Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1768
3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo
Markus J. Pflaum
Analytic and Geometric Study of Stratified Spaces
123
Author Markus J. Pflaum Department of Mathematics Humboldt University Rudower Chaussee 25 10099 Berlin, Germany E-mail:
[email protected] Cataloging-in-Publication Data applied for
Mathematics Subject Classification (2000): 58Axx, 32S60, 35S35, 16E40, 14B05, 13D03 ISSN 0075-8434 ISBN 3-540-42626-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10852611
41/3142-543210/du - Printed on acid-free paper
Contents
1
Introduction
11
Notation
1
1.1
Spaces and Functional Structures Decomposed spaces
1.2
Stratifications
1.3
Smooth Structures
1.4
Local
1.5
The, sheaf of
1.6
Rectifiable
1.7
2
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Extension
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22
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34
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Whitney conditions
functions
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regularity Whitney functions .
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42
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44
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53
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63
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on
regular
Singular Spaces Whitney's condition (A)
2.4
Metrics and
2.5
Differential operators
2.6
Poisson structures
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space structures
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spaces
63
on
Differential forms and stratified cotangent bundle
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68
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80 83
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95
91
Theory
neighborhoods point distance and
3.1
Tubular
3.2
Cut
3.3
Curvature moderate submanifolds
3.4
Geometric implications of the
3.5
Existence and
3.6
Tubes and control data
3.7
Controlled vector fields and
maximal tubular
uniqueness
3.8
Extension theorems
3.9
Thom's first
on
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91
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101
conditions
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112
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117
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125
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Whitney
theorems .
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neighborhoods .
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integrability
controlled spaces
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134
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140
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143
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147
Differentiable G-Manifolds
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151
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153
Orbit
4.2
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2.3
4.1
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2.2
Control
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Stratified tangent bundles and Derivations and vector fields
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for
length
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and
theory
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Triviality Whitney curves
...
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and the
3.10 Cone spaces 4
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Differential Geometric Objects 2.1
3
15
Stratified
isotopy lemma .
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151
Spaces
Proper Group
Actions
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VIII
5
4.3
Stratification of the Orbit
4.4
Functional Structure
5.2
5.3 5.4
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6.4
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158
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162
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171
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173
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177
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169
DeRham theorems DeRham
on
cohomology
orbit spaces of
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Whitney functions
of Smooth Functions and their modules
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for
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183
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186
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189
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195
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Sup plements from linear algebra and functional analysis The vector space distance
A.2
Polar
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201
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201
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202
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203
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207
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207
Kiihler differentials
B.2 B.3
205
The space of Kdhler differentials
Topological version Application to locally ringed .
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spaces
Jets, Whitney functions and a few eOO-mappings C.1 Fr6chet topologies for e'-functions C.2 C.3 C.4
Jets
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169
183
topological modules Continuous Hochschild homology Hochschild homology of algebras of smooth functions
A.1
B.1
C
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complex on singular spaces DeRham cohomology on e00-cone spaces
decomposition A.3 Topological tensor products B
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The deRham
Homology of Algebras 6.1 Topological algebras 6.2 Homological algebra 6.3
A
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Space
DeRham-Cohomology 5.1
6
Contents
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Whitney functions Smoothing of the angle .
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205
209
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209
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210
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212
For
Stephanie and
Konstantin
Chapter
1
Stratified
and Functional
Spaces
Structures
Decomposed
1.1 1.1.1
Let X be
a
spaces
paracompact Hausdorff space with countable topology, and Z
partition of X into locally closed subspaces S C X. better the pair (X, Z) a decomposed space with pieces S E Z and X, if the following conditions are satisfied:
locally
finite
(DS1) Every piece
S E Z is
a
(DS2) (condition of frontier) We write in this One checks
pieces
of
X,
case
immediately
smooth manifold in the induced If R n
9 =/= 0 for
a
Then Z
a
one
calls X
decomposition
a or
of
topology.
pair of pieces R, S E Z, then R c -9. S, or a boundary piece of S.
R < S and call R incident to
that the incidence relation is
hence the notation R < S is
an
order relation
on
the set of
justified.
Explanation The notion "locally closed" will appear more often in this work. us briefly recall its meaning. By a locally closed subset of a topological X understand a subset A c X such that every point of A has a neighborhood we space U in X with A n U closed in U. Equivalently, A is the intersection of an open and a closed subset of X, or in other words A is open in its closure. Obviously, the finite intersection of locally closed subsets is again locally closed. Submanifolds lie always locally closed in their ambient manifold. By the boundary aA of a locally closed subset A C X we will understand the closed subspace X \ A, which in general does not coincide with the topological boundary X n CA. If X is a decomposed space, and S C X one of its pieces, then aS bdr (A) consists of all boundary piece's R < S. Let us note that the notation aA will not lead to any confusion with the boundary aM of a manifold-with-boundary M. Namely, if M is embedded as a closed subspace of some Euclidean space Rn, then the interior M' of M is locally closed in Rn and the boundary aM of the manifold M is just the boundary aM' of the locally closed 1.1.2
Therefore let
=
,
subset M' C Rn
as
defined above.
M.J. Pflaum: LNM 1768, pp. 15 - 62, 2001 © Springer-Verlag Berlin Heidelberg 2001
Stratified
16 1.1.3 Remark As X is many
Spaces
separable, the decomposition Z
and Functional Structures
contains at most
countably
pieces.
1.1.4 Remark Instead of manifolds
(DS1) any object of arbitrary topological spaces. Thus one obtains the so-called Tdecomposed spaces. As an example for T let us name the category of real or complex analytic manifolds, or the category of polyhedra. one can
take in condition
category T of
an
introduce the category E-Tar of E-manifolds, the objects of topological sum of countably many connected smooth and sepaThe morphisms of E-Tar are the continuous and on every component
In this context
we
which consist of the rable manifolds.
smooth functions between E-manifolds. If the dimension of the components of a Emanifold M is bounded, we will say that M has finite dimension, and denote the supremum of these dimensions
A
1.1.5
decomposition
The dimension of
In most
by dim M.
of X into E-manifolds is called
applications
a
(X, Z)
decomposed
space
dimX =sup
f dimSJ
E-decomposition.
a
S E
is defined
by
ZJ.
will consider
only finitely dimensional decomposed spaces. (X, Z), where k E N, we denote the decomposed space
we
By the k-skeleton of
Xk
U
S
SEZ,dimSO,y
=:sin X
topology by R2 is not locally connected, but a a decomposed space S2. Hereby S, < S2, but simultaneously dimS, dimS2- Such
and the induced
with
pieces S,
and
=
kind of spaces should not be included in
further conditions
impose
Even "more
where
R,
R,
=
R2
=
f (0, 1j, Z )
we
will later
a decomposed space which will exclude such examples. is the decomposed space Y R, U R2 C R3 with pieces =
E
WI
jj2
Z2
+
(x,ij,z) EWlx>O,lj
0 is transcendental.
