Stratified Mappings — Structure and Triangulability

175 Pages·1984·1.898 MB·English

by  Andrei Verona (auth.)

Lecture Notes ni
Mathematics
Edited by A. Dold and B. Eckmann
Subseries: Mathematisches Institut der ti~tisrevinU und
Max-Planck-lnstitut fur Mathematik, Bonn - vol. 4
:resivdA E Hirzebruch
2011
Andrei anoreV
Stratified Mappings
Structure
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Berlin Heidelberg New York oykoT 1984
Author
Andrei Verona
Department of Mathematics and Computer Science
California State University
Los Angeles, CA 90032, USA
AMS Subject Classification (1980): 57 R05, 58A35; 32B25, 32C42,
57R35, 57R45, 57S15, 58C25, 58C27
ISBN 3-540"13898"6 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-13898-6 Springer-Verlag New York Heidelberg Berlin Tokyo
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INTRODUCTION
For several reasons, most of them stemming from algebraic topology,
it is important to know whether a topological space, or more generally a contin-
uous map, is triangulable or not° Cairns [Ca] proved the triangulability of
smooth manifolds; another proof, also providing a uniqueness result, is due
to J. H. C. Whitehead [Wh]o First attempts to prove the triangulability of al-
gebraic sets are due van der Waerden [W], Lefschetz [Le], Koopman and Brown
[K-B] and Lefschetz and Whitehead [L-W]~ Rigorous proofs, in the more general
case of semianalytic sets, were given by Lojasiewicz [Lo] and Giesecke [Gi]o
Later, Hironaka [Hil]and Hardt [Ha2] proved the triangulability of subanaly-
tic sets. The most general spaces known to be triangulable are the stratified
sets introduced by Thom IT2] and the abstract stratifications introduced by
Mather [Mal] (Mather's notion is slightly different from Thom's one, but it
is more or less clear that the two classes of spaces coincide, at least in the
compact case); they include all the spaces mentioned above and also the orbit
spaces of smooth actions of compact Lie groups° Their triangulability was pro-
ved by several authors (Goresky [GI], Johnson [J2 ,] Kato [Ka] and Verona [Ve3]
to mention only the published proofs. A more detailed discussion of these
proofs and of others can be found in the introduction of Johnson's paper or
at the end of Section 7 of the present work). The more difficult problem of
the triangulability of mappings was considered by much fewer authors: Putz
[P] proved the triangulability of smooth submersions, Hardt [Ha 2] proved the
triangulability of some, very special, subanalytic maps and I proved in
[ Ve3] the triangulability of certain stratified maps. In IT1] Thom con-
sidered the problem of the triangulability of smooth maps and (implicitely)
conjectured that "almost all" smooth mappings are triangulable. It is the
aim of this paper to prove this conjecture. To be precise, we shall prove
Theorem. Let M and N be smooth manifolds without boun-
dary. Then any proper, topologically stable smooth map from M to N is
triangulable.
VI
Since the set of proper and topologically stable smooth
maps from M to N is dense in the set of all proper smooth maps from
M to N (Thom-Mather theorem*) we obtain a positive answer to the
above mentioned conjecture.
As a matter of fact, we prove a more general result concern-
ing the triangulability of certain stratified mappings (Theorem 8.9)
which implies the theorem stated above and also the following result
(first proved by Hardt [Ha2]): any proper light subanalytic map is trian-
gulable (light means that the preimage of a discrete set is discrete).
Our Theorem 8.9 is not as general as one would expect it. It applies only
to proper and nice abstract Thom mappings (nice means that the mapping is
finite to one when restricted to a certain subspace). It is natural to con-
jecture that the theorem is true for any proper abstract Thom mapping. A
positive answer would solve another conjecture of Thom [TI] : any proper
Thom mapping (in Thom's terminology "application sans eclatement") is
triangulable. The main difficulty in proving this more general version of
Theorem 8.9 is explained in Section 8.15.3.
