus cBr A
go to the same p o i n t in
must n e c e s s a r i l y be because there is an element
169
in
by an a r b i t r a r y isometry
the o r d e r e d brouwer sphere
o b t a i n i n g an a p p r o p r i a t e c e l l of
it
I
CZL
@E O(j)
under Br A , such
8.10
that with
~ induces an ordered s i m p l i c i a l isomorphism ~L (~')÷L ~K(~j)K -i @(¢L ~ ) : ,K~. But then ~K ~ L is an isometry WL ÷ WK w h i c h l
-1
induces the corresponding isomorphism ~ + ~' with ~K ~ L ~ = ~" -i "L ~K Now e x t e n d ~K ~+L t o an e l e m e n t ~ of O(n+k) with ~U L : UK. Obviously
~L : K.
Moreover, since
a~ : ~, i t f o l l o w s , by the caL , caK , consistency of the i d e n t i f i c a t i o n s CZL cZ ~ CZ L, C~ K ~ K with orthogonal t r a n s f o r m a t i o n t h a t ~ = ~. Thus ~ and ~ are i d e n t i f i e d by the a c t i o n of
0(n+k)
The remaining p o i n t is
that
J:F.c°s
on
and hence x = y. ~v n,k is an open map. This is l e f t
u
to
the reader as an e x e r c i s e .
v
The argument f o r
is somewhat more d e l i c a t e .
First,
let
"T
denote an isomorphism class of ordered t r i a n g u l a t e d ( j - 1 ) - s p h e r e s , and l e t sion
~/T
j
links
be the union of a l l
with L
EL E T.
of dimension
U~(n,k) ~T n,k T (n,j;j) i s to
(n+l,k;j)
it
if
~(n,k)
(n,k;j)
topology,
with
its
is the set of a l l
suspensions
namely
T h u s ~/T 0(n+k)
links.
that
which
formal
w h i c h are
Now of
was used
in
course O~T(n,k) obtaining
the
acquires the weak topoloqy of union.
on l i n k s of dimension
as an i n v a r i a n t subspace.
of the i n f i n i t e
of dimen-
with
(n,k+1;j)-dimensional
topology of ~ ¢ o r d n,k Now the a c t i o n of
L
~ E T, then ~L being understood t h a t a l i n k of dimension
be i d e n t i f i e d
and
has a c e r t a i n
T(n,k)
I.e.
ordered formal l i n k s
orthogonal group
(n,k;j)
In the l i m i t , 0
on ~F~ T.
preserves
we have an a c t i o n
We wish, f i r s t
of a l l
to c h a r a c t e r i z e the homotopy type of / ~ / / 0 . T 8.5
Lemma. ~ / 0 Proof:
Let
is n
(weakly) c o n t r a c t i b l e . and
k
both be large compared to the number of
v e r t i c e s of a r e p r e s e n t a t i v e of
oK~T( n , k ) / O ( n + k ) . n+k standard R
Let and l e t
Rj + k
T.
denote
]7"(n,k) ~T
We wish to study
the
standard
be t h e
170
space
j+k-space of
those
in
the
L C c<~(n,k) I
8.11
such t h a t 8.6
U = Rj+k. L
Sublemma. Any
some element of
L(~(n,k)
7T(n,k).
Moreover any two elements of
in the same O ( n + k ) o r b i t i f O(j+k)
orbit.
Proof: f i n d an
is in the same O ( n + k ) o r b i t as
and only i f
~T(n,k)
are
they are in the same
Therefore ~ T ( n , k ) / O ( n + k ) = J T ( n , k ) / O ( j + k ) .
F i r s t of a l l ,
~ C O{n+k)
f o r any
L = (UL,~,)~ ~ ( n , k )
RJ+k
so that
L
T
we may
JT
aU = , thus aLc (n,k). Next, L if L ' L 2 ~ ,v' ~ ( n ' k ) and L = ~L for some ~ d O(n+k) then 1 2 1 ' Rj+k is an i n v a r i a n t subspace f o r ~ and thus = ~ (~) ~ E O(j+k) + O(n-j) 1 2
(R j+k ) I .
But then
where the l a t t e r
summand acts on
~'
: ~ ~ I has the same e f f e c t on L as i n-j I d i d , thus L d i f f e r s from L by an a c t i o n of O(j+k). The 2 I remainder of the sublemma f o l l o w s immediately. Note t h a t throughout the proof of 8.5 we have kept and, moreover, and
j.