11
and
=sin
(1)
dim R2
holds, though
R2-
Spirals
1.1.13
The
XSpirr and the slow
both
fast spiral (Fig. 1.3) =
f0j U
I (T
0, T COS 0) 1
T
f(Tsin 0,TCOS 0)1
r
sin
=
e-E)2 ,
E)
>
01CW
spiral (Fig. 1-4)
X ;pj . are
on
pathological"
considerations. Therefore
our
decomposed
=
{0} U
spaces with the
Note that both the slow
as
well
as
origin
=
e-0,
0 >
01
C
W
piece and the rest as second piece. spiral turn infinitely often around the
as one
the fast
origin. Simplices and polyhedra By an affine simplexof dimension point set s C Rn with n > m of the form
1.1.14
stands
s
a
=
s[vo,vi,
-
-,
v,,,]
tE Aj,j E j=0
j=0
Aj
=
1 and
Aj !
0 for
j
=
0,
m one
m
under-
I
1.1
Decomposed
1.3: Fast
Figure where vo,
of
19
spaces
v,,
are
Spiral
Figure
affinely independent points
With the standard basis
s.
standard simplex s. := s[0, el, For every simplex s the (m
-
(el, -
..
)
Spiral
of R1, and will be denoted
of RI
one
vertices
as
obtains the so-called m-th
e,,,].
-
,
-
e
1.4: Slow
k)-dimensional
manifolds
n
sjo,.-.,jk
ENjvj
:=
E s
I Aj,,,
-
-Aj,
>
0 and
Aj
=
0 for
Aj :A Ajo,
-
-
-
,
Ajk
j=0
where k
runs
through
through the natural numbers from 0 to
all sequences of the form 0
max(n, h),
L(M)
c
and
0, 1
H
=
(CD4) M,, Hereby
dimension M
:
Rm such that the
on
0 for all
o
a
=
(d,(, )ijj )jEI
tions i1j, which define
(CD3)
boundary of topological embedding L
a
j
E
By a corner together with a following holds: )
---
m.
Rn
J1.
For every I C J define
M,
Two
Whitney Cusp
manifold with
topological
a
we
1.6:
(11j)jEj
MI,
and
is
1,
diffeomorphism bijective mapping a
0 for all
j
E I
and
llj(L(-y))
>
0 for all
of cotangent vectors
corresponding
linearly independent
at
( j)jEj H oc:
:
0 J
of M --
-4
j
are
L(x)
for all
j
Ij.
to the funcX
E
MI.
called equivalent, if there is
an
(5
between open subsets of RN with such that
L.
M, for all I
c
J.
have embedded Rn resp. Rf' into RN via the first
coordinates, and extended correspondingly. By a manifold-with- corner we now understand a topological manifold M with boundary together with an equivalence class of corner data. The family Z (Mj)jcj then is independent of the special choice of the corner datum in the equivalence class and comprises a decomposition of M. By (CD2) every L(MI) is a submanifold of Rn, hence its manifold structure can be carried over to M, via the embedding L. By (CD3) this manifold structure is independent from the particular corner datum. Moreover, MI, nVj-:A 0 implies P c I, hence MI, C MI. So the condition of frontier is satisfied as well, and M is a decomposed space indeed. Examples of manifolds-with-corners are given by the simplices defined above. L
we
and 1
=
Stratified
22
Spaces and Functional Structures
1.1.20 Glued spaces A method for the construction of decomposed spaces is given by iteratively gluing together manifolds-with-boundary along their boundary. The corresponding construction is found in the work of THOM [169, See. C] and comprises
essential component of THOM's notion of an ensemble stratifi6
one
is the notion of
component
point).
at this
We will
give
an
incidence
but
scheme, decomposed we
(the
other essential
will not discuss this further
the thus obtained
namely general than the one of THOM. The larger generality lies in the fact that we allow as pieces even manifolds not diffeomorphic to the interior of a compact manifold-with-boundary spaces. Let
glued
(cf.
also
mention that
us
construction is
our
a
spaces
a new
little bit
name,
more
[751).
glued spaces we first need a finite family (Mi)1 1 are with boundary. Recursively M, one then defines topological spaces Ei by El Ei Uhi+l Mi+1 where the M, and Ei+j functions continuous satisfying the following hi+1 : 3Mi+l -) Ei are a priorily given gluing condition As
ingredients
for
is without
where
=
=
,
(GC) Thus
M? n we
hj(aMi)
obtain the
=
M? for all
glued
space X
pieces Si
M?. The functions hi z
PROOF:
Axiom
j =
> i
with M? n
Ek- It
is
belonging
a
space and possesses the
decomposed
to X
are
(DS11) is satisfied trivially by (DS2). According to definition the
to show axiom
hj(aMi) :A 0. called its
gluing functions.
definition of sets
Uj>i Mj,,
X; hence are
it remains
open in
5(,
that
Let Mi n Mj, :A 0 for j < i, and x E Mj' arbitrary. Moreover, neighborhood of x. By the gluing condition (GL) h;-1 (U n MO) has to be nonempty, hence by construction of X there is an open V C Mi with V n a Mi and 0 * V n Moi C U. But that means U n Moi :A 0, hence h;-1 (U n M,0) I means
Mi
Uj 0.
=
0,
=
(t, hi+,(m))
CM, and CEi+j CEi UChi+l Now, the reader will convince himself easily that CE, this with CX i 1. for k CEk implies that CX is a glued 1, Together CMi+j =
=
-
=
=
space.
Stratifications
1.2 Let
us
of X
consider
one can
a
decomposed space (X, Z). Then within the class of all decompositions decompositions which differ from Z only slightly, as locally around
find
1.2 Stratifications
.