Since we are dealing with stratified spaces as introduced by
Mather in [Mal] and since these lecture notes have never been published,
I thought it would be useful to collect in a first part of the present
work (Chapters ,i 2, and )3 the main results of the theory. Some of the
proofs presented here are new and simpler than the original ones. For tech-
nical reasons, we are obliged to work with certain manifolds with corners,
called here manifolds with faces. The necessary facts concerning them are
presented in Chapter 4. In Chapter 5, we extend the theory of abstract strati-
*) This result was conjectured by Thom [TI]. In [T3] and [T2] he outlined
a possible proof. Later Mather ([Ma2] and [Ma3]) filled in the details, made
it rigorous and, by slightly changing some of Thom's concepts, cleaned up
many technical points. Another proof (only slightly different) is presented
in [ Gib]o
fications and abstract Thom mappings to the case when the strata are allowed
to be manifolds with faces; most of the proofs are copies of the proofs
presented in the first three chapters and so they are omitted. In Chapter 6,
we prove some theorems concerning the structure of abstract stratifications
and of abstract Thom mappings. In some sense, they can be viewed as a kind
of "resolution of the singularities" in the ~ C -case. For example, Theorem
6.5 can be interpreted as saying that any abstract stratification of finite
depth can be obtained from a manifold with faces by making certain identi-
fications on the faces. Chapter B contains the main results of the paper
(they were mentioned above). In an appendix, I have collected some facts
from PL-topology which are needed in Chapters 7 and 8.
Acknowledsement. For their support and/or hospitality during
the preparation of this work, I wish to express my gratitude to the Institut
des Hautes Etudes Scientifiques in Bures-sur-Yvette, the Alexander yon Hum-
boldt Foundation, the Sonderforschungsbereich Theoretische Mathematik of
the University of Bonn, the National Science Foundation (Grant MCS-8108814
(A01), the Institute for Advanced Study in Princeton and, last but not
least, the Max-Planck-lnstitut fur Mathematik in Bonn. Especially, I wish
to thank Professor F. Hirzebruch, whose confidence and understanding during
the preparation of this paper were of great helpo
TABLE FO STNETNOC
O. NOTATION DNA SNOITNEVNOC
1. TCARTSBA STRATIFICATIONS
1.1. Tubes
1.2. Definition and elementary properties of abstract stratifications
1.3. Existence of partitions of unity
1.4. The sheaf C: is fine
Notes
2. DELLORTNOC VECTOR FIELDS 9
2.1. Definition and elementary properties 9
2.2. Flows Ii
2.3. Existence of flows 12
2.4. Lifting of vector fields 14
2.5. Locally trivial controlled maps 16
2.6. First isotopy lemma 16
2.7. The local structure of abstract stratifications 17
2.8. Saturated subsets 17
2.9-2.10. Preimage of saturated subsets 17
2.11. Fibre products 19
2.12. The abstract stratification ~IS~ 19
Notes 20
3. ABSTRACT MOHT MAPPINGS 20
3.1. Regular squares 20
5.2. Definition and elementary properties of abstract Thorn mappings 21
3.3. Examples 22
f
3.4. The sheaf BX 23
3.5-3.6. Lifting of controlled vector fields 24
3,7. The second isotopy lemma 27
Notes 28
IIIV
4. MANIFOLDS WITH FACES 29
4.1. Definition and elementary properties 29
4.2. Maps compatible with the faces 31
4.3. Examples and remarks 32
4.4-4.5. Vector fields parallel to the faces 36
5. ABSTRACT STRATIFICATIONS (WITH FACES] 37
5.1. Definition and elementary properties 37
5.2. Maps compatible with the faces 39
5.3. Examples, remarks and constructions 40
5.4. Controlled vector fields 48
5.5. Abstract Them mappings 49
5.6. Lifting of controlled vector fields 50
5.7. Second isotopy 1emma 51
5.8. First isotopy 1emma 51
5.9. More examples and remarks 51
5.10-5 .11. Applications of the isotopy lemmas 54
5.12. nO the controlability of the 0 functions 56
6. EHT STRUCTURE OF ABSTRACT MOHT MAPPINGS 57
6.1. Decompositions of abstract stratifications 57
6.2. Existence of decompositions of abstract stratifications 65
6.3. Total decompositions of abstract stratifications 69
6.4. Restriction of decompositions 70
6.5. Existence of total decompositions of abstract stratifications 71
6.6. The core of an abstract stratification 71
6.7. Admissible squares 72
6.8. Decompositions of abstract Them mappings 83
6.9-6.10. Existence of decompositions of abstract Them mappings 96