JT(n,k) _
is
independent of
Thus we modify n o t a t i o n and w r i t e
the l i g h t of 9.6 we need only show t h a t
n
fixed
and depends only on
~T(k)
lim k+~
j
for
~T(n,k).
(k)/O(j+k)
k In
is weakly
T
contractible. Let
(J~T : lim~uT(k),
i n t e g e r span L
LE ~T ~J "
Define the p o s i t i v e
to be the dimension of the vector space spanned by
the v e r t i c e s of is
and l e t
ZL,
taken as vectors of Euclidean space.
independent of the choice of the ST(k)
Span L
in which to regard
L
as
belonging, and is obviously i n v a r i a n t under the a c t i o n of
O(j+k).
8.7
under the
Sublemma.
a c t i o n of L
For
O(j+k)
L e ~T(k), on
JT(k)
is
under t h i s a c t i o n is of type
(j+k-span L)-connected.
the i s o t r o p y group of O(j+k-span L);
thus the o r b i t of
O(j+k)/O(j+k-span L)
Thus, in the l i m i t
contractible.
171
YT'
L
and is
the o r b i t of
thus L
is
8•12 Proof:
Z be the subspace spanned by the u n i t vectors L ~ is in the i s o t r o p y Qroup of which are the v e r t i c e s of ~L • I f L,
it
Let
must leave
ZL
pointwise f i x e d since no vertex of
~L
may be
moved.
On the other hand, so long as t h i s r e s t r i c t i o n is met, any L action i s possible on the i n v a r i a n t subspace Z w i t h o u t compromisL I . e . the i s o t r o p y group ing membership in the i s o t r o p y group. c o n s i s t s of a l l
elements of
O(j+k)
leaving
Z pointwise f i x e d , L The remainder of 9.7 f o l l o w s
i.e.
O(j+k-dim Z ) = O(j+k-span L). L immediately.
8.8
Sublemma. ~T Proof:
S~
J T
is
is,
(weakly) c o n t r a c t i b l e .
e s s e n t i a l l y the space of embeddings of
as an a d m i s s i b l y - t r i a n g u l a t e d subsphere, where
r e p r e s e n t a t i v e of
T.
The c o n t r a c t i b i l i t y
Z
Z
in
is a f i x e d
of t h i s space was
demonstrated in Lemma 7.2 To summarize, T/O.
~/TIO = lim~7#T(n,k)/O(n+k) : lim J T ( k ) / O ( j + k )
is weakly T c o n t r a c t i b l e , while the o r b i t of every p o i n t is weakly c o n t r a c t i b l e • It
In t h i s l a s t q u o t i e n t space,
follows that
~T/O
the numerator
:
j
is c o n t r a c t i b l e , at l e a s t weakly, thus, our
lemma 8.5 f o l l o w s . Returning to the proof t h a t u is a homotopy equivalence, we ~cord ~/co r d i n v e s t i g a t e the geometry of and i t s o r b i t space /0. Just as in the previous chapter, ~ T ( n , k ) union in z~ c°rd
taken
of cone points of
"~ n,k over a l l L ~ S( n , k ) . ~ cord
e
L
may be i d e n t i f i e d with the
= image
Also as b e f o r e ,
C~L,
~T(n,k)
the union being
has a n e i g h -
borhood in ~
, denoted N where N is the image of a n,k T T (trivial) j - d i s c bundle P over ~ . (Here j = dim T; the f i b e r T T Let of PT over L E~.T i s , of course, i d e n t i f i e d with ) L" (j) ~ord T h e n we see t h a t Q(j+I (n,k) = 1NTC~n is
Q(n,k)
dim T ~ -
,k "
(j)
plus some spaces of the form ~ T ( n , k ) x D J + l attached by maps Q(n,k) . j) ~T(n,k)×S3 ÷ Q( . Passing to the l i m i t , ~/cord, we e a s i l y see t h a t
172
8.13 ~T :
U ~(n,k) is embedded and has a neighborhood ~ which is the n,k T T image of a t r i v i a l j - d i s c bundle ~C1) over ~ . Again, i f we l e t
Q( j )
: lim Q ( j ) (n,k) n,-~
c h a r a c t e r i z a t i o n of Q(j~I) Q(J)
~ i~ dim T~j-
:
Q(j+I)
~_×D dim T : j T
= Q(j) u
c ~£cord ,
I
we see t h a t the same
holds, v i z .