23
point they look like Z, and decompositions which differ from Z in an essential an example for that take R with the decomposition Z into to}, Wo and R"--O. Other decompositions of R are given for example by Z, ff0j, R \ 101} and {R}. Intuitively it is clear that Z, looks similar to Z, where is really different from Z. Let us formulate this phenomenon in more precise mathematical terms and introduce the notion of a stratification according to MATHER [123]. Stratifications in MATHER's sense generate equivalence classes of decompositions of X. It will turn out that within every such equivalence classes there exists a coarsest decomposition; its pieces are the so-called strata of X. every
way. As
=
1.2.1
Before
notion of
a
we
=
introduce in this section stratified spaces let
us
recall the
briefly
set germ.
by X a topological space, and let x E X. Two subsets A and B of X equivalent at X, if there is an open neighborhood U c X of X, such that A n U B n U. This relation comprises an equivalence relation on the power set of X. The class of all sets equivalent to A C X at x will be denoted by [A],, and is called the set germ at'x. If A c B c X we sometimes say that [A]x is a subgerm. of [B]X, in signs [A]x c [B].,. One checks easily that two sets A, B C X are equivalent at x, if and only if the function germs IXAI.., and IXBI-x of the characteristic functions of A and B at x are equivalent. Denote called
are
=
1.2.2 Definition
By
stratification
a
8 which associates to
ping following
(ST1)
X
of
a
topological
E X the set germ
8.X
of
a
space X
we
understand
a
map-
closed subset of X such that the
axiom is satisfied:
For every
x E
X there is
a
neighborhood U of x and a decomposition Z of 8,, coincides with the set germ of the piece
such that for all Ij E U the germ Z of which V is an element.
U of
pair (X, 8) is called a stratified space. Every decomposition Z of X defines a stratification by associating to X E X the germ 8.x of the piece of which x is an element. In this case we say that 8 is induced by Z. By definition a stratification is always induced at least locally by a decomposition The
Z.
A continuous map f
:
X
--
Y between stratified spaces
morphism of stratified spaces or shortly exist neighborhoods V of f (x) and U c of U and
of V
inducing Sju
resp.
JZjv
(X., 8)
and
(Y, _T)
is called
a
stratified mapping, if for every X E X there f-'(V) of x together with decompositions Z
a
in the
sense
of
(ST1)
such that the
following
holds:
(ST2)
For every ij G U there is an open neighborhood 0 C U such that the map restricted to the open subset S n 0 of the piece S E Z containing -Y has
fisno
image
in the
piece R
E
containing f (y)
and such that f isno is
from S n 0 to R. In
particular f (8.x) then
is
a
subgerm of gZf(.,, morphisms
The stratified spaces and their
form
a
category (5sp,,t,,,t.
a
smooth map
Stratified
24
and Fbnctional Structures
Spaces
1.2.3 Remark Similarly like for decomposed spaces one can form the notion of a T-stratification, where T is a category of topological spaces. More precisely, by a T-stratification we understand a mapping x -4 & such that 8x is locally induced by Now it is clear what to understand by a a T-decomposition in the sense of (STI).
E-stratification.
60strat has finite products and sums. Moreover, one can define subobjects 1Ezp,,t,,,t. By a stratified subspace of (X, 8) we mean a topological subspace Y C X such that for every X E Y there is an open neighborhood U in X and a decomposition Z inducing Sju such that (Y n U, Z n Y) is a decomposed subspace of The category
1.2.4
in
(U, Z).
In this
case
We want to
pair(Y, 8 n Y)
the
regard
is
again
a
stratified space.
decompositions Z, and Z2 by them are the same. In
two
the stratifications induced
of X
as
such
essentially
a case we
the same, if
call
Z, and Z2
equivalent.
(MATHER [123,
1.2.5 Lemma
Z, and Z2
on
X then for all
x
E
2.1])
Lem.
We show
by
induction
interchanging Z, and Z2
to be shown.
So let
us assume
=
dp Z2 N
dpz, (x)
on
dpz, (x) After
are
equivalent decompositions
two
X
dpzl (x) PROOF:
If there
(1.2.1)
'
that
:! dp Z2 N
-
the claim then follows.
dpz, (x)
that
=
If
dpz, (x)
=
0
nothing
has
k + 1 and that the claim holds for
< k. Let X E S < S k+1 be a maximal sequence of pieces from So < S 1 < Z, and R the piece Of Z2 with x E R. Then there exists an open neighborhood U of R n U such that U meets only finitely many pieces Of Z2. By X E -j x with S n U there then exists a piece R, Of Z2 such that X E S, n R, n U. Then R < R, follows. After the choice of an element Ij E S, n R, the induction hypothesis entails
dpz (-y)
=
*
'
*
=
dpzl (x)
=
dpz (ij)
+ I
1. Let y be a further point of depth k. Then there exists an open neighborhood U of ij, a decomposition Z of U defining 8 and a not extendable < Sk of pieces of Z. As Ij E Sk-I and Sk-I C TI, the relation chain'tj E So < S1 < 0 hence Tj_. This finishes the proof. C E T, Tk follows, Ij k.
*
-
1.2.7 Proposition Any stratified space (X, 8) has a decomposition Z8 with the following maximal property: for every open subset U C X and every decomposition Z inducing 8 over U the restriction of Zs is coarser than Z.
The
unique decomposition Z8
space
(X, 8)
decomposition of the stratified pieces of 8 are called the strata of
is called the canonical
by 8.
and will often be denoted
The
X.
decomposition Z8 inductively. To decomposition of To inducing 8 over To. Now, let us suppose that we have for all T, with 0 < 1 < k open submanifolds S C T, such that the following holds:
Let
We construct the
PROOF:
be the union of all d-dimensional connected components of
family Zk
(S)SE, ,I'I
a
To of all points of depth 0 by X'. If stratum, the so-called top stratum In this case .
not every connected stratified space needs to have
a
top
stratum.