6.11. Total decompositions of abstract Them mappings 111
6.12. Existence of total decompositions of abstract Them mappings 117
6.13-6.15. More cores i19
XI
.7 TRIANGULATION FO ABSTRACT STRATIFICATIONS 120
7.1. Triangulation o£ relative manifolds 120
7.2-7.5. Good triangulations of cores 121
7.6. Smooth triangulations of abstract stratifications (definition) 126
.7.7 Extending a good triangulation of the core to a smooth triangulation
of the corresponding abstract stratification 126
7.8, Existence of smooth triangulations of abstract stratifications 127
7.9. Triangulation of subanalytic sets 127
7.10. Triangulation of orbit spaces 128
Notes 128
8. TRIANGULATION OF NICE ABSTRACT THOM MAPPINGS 129
8.1. Notation 129
8. .2 Remarks on good triangulations of c(f) 129
8. .3 Regular triangulations of c(f) 130
.8 .4 Nice abstract Thom mappings 131
.8 5-8.7. Properties of regular triangulations of c(f) 133
8. 8. The canonical extension of a regular triangulation of cCf) 135
.8 .9 Triangulation of proper nice abstract Thom mappings 138
.8 .01 Topologically stable mappings 138
.8 .11 Triangulation of proper topologically stable mappings 139
.8 .21 Triangulable smooth mappings are generic 140
.8 .31 Triangulation of proper light subanalytic mappings 140
.8 .41 A remark of Hironaka 140
8. .51 A possible generalisation 140
9. APPENDIX 141
9.1. Simplicial complexes and triangulations 143
9.2. Product of triangulations 143
9.3. Fibre product of triangulations 145
9.4. Mapping cylinders 146
9.5. Two lem~as 151
9.6. Piecewise linear maps 152
SECNEREFER 351
SUBJECT INDEX 156
LOBMYS INDEX 158
O. NOTATION AND CONVENTION
O.I. A topological space is called nice if it is Hausdorff, locally compact,
paracompact and with a countable basis for its topology.
0.2. If A is a topological space and X c A , then ClA(X) (resp. intA(X))
denotes the closure (resp. interior) of X in A .
0.3. If A and B are topological spaces, then B A[./ denotes their dis-
joint union.
0.4. For any set A A , I (or id A ) denotes the identity map of A .
0.5. Smooth means always differentiable of class ~ . C
0.6. The connected components of a smooth manifold may have different dimen-
sions. Given a smooth manifold M we denote by i~T its tangent bundle. If
x C M , TM denotes the tangent space of M at x . If N is another smooth
x
manifold and f : M ~ N is smooth, df : TM ~ TN denotes the differentlal of
f ; if x C M , then df x : TM x ~ TNf(x) denotes the restriction of df .
The smooth map f is called submersive if df is surjective for any x 6 M.
x
0.7. Let A be a topological space and let X , Y and Z be subsets of A.
Let f and g be maps defined on X and Y respectively. We say that f
equals g near Z (denoted f = g near Z ) if there exists a neighborhood
U of Z such that lf X N U = Ig Y N U . The same terminology is also used in
other similar situations (for example X = Y near Z means that there exists
a neighborhood U of Z such that X N U = Y N U .)
0.8. R denotes the field of real numbers ; R = {r C R ; r > O} and
+
~ = R + {r 6 R ; r > O}
.I ABSTRACT STRATIFICATIONS
i.I. Let A be a nice topological space and tel X c A be a locally
closed subset. Let X T be an open neighborhood of X in A,
~X X : T > X be a continuous retraction (i.e., ~rxIT X = IX) and
PX X : T >---- R+ be continuous and such that pxl(0) = X. Given
c, 6 : X ~ R we shall use the following notation c < G if
¢(x) < 6(x) for any x E X; if a < ~ set
X × (s, {) : {(x, t) E X × R; ~(x) < t < 6(x)}
X x {~} = {(x, t) E X x R; t : ~(x)}
X × [~, 6) : (X × (~, ~)) U (X × {~})
X × (~, 16 = (X × (~, 6)) U (X × {6])
X × [~, ]~ : (X × [~, 6)) U (X × {6})
eXT = {aC TX; 0x(a) < a(~x(a))}
x S : {a E TX; Px(a) : ~(~x(a))}
In such an expression a real number will be considered as the corresponding
constant function on X. From now on, unless the contrary is specified,
(or el' ~2 ~' 61 .... ) will be a continuous function taking
. . . . .
values in R+, its domain being determined by the context; if in addition
the domain is a smooth manifold, then ~ will be assumed to be smooth.
When no confusion can arise we shall denote the restrictions of ~X and
e
PX to X T by the same symbols; otherwise we shall use the notation
~X and XP respectively.
Let T X, X~ and PX be as above. One can verify that, possibly
after shrinking TX, the following assertions are true:
e
(1.1.1) if X c U cA with U open, then T XC U for some s;