j+l
by maps on ~ / xSj T Now, when we pass to
I
~
where the p r o d u c t s are a t t a c h e d
ord
I0
it
the same sort of p i c t u r e emerges:
to
is i n t e r e s t i n g to note t h a t
The image of
~T'
denoted by ~ T
is a j - d i s c bundle over /~T/O, mod i d e n t i f i c a t i o n s on the b o u n d i n g I
sphere bundle. over ~T/O
That i s ,
is n a t u r a l l y
T
w h e r e 2T = ~TT/O.
f a c t t h a t the a c t i o n of i n t e r i o r of the same some fashion i f /~°rd/o
~
it
0
induced
cannot i d e n t i f y two points w i t h i n the
eL,
while i t
identifies
L
must i d e n t i f y
with
K.)
PT
Thus we have once more a f i l t r a t i o n :
is defined as
~C) T
(This claim e s s e n t i a l l y reduces to the
via a map which is a homeomorphism o f f
bundle.
from a bundle
,
e
with e in L K maps n a t u r a l l y i n t o
the bounding sphere
namely
~ ( J ) ~ ~Z c°rd
/0
and once more we have
dim T < j - 1 T ~(j+I)Q ing
:
(J)~
U ( /O)xO j + l dim T=j T
via a t t a c h i n g maps on the bound-
(~T/O)×SJ We are f i n a l l y
able to f i n i s h the proof of 8.4 by showing ~ oral be a homotopy equivalence. Recall t h a t A has one j - c a l l f o r each equivalence class of ordered t r i a n g u l a t e d ( j - 1 ) - s p h e r e s . be such a class:
then t h i n k of the c a l l
corresponding to
where, by s l i g h t abuse of n o t a t i o n we confuse tive.
In the space __~T/OxDJ =~T
graph, each t i b e r i s , way, i . e .
in f a c t ,
serves the decomposition of
.~ord
Let as
T cT
with a representa-
mentioned in the preceeding para-
i d e n t i f i e d with
we have a canonical map
T
T
to
cT
in a s p e c i f i c
m : /~/~/O×DJ + cT. This map preT T ord 10 and A respectively into
173
8.14 t h e spaces
~1~/OxDj and cT. That i s , the i d e n t i f i c a t i o n s on T ~ / /OxD are c o n s i s t e n t , under m , w i t h the i d e n t i f i c a t i o n s on T. T T ~cord ord Thus the m taken t o g e t h e r y i e l d a map /0 + A , which i s T in f a c t , u. Now assume i n d u c t i v e l y t h a t ulQ ^ (j) i s a homotopy J
A(j)
e q u i v a l e n c e o n t o the j - s k e l e t o n Q(J)
for
some j - d i m e n s i o n a l
T,
of
A°rd
Attach ~T(j)t°
while attaching
cT
to
A
~(j) Clearly
the e f f e c t
(j+l)-cell
to
tractible
of adding
~(J) ,
T
to
Q
is
up to homotopy, s i n c e
space crossed w i t h
Dj+l
i s a ( w e a k l y ) conT c l e a r l y maps t h i s
m T by a degree one map to the c e l l
homotopy c e l l
m e r e l y to add a
Now
cT.
C l e a r l y then,
(J
i s h o m o t o p i c a l l y the same as O cT since the c e l l s T were added, in e i t h e r case, by e q u i v a l e n t a t t a c h i n g maps. The same i
argument s t i l l ~(j+l) obtain Q u
holds
when we a t t a c h a l l
the ~ s i m u l t a n e o u s l y , to T ~(j) ~<~ord lim Q = /0, we see t h a t
Taking the l i m i t
must be a homotopy e q u i v a l e n c e , as d e s i r e d , and 8.4 is
thus
proved. Continuing with
the p r o o f of
cos F
F
÷
cord
/~/Co s
~ord
÷
8.1, +
+
we examine the d i a g r a m
x~cos /0
/ccor d
cos cord F and F are the f i b e r s X X/ maps ~ ÷/0~ O, X = cos, c o r d . where
8.9
Lemma.