For
S in
a
checks
exists are
a
one
calls the union of all strata R
signs EtX(S). Obviously,
Given
1.2.10 one
stratum S C X
in
X,
morphism f
a
stratum
X
--
Etx(S)
is
Rs,,
of Y with f (So)
(resp. submersions),
S the star
>
again
C we
Rso
and
call f
a
f1so
E
So of
a
e00 (SO)
stratified
(or
6toile
)
of
stratified space.
a
Y between stratified spaces
that for every connected component
easily
immersions
:
the star
-
(X, 8)
and
(Y, 3z)
stratum S of X there
If all restrictions
immersion
f1so (resp. stratified
submersion). 1.2.11 Example At the end of this section let me give an example of a stratified subspace such that the canonical injection does not map strata into strata. Obviously such an example does not stand in contradiction to the fact that connected components of strata are mapped into strata. Consider the stratified space X c R1 given by the union of the cube W [0, 1]3 x f (0, 0) 1 and the square Q {(0)0, 0) 1 X [0, 1] 2. The set of all closed edges K U f(O, 0, 0, 0)} X [0, 11 lying on the [0, 11 x f (0, 0, 0, 0)} U coordinate axes is a stratified subspace of X and consists of a 0-dimensional stratum =
==
=
and
a
1-dimensional stratum. Both
of the cube have
Analogously not lie
on
1.3 1.3.1
of class
are
not subsets of
in contrast to the
the vertices of the cube have
depth
ones
a
stratum of
X,
as
the
of the square which have
depth 3,
where the vertices of
Q
edges depth 1.
which do
2.
Smooth Structures Let X be
C',
image x(U a
depth 2,
W have
n
a
m (=-
set U c X to
is
...
a
S)
stratified space, and 8 the
N"
U
locally is
a
diffeomorphism
fool
family of its strata. A singular chart homeomorphism x : U -- x(U) c R1 from an open subspace of R' such that for every stratum S E 8 the
is
closed
a
submanifold of Rn and the restriction Xiuns : U n S -- x(U n S) of class Cm. Sometimes we call the domain U of a singular chart
1.3 Smooth Structures
chart domain.
shortly
27
Moreover,
we
often
0 C Rn to express that 0 Two singular charts x : U -4 Rn and U
use
for
charts
singular
x(U)
C Rn is open and
notion of the form
a
C 0 is
locally closed. compatible if for every x E U n ft there exists an open neighborhood U, c U n ft, an integer N > max(n, ii), open neighborhoods 0 C RN and 6 c RN of x(Ux) x f0j resp. R(U,.) x fO}, and a H LN xlu.,. Hereby we diffeomorphism H : 0 --1 (5 of class C1 such that L l Rjux have denoted by LN for N > m the canonical embedding of R' in RN via the first m coordinates. We call the diffeomorphism H a transition map from x to R over the domain U,,. To keep notation reasonable we will identify singular charts x: U -4 Rn N in the following with their extensions Ln x: U -4 RN, N > n. Like in differential geometry one defines the notion of a singular atlas on X of class C' as a family (xj),,j of pairwise compatible singular charts xj : Uj -4 Rni of X. Often we will denote such a singular atlas class C' on X such that Ujc Uj :j this will emphasize the domains Uj and will express that the Uj by U (Uj, xj)j,j; provide a covering of X. Sometimes we will say that U is a covering by charts. Two atlases U and ft of X are called compatible, if every singular chart of U is compatible with every singular chart of ft. x :
--
:
fl
--+
RF'
are
called
-
=
,
.
-
-
n
n
M
-
=
=
1.3.2 Lemma The
compatibility of,singular atlases
is
an
equivalence relation.
Obviously the compatibility of atlases is reflexive and symmetric. It transitivity. Let U, ft and ft be three singular atlases such that U and ft are compatible as well as ft and it. We have to show that every chart x : U -4 R' out of U is compatible with every R : U -- Rf' out of ft. For x E U n fl choose a sufficiently small open neighborhood U, c U n ft and a chart R: U, -- Rf' A h and out of ft. After shrinking Ux and enlarging n, h and h we can suppose n (5 can find over U, transition maps H : 0 -4 (5 C Rn from x to R and c Rn PROOF:
remains to prove
=
H
from R to R. But then
U, hence
domain
x
and
are
:
0
is
a
transition map from
Like for differentiable manifolds the set theoretic inclusion induces
atlases.
between
to R
x
over
the
compatible. an
order relation
all charts of all atlases in
a compatible singular Now, combining equivalence class one obtains a maximal atlas containing all other atlases of the equivalence class as subsets. In particular the maximal atlas determines the equivalence class uniquely.
fixed
1.3.3 Definition A maximal atlas of
X is called
a
C!'-structure
on
singular
X, and for the
charts of class C'
case
that
m
=
oo a
on a
stratified space
smooth structure
on
X.
1.3.4 Remark In the mathematical literature to define
"differentiable"
proaches though not necessarily stratified
one
can
already
find various ap-
"smooth" functional structures
on singular, Probably SIKORSKI [159, 160] was the first who worked in this direction and introduced the notion of a differential space. Mainly for the purpose to study singular complex spaces from a differential viewpoint SPALLEK developed in [163] his concept of differenzierbare Rdume. Finally there are -
or
spaces.
the subcartesian spaces which go back to the work of ARONSZAJN
[3]
and which have
Stratified Spaces and F znctional Structures
28
been used
him to consider
by
information
on
in
analytical questions
subcartesian spaces
see
aspect of these approaches and the
one
[4]
a
For further
singular setting.
MARSHALL'S paper
or
introduced here is that
[120].
they
The
common
all embed
a
sin-
Euclidean space. The differences become apparent in the gular space additional conditions imposed on these embeddings or on the transition maps. in
locally
some
In the context of orbit spaces
SJAMAAR-LERMAN
[162]
structure for stratified spaces
in this work.
By
(see
for
example SCHWARZ [156], BIERSTONE [14],
and HUEBSCHMANN as
well,
but it is
smooth structure the
a
of smooth functions
a
[931)
one can
weaker
one
find
a
notion of
just named authors understand
nevertheless, the algebras of smooth functions constructed 162, 93] always give rise to singular atlases in the sense as defined above. we
smooth
an
algebra
stratified space such that the restrictions to the strata
on a
smooth. But
of this fact
a
than the notion introduced
in
are
[156, 14,
For
a
proof
refer the reader to Section 4.4.
1.3.5 Remark In the definition of
R' we have required a singular chart x : U -locally closed subset of Rn. This property is indispensable when we later want to apply the rich theory of Whitney functions to the study of stratified spaces. But for many applications, in particular for the definition of smooth functions and notions connected with that the local closedness is not absolutely necessary. To be able to allow a greater generality when needed we therefore will speak of a weak singular chart and correspondingly of a weak smooth structure, if all axioms besides
x(U)
that
the
one
is
a
of local closedness in Rn With the
1.3.6
we can now
help of a
are
satisfied. on X represented by the maximal atlas U, the so-called sheaf of smooth functions. C', X
smooth structure
construct the structure sheaf
Let U c X be open. Then one defines COO(U) as the set of all continuous functions X g : U -4 R such that for all x E U and all singular charts x : ft -4 Rn from U with x
ft
E
there exists
an
open set
U,
c U n
ft
and
a
smooth function g
:
R1
-4
R with
are the sectional spaces of a xlu.,. One now checks easily that the C' X (U) sheaf C'. In case no confusion is possible we will often denote 12' C'. Moreover, X X by
g lux
g
-
immediately by definition that for canonically isomorphic to smooth functions vanishing on x(U).
it follows
algebra of all
C' (U) is
every
singular
chart
x:
U
--)
0 C Rn the
C' (0) /J, where J is the ideal c COO (0)
On
a stratified space X with a C'-structure one can define analogously for every N, k < m the sheaf CkX of k-times differentiable functions on X by pullback of the Cx. In most cases corresponding sheaves on the Rn. Obviously we then have C'X we will restrict ourselves to consider only smooth structures. Now let us come back to stratified spaces with a smooth structure. Every stalk C?' of the structure sheaf with footpoint x E X has a unique maximal ideal M, namely the ideal of functions vanishing at x. In other words nix is the set of all germs [glx E COO with g (x) 0. Thus the pair (X, C') becomes a locally ringed space which we will also call though formally not quite correct a stratified space with a smooth structure.
k E
=
X
X
=
-
1.3.7
-
Let
(X, C') X
continuous map f
:
and
X
--
(Y, CI) Y
be two stratified spaces with smooth structure.
Y is called
if f,:Cl C C' X Y
smooth,
with V C Y open the relation g g E COO Y (V)
-
f (=- C' X
or
(f-'(V))
A
in other words if for all
holds.
Analogously,
one
29
1.3 Smooth Structures
Cm. Note that a smooth map between calls the map f of class C', m E N, if f,,(!' X C Y a'stratified map between such that and stratified need be not stratified spaces a map, spaces need not be smooth.
composition of smooth
definition the
By
maps is
Therefore the
smooth.
again
stratified spaces with smooth structures with the smooth maps
as
morphisms
form
a
category (Esprw.
Proposition A
1. 3.8
(X, ff) X U
x :
and
-4
map f
0 C Rn around C U
neighborhood U,,
:
X
Y between stratified spaces with smooth structure only if for every x E X and singular charts
--
is smooth if and
(Y, (!') Y
(5
and y
x
f (U.,
with
C
ft
R' around f (x) there exists
C
and
smooth
a
mapping f
:
Rn
an
open
R1 such
-4
that Y
-
flu.
f
=
-
XIu.-
problem is a local one, we can suppose without loss of generality (5 are locally closed subsets. By y', y' we denote the coordinate functions of RN. If now f : X -4 Y is smooth, we can find functions N such that fi IX y' f Now, choose a smooth function fi E C' (0), i 1, 1 on an open neighborhood 0,,'c 0 of y : Rn -4 [0, 11 with supp y C 0 and ylo.,, As the
PROOF:
that X C 0 and Y C
=
-
-
-
Then f
:
Rn
--)
-
-
.
RN with y (x)
f(X)
(f, (x),
-
-
-
,
for
fN (x))
E
0,
else, properties. As the
smooth map with the desired
a
x
=
0 is
-
,
,
x.
-
inverse
implication
is
obvious, 11
the claim follows. The
proposition just
proven allows
mersions to smooth maps f
More
by
precisely,
domains of
fj : Oj
-4
we
call f
singular
a
X
:
-4
us
to transfer the notions of immersions and sub-
M between
Uj A Oj
charts
M such that for all
are
case
that all
submersive,
1.3.9
Next
we
is finite
indeed, homomorphism
we
fj
can
call f
a
stratified space and
C Rni
and
a
a
a
manifold M.
covering (Uj)jEj of
family (fj)jEJ of
=
fj
-
point
X
E
X its rank rk
x
=
-4
dim(m,,/m.x,).
because every chart x : U -- Rn of X around x induces x* : mx( x) -4 mx between the maximal ideals of the stalks dim
X
immersions
Xj.
be chosen submersive and all restrictions f Is smooth stratified submersion from X to M.
associate to every
C', which implies
X
i
flui In the
a
smooth immersion, if there is
(mx/m.1, )
: dim
(mx(.)/Mx,(X))
=
a
E
8
The rank
surjective
CR' n',x(_)
The rank has the
n.
M, S
and
following
interpretation. 1.3.10
Proposition
structure there exists
PROOF:
already
Let
x :
U
know that rk
For every point x of a stratified space (X, chart around x of the form X: U ___) Wk X.
C')
with smooth
a
-4 x
Rn be
Rn
x.
proposition and the last corollary n
such that
a
one
interpret the rank of
can
neighborhood
of
x can
x
as
the
be embedded into Rn
chart.
singular
The sheaf of smooth functions
on
Rn is fine. Via
singular
stratified space with smooth structure theorem.
over
following
from the
pair of singular charts
E X there exists
H
=
11
two corollaries follow
Corollary
around
be carried the
of functions
basis of
a
rkxand
1,...
smallest natural number via
form
this contradicts the
Corollary
the
01
=
erat demonstrandum.
such that y
By
[H,,Io
has to be
< n
x
for every
1.3.12
Rrk
glx(u)no
Hn)
for i
following
means
[Hilo,
yrk x)
(Y1,
The smooth function h
go + mo. Now, by dim (mo/mo)
mo n
Hi-xlu. =y' =
(!-(0) 1
E
that the germs 0 the map H
Y
real numbers.
are
fg
=
to
a
1.3.13 Theorem The structure sheaf C!' of X
a
as
charts this property can is shown in the proof of
stratified space X with e,'-structure
is fine.