F
Proof: in
cos
isotropy
cord
We f i r s t
particular, First
+ F
of
group
at
look at
cBr
ord /0 ~ A of
the r e s p e c t i v e q u o t i e n t
a homotopy e q u i v a l e n c e .
the a c t i o n o f
O(n+k)
on
cos n,k
and
,
the o r b i t
all, I
is
HA
it x
of
t y p e s which occur under t h i s a c t i o n . cos ~s c l e a r t h a t i f x E int e c , then the L ~ n+K
~
x
is
exactly
174
the i s o t r o p y
group
I
L
of
L
8.15 itself
under
links.
the
action
However, i t
link
L,
I
L of a l l
first
of
c o n s i s t s of leave
O(k)x0(n-j),
straight for
UL
CZL,
pointwise fixed.
a
t r u e of
of dimension
of
straight
I
x
J l 'J2
.
IL
is
ordered
e
which, WL,
thus c o n j u g a t e
Now l e t
L,K
be
r e s p e c t i v e l y with
L = K a we wish to
of
~ . Then, c l e a r l y ! C I ; -K K L i n c l u s i o n , and we see, as the r e a d e r may e a s i l y
t h a t up to conjugacy i t
where the map is
space
i n v a r i a n t and which, moreover, leave
ordered l i n k s
characterize this check,
the
those o r t h o g o n a l t r a n s f o r m a t i o n s
so the same i s
some s i m p l e x
on
is e a s i l y seen t h a t f o r a j - d i m e n s i o n a l s t r a i g h t
the j - p l a n e c o n t a i n i n g to
O(n+k)
looks l i k e
given by the i d e n t i t y
standard i n c l u s i o n
O{n-J2) C 0(n-J1 )
y E int
eL
e K, x e i n t
on the
0(k)-factor
on the o t h e r .
and the
Thus,
if
C I may be s i m i l a r l y c h a r a c t e r i z e d . y x Hence the c o n n e c t i v i t y of the i n c l u s i o n I C I is a f u n c t i o n y x s o l e l y of n, k, J l and J2' and, assuming j l , j 2 f i x e d , t h i s connectivity
goes to
Extending t h i s see t h a t ,
for
~
I
with
n
and
k. cord
a n a l y s i s to the a c t i o n on
x ~ int
Then we see t h a t
then
0 ( k ) x O ( n - j 2) + 0 ( k ) x 0 ( n - j 1)
eL,
Ix = IL.
Let
j
n,k
= dim L
,
once more we s = span L.
is c o n j u g a t e to 0(k+j-s)x0(n-j). Again, i f L L = K where the r e s p e c t i v e dimensions are J 'J2 and spans kl,k , a 1 2 then I CI is c o n j u g a t e to the p r o d u c t of the s t a n d a r d i n c l u s i o n s K L 0(k+J2-s2) c 0 ( k + J l - S l ) and 0(n-J2 ) C 0(n-'31) (note in t h i s regard that
I
s2-J 2 ~ s l - J l ) .
Thus,
if
x E int
eL
and
y ~ int
eK,
then
the i n c l u s i o n
I C I may be s i m i l a r l y c h a r a c t e r i z e d . As b e f o r e , y x we see t h a t the c o n n e c t i v i t y of I ~ I depends o n l y on a few y x p a r a m e t e r s , namely n, k, J l ' 32' Sl' s2' and, keeping the l a s t four (which depend only on the geometry of that this
c o n n e c t i v i t y goes to
Passing, t h e n , we see t h a t a l l
to the l i m { t
orbit
~
with
ZL
and n
a c t i o n s of
and 0
ZK,
c o n s t a n t we see
k. on ~ c o s
types are homotopy e q u i v a l e n t , i . e .
175
#cord
and ~ I
y
C__ I
x
,
8.16 is
a homotopy e q u i v a l e n c e when
homotopy t y p e o f
the o r b i t
be the unique p o i n t of we see t h a t
O/I x
space in
for stable
the f i b r a t i o n
theoretic
fiber
x E int
type
O/I
the O - c e l l
e is
x
and
L
y~ e L o
constant.
Thus the
By t a k i n g
x
to
e
for L a O-dimensional l i n k L O ( n + k ) / O ( k ) × O ( n ) = BO, the usual c l a s s i f y i n g
= lim n,k vector bundles.
It is thus
cos ~os ~/cos F + + /0 cos F is i d e n t i f i a b l e
possible
the a b s t r a c t with
to deduce homotopy-
BO - O/I
cord
~cord
that
for all X
~ord
The same holds t r u e f o r the f i b r a t i o n F ÷ + cos cord of course, F + F i s a homotopy e q u i v a l e n c e . This
x.
/0
and,
completes
8.9. It
follows
immediately that,
g i v e n a diagram
Y X
where
X
is
a
C-W
/~COS
~ >~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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