PROOF:
compact, is
a
fine
First note that every stratified space with a C'-structure must be locally every locally closed subset of Rn is locally compact. To prove that C'
as
X
sheaf,
finite open
it suffices
covering U
=
the paracompactness of X to construct for every (Uj)jEJ of X a subordinate partition of unity ( Oi)iEi
by
locally by C'of generality
functions (pj : X -- R. After refinement of U we can assume without loss that every Uj is the domain of a singular chart xj : Uj --4 Rni. As X is normal
topological
space, there exists
an
open
covering (Vj)jEJ of X
with
Vj- cc Uj
as a
for every
31
1.3 Smooth Structures
j
C- J. Now choose
and for every
x
for every j E J an open subset Oj E R'j with xi (uj) n oi xi (vi) an index J,, with x E Vj, Next choose for every x a relatively =
E X
neighborhood Wx" CC Vj.. By paracompactness of X we can then locally finite open coverings (Wx)-,Ex and (W,')XEX subordinate to (W.")XEX such that Wx C W,' C W,', C Wx". The W, have to be compact, hence there exist 1 and smooth functions px : Oj.. [0, 1] with compact support such that Oxlxjx(w') compact
open
find two
=
supp qx n
xjx (vij
C
xjx (W.,).
Now let
oj
set
us
E
:=
0-
-
Xj",
fxEXlj.=jl
where q,
o
xjx is set to 0 outside Vj..,. Then supp ( Pj)jEJ with
p-j
C
7j C Uj
(supp ioj)jEji
and
is
a
covering of X. Hence
oj (Y-)
=
E
j (Y-) jN
X,
X E
,
jEJ
comprises
partition of unity by C'-functions subordinate
a
If the situation
1.3.14
that X possesses
occurs
a
global
to
(Uj)jEJ-
chart
X
x :
--4
R1 of class
will say that X is Euclidean embeddable. In most applications the regarded el, stratified space will be Euclidean embeddable. In the following we will provide criteria we
which guarantee the existence of a global singular chart. Hereby it will turn out useful to have a new name for injective smooth maps f : X --+ M between a stratified space X with smooth structure and
a
manifold M such that f is proper and such that the
pullback f* is
surjective.
1.3.15
:
Proposition Every
proper
e,(X)
e'(M)
We will call such maps proper
embeddings.
embedding
f
:
X
R' is
--4
a
global singular
chart
for X. PROOF:
As f is continuous, proper and is closed in R.
and the
injective, f is a homeomorphism onto its only remains to show that f is
Therefore it
image image, compatible with all singular charts of an atlas of X. Let x : singular chart around x E X. By smoothness of f there exists 0
a
smooth function f
:
0
-1
R' such that f
and 3F:
-
x
=
As
flu.
U
--4
after
0 C Rrk X be
shrinking
fx* : mf(x) /m',, f( )
m,/m,2,
--4
a
U and
mx/mx2
isomorphism, the is injective. Hence, after shrinking 0 further derivative Dx(-,)f of f at f is an immersion, and f (0) a submanifold of R". After shrinking 0 a last time one and (5 C W can find a diffeomorphism H : 0 x V --4 (5 C R, where V C W-rk is
f, the point x(x)
surjective by assumption
on
m),( x)/mx2(.X)
--4
an
'
are
open
subsets,
such that
Hloxfo}
=
f. Therefore f is
compatible
El
claim follows.
1.3.16
Uj
-4
Proposition Assume E N, of class e,'
Rvi, j
with x, hence the
that X has such that
Nj
a
countable atlas of
< N for
a
singular charts xj number N E N and all j E N.
32
Stratified
Then X
be embedded into R?N+' via
can
compact stratified space with We divide the
PROOF:
a
a
proper
Spaces
singular
and Functional Structures chart. In
particular,
every
smooth structure is Euclidean embeddable.
proof in
two
steps, and will prove the claim in the first step
for the case, where X is compact. Then we will extend this result in the second step to the general case. In the course of the argumentation the reader will notice that the is very close to the
proof
1. STEP:
Uj A Oj
domains
one
of the WHITNEY
By compactness of X there C
RNi. Let
(Vj)jA'
exists 1
be
Embedding Theorem. finite covering (Uj), =,
a
an
open
of X
covering subordinate
to
by chart
(Uj)jk
Then first there exist smooth functions yj : X -- R with support in Uj and identical to 1 over Secondly fix maps yj E C- (X, Rni ) by yj (x) yj (x) xj (x) for x E Uj and 0 for x Uj. Using these functions we can now define a map x: X -- Rn with
7j.
=
=
n
=
N, +
-
-
-
+
Nk + k by the following: X
=
(YI,
-
-
-
)
(Pk)
Yk) Y1)
Obviously x is smooth and injective. Moreover, the map x is proper, hence a homeomorphism. onto its image. As for every j the restriction of yj to Vj comprises a singular -4 ff is surjective for all x E X. By the proof of Proposition 1.3.15 chart, 4 : C' X(X) then x is a global singular chart of X. By Corollary 1.3-12 and the compactness of X there exist submanifolds U MI. Let us now M, c Rn of dimension < N such that X C M, U M1, suppose there exists a vector V E Rn which for every pair y, z E x(X), y :A z is not parallel to -y z and which is not tangent to any of the submanifolds Mj. The composition R of x with the projection from Rn to the hyperplane v' _ Rn-1 then is iniective and induces surjective morphisms 4 : E!R'(, -- ex`7 x E X, hence is a global singular X
-
-
-
...
,
-
chart of X. If
we can now
will obtain after
a
prove that for
finite recursion
n
>
2N + I there exists such
a
vector v,
we
chart of X with values in R2N+1
global singular
a
Consider the maps aj : TMj \ Mj -4 Rpn-1, j 1 which result from 1, assigning to every nonvanishing tangent vector of Mj its equivalence class in the projective space Rpn-1. Denote the diagonal in Rn x Rn by the symbol A and consider the maps -rij : (M, X Mj) \ A __ pj?n-1, i, i 1, 1, which assign to every pair with line the Both the well z z. as as the -rij are smooth and through y (y, z) ij 7 aj defined on manifolds of dimension < 2N. As long as 2N < n I holds, the images of the uj and rij are of first category in Rpn-1 by the theorem of SARD, and so is their union. Consequently its complement is nonempty, hence there is a vector V E Rn with the desired properties. This proves the claim for compact X. 2. STEP: Now we drop the assumption that X is compact. Without loss of generality we can assume that all chart domains Uj have compact closure. Then we choose a locally finite smooth partition of unity ( pj)jErq subordinate to (4i)iEr4. =
-
-
-
,
=
-
-
-
,
-
-
and set
A(x)
A-'
=
EjEq
j pj (x).
Then A
:
X
-4
R is smooth and proper.
Let
Vj (Ij 1/4, j + 5/4 [) Kj A-'( U 1/3, j + 4/31 ). Then Vj is open, Kj is compact and Uj- C Kj*. Moreover, all K2j are pairwise disjoint, just as the K2j+,. According to the 1. Step we can choose smooth functions gj : X --> R2N+1 =
and
-
with bounded
=
image such that the
such that supp gj C
Kj.
We
now
-
Vj- is a singular chart and EiEN 92i) XO := EjEN g2j+1 and set
restriction of gj to
define x.
:=
33
1.3 Smooth Structures
WN+1
R2N+1
smooth, proper and injective. By by definition of x every one of the induced maps 4 : Cx'(-,) --) Cx'(xp X E X has to be surjective, hence by the proof of Proposition 1.3.15 x is a singular chart of X. Analogously to the arguments in the 1. Step one can now find by the theorem of SARD a (21Y + l)-dimensional hyperplane H C R4N+3 such that the composition of x with the orthogonal projection 7rH onto H is a again a proper singular chart of X. To guarantee the properness of 7rH x one has 4N+3 does to choose H such that subspace R4N+3 generated by the last coordinate of R not lie in the kernel of 7rH. But this is possible indeed, as R4N+3 is of first category in R4N+3 So finally we obtain a global singular and proper chart of X with values in a
X
(Xe) Xo) 1 )
:=
:
the fact that the gj
X
Then
R.
x
singular charts
are
is
x
Vj-
on
and
-
.
(2N
l)-dimensional
+
For the
case
E3
vector space.
that X is not Euclidean embeddable
has
one
special atlases
at ones
disposal which in many cases achieve almost the same like global charts. But before we can explain this in more detail let us briefly recall the notion of a compact exhaustion of X this is a family (Kj)jEN of compact subsets of X such that Kj c Kj'+, and UjEN Kj X. Such a compact exhaustion of X exists, as X is locally compact Hausdorff with countable topology. If one now chooses a compact exhaustion (Kj)jEN of X, a singular atlas (Xj)jEN of X consisting of charts of the form xj : Kj'+, --i Oj C Rni is called inductively embedding with respect to (Kj)jEN) if nj+l ! nj for all j, and if there are relatively compact open neighborhoods Uj CC Kj'+, of Kj such that =
Xj+1 N
=
Lnj+l .xj (x) for all nj
X
E
Uj
-
compact exhaustion there is
1.3.17 Lemma For every
an
inductively embedding
atlas. As all
PROOF:
singular charts yj neighborhoods of
:
Kj
are
Kj+2
Kj_1
Kj_J
C
-4
in
compact, there exists by Proposition 1.3.16 Rli. Let
Kj'+,
Uj_J
now
Kj', ojlw.
1, supp%
3
XO
xj
=
:
If for
YO.
Kj0+1
--)
Rni
Xj N
some
:=
Rmi
c
Uj
be
an
atlas of
relatively compact
open
with
CC
Vj
CC
Then there exist smooth functions oj =
Vj, Wj
and
:
X
Kj'+, \ Uj_1
Wj --
CC
[0, 11
and
Kj'
CC
and
%,X V-
Uj :
1.
X
Kj'+11
CC -
[0, 11
with supp Yj CC : K' -) R'0 by I
Define xo
i
j are already determined, then fix R2 recursively by
index j all xi with i x
Rni-1
x
(yj (x) xj_1 (x), % (x) yj (x), 1 (0, Yj (X), 1 Yj (X), % (X)),
-
yj (x),
% (x))
-
for
x
E
for
X
E
Kj, Kj'+, \ Kj.
nj+1 by definition of xj the relation Xi+11Ui j .X.3 Juj holds. Moreover, one checks easily that xj is injective, a homeomorphism onto its image, and compatible with the singular charts xj_1 and yj. By induction one thus obtains a singular atlas with the El desired properties.
Then
1.3.18 corner
=
Example Manifolds-with-corners possess a smooth structure induced by their Moreover, manifolds-with-corners are Euclidean embeddable by defini-
data.
tion. The smooth functions with
smooth functions
on
respect
to this smooth structure coincide with the
manifolds-with-corners in the usual
sense.
34
Stratified
Spaces
1.3.19 Example Examples 1.1.12 to 1.1.17 inherit locally closed subspaces of Euclidean space.
1.3.20
Example Every triangulation of
Note that the smooth structures defined
compatible
and F inctional Structures
canonical smooth structure
a
as
polyhedron provides a smooth structure. by two different triangulations need not be a
with each other.
R -- R is not a smooth function, if R carries Example The absolute value I ordinary smooth structure. But it is possible to interpret R as a stratified space with the decomposition JR into Ro R \ f 01, and then embed this space f0j and R, by x F--4 (x, jxj) into R2. The stratified space (R, 3Z) then inherits from W a smooth 1.3.21
the
=
structure with
(R, 3Z) 1.3.22
is
to which the absolute value is
respect
in
diffeomorphic
Example
=
The
canonical way with the
a
comb possesses
cone
a
smooth map.
a
Incidentally
edge XEdge-
natural smooth structure, but
as
an
infinite dimensional stratified space it is not Euclidean embeddable. Starting from the cone comb one can even construct an example for a stratified space with smooth
having only
structure
strata, hence being finite dimensional, but which is
two
Euclidean embeddable. Let
explain this in more detail. definition arises Xcm,, By by appropriately gluing the cones V:0. Now set U,, := CS'U In 3/4, n + 3/4[, and note that
not
us
CS' to the half line
-
CSn
can
be
regarded
:
Xn
Un
stratified
as a
Rn+3
__
-
R
X
=
f (tjj' t)
G
Rn+21 1j
subspace of Rn+2
R7+2'
X
ifxc-ln-3/4,n+3/4[,
(n, x)
comprises
a
chart of
singular
Xcm, and
the
Sn
Then
.
(X'0)
-4
E
if
X
CSn
E
family (Xn)nEN
is
a
\ 100,
singular
atlas. On the
other hand the set CSn F where "F" stands for
:=
f (t1j' t)
frame,
E
W+1
is for every
x
n
[0, 1[ 1 y
E
fel,
canonically
a
-
*
I
en+11j)
stratified
subspace
of CSn of
dimension 1. Hence
XFCmb:= becomes
from
a
XCmb
stratified a
fY. E XCmbl x E R
or x
E
CSn for F
an n
E
NJ
subspace Of XCb of dimension 1, has only two strata, and inherits Obviously, XFCmb together with this smooth structure
smooth structure.
is not Euclidean embeddable.
1.4
Local
Triviality
and the
Whitney conditions
Several of the decomposed spaces introduced in Section 1.1 have properties which seem unnatural, like for example the space Y from 1. 1. 12, which satisfies dim S 1 > dim S2 7
1.4 Local
and the
Triviality
conditions
Whitney
35
boundary piece Of S2- Such and other "pathological" stratified spaces consideration, so in the course of the formation of stratification theory people have tried to find criteria which exclude such unwanted spaces. Usually the conditions on stratifications appearing in the mathematical literais
though S,
a
should not be admitted for further
ture
impose further restrictions
to the behavior of
a
stratum
near a
boundary
stratum.
remaining stratified spaces have nice properties which admit further topological, geometric or analytic considerations. First in this section we will introduce topological local triviality of a stratified space. As already explained in the introduction this condition says that a locally trivial space is locally around each of its points isomorphic to a trivial fiber bundle over the stratum of the point. Often it is supposed additionally that the typical fiber is This should guarantee that the
given by the
cone over a
we
spaces
In
certain
a
as
not uniform in
study examples of stratified typical fibers "cone spaces".
will treat such spaces in detail.
we
sense one can
regard topological
local
reasonable stratified space. Therefore
a
to
cones are
In Section 3.10
ment to
compact stratified space, but the literature is
important nontrivial and well have named locally trivial spaces with cones
this point. As
in the definition of
(cf.
stratified space
a
e.g
triviality
some
authors
as a minimal requirerequire local triviality
[64, 162]).
given (locally) as subspaces of manifolds we will afterwards introduce in 1.4.3 the famous Whitney conditions (A) and (B). These conditions essentially impose restrictions on the behavior of the limit tangent spaces of a higher stratum when approaching a boundary stratum. The Whitney conditions have far reaching implications, in particular condition (B) guarantees that the considered stratified space is locally trivial (see Corollary 3.9.3). The corresponding proofs are quite involved and have led to the control theory of J. MATHER which will be explained in For
a
Chapter
stratified space
3.
theory many more conditions have been imposed on "good" stratified spaces. goal is to formulate criteria, which are as easy as possible to prove and which entail essential but more difficult properties like locally triviality, a particular metric structure or even geometric features. At the end of this section we will introduce some of these further criteria and explain their meaning. In the evolution of stratification
The
1.4.1
triviality A stratified space X is called topologically locally there exists a neighborhood U, a stratified space F with every 81% a distinguished point o G F and an isomorphism of stratified spaces local
Topological
trivial, if for stratification
x
E X
h: U such that h-1 (Aj,
{oj. Hereby, over x.
o)
=
-4
cases
F is
given by
a
is
locally
and
x
F
S.
S. Sometimes
cone. F
=
CL
we
over a
is the germ of the set call F the typical fiber
compact stratified
space
one
so
on,
we
obtain
a
class of stratified spaces
spaces of class (!'. For a precise definition of the notion of explanations we refer the reader to Section 3.10.
cone
some
G
x
for the links of the points of the link and
called
U)
says that X is locally trivial with cones as trivial with cones as typical fibers and if that holds again
L. Then L is called the link of x, and
typical fibers. If L
n
ij for all -y E S n U and such that
S is the stratum of X with
In many
(S
a cone
space
36
Stratified Spaces and Functional Structures
1.4.2
It is
Example
relatively easy to prove that manifolds-with-boundary or are locally trivial, in particular so are simplices and polyhedra. It is much more difficult to see that (real or complex) algebraic varieties possess topologically locally trivial stratifications, more generally even all semialgebraic, semianalytic and subanalytic sets. This follows from the fact that all these spaces have an essentially unique Whitney stratification (see Example 1.4.10 for references) and that Whitney stratifications are locally trivial according to THOM [169] and MATHER [122]. In the course of this monograph we will show explicitly local triviality for Whitney stratifications in Corollary 3.9.3 and for orbit spaces in 4.4.6. manifolds-with-corners
Local
locally other
triviality alone
trivial
well. In
as
does not case
automatically imply that
all the different trivializations
the fibers
are
or
links
are
compatible with each
show that local
triviality does also hold for the fibers and links. The right that, an appropriate definition of compatibility and the corresponding implications is given by the control theory of MATHER (see Chap. 3). one can
axiomatics for
Whitney conditions
The
1.4.3 as
well
submanifolds R and S.
as
(A)
condition
at
x
R,
E
or
that
In the
following
will consider
we
On says that the pair
(R, S)
(A)-regular
is
(R, S)
a
manifold M
fulfills the
at x, if the
following
Whitney axiom is
satisfied:
(A)
0Jk)kE1,j be a sequence of points'Yk
Let
E
S
converging to x such
that the sequence
of tangent spaces T,,,S converges in the Gra6mannian of dim S-dimensional subspaces of TM to -r C T.,M. Then T,,R C -r. If x: U is
Rn is
--)
fulfilled,
one
to the chart
(B)
a
(R, S)
satisfies the
and
fulfilling
the
(131)
0 ijk
Xk
following
be two sequences of three conditions:
and lim xk
The sequence of space to
(133)
condition
following
(B)
at
axiom
(B)
x
with respect
lim ijk
-*=
(T-,x)-'(f)
Now the
xk E R n
U,
Yk E S n U
x.
k-400
connecting lines X(Xk) X(IJ-k)
The sequence of tangent spaces
Then
points
C W converges in
projective
line f.
a
subspace
nate
E R such that the
(lgk)kEq
k- oo
(132)
X
Whitney
x.
(Xk)kEq
Let
smooth chart of M around
says that
r C
T,,,S
converges in the Gra6mannian to
a
TxM.
C r.
question arises how Whitney's condition (B) transforms under
a
coordi-
change.
1.4.4 Lemma If
chart
x :
U
satisfied
as
--
(R, S)
satisfies the
R, and if y
:
U
--i
well with respect to y.
Whitney condition (B)
R' is
a
at
x
with respect to the
further chart of M around x, then
(B)
is
1.4 Local
37
Whitney conditions
and the
Triviality
and (IJk)kEN be two sequences of points Xk E R n U, ijk E S n U such that the sequence of secants 4 Y(Xk) YNJ converges to the line f and such that the condition (B3) is satisfied. Let further H : 0 -- RI be an open PROOF:
Let
converging
to
(Xk)kCN
:--
x
embedding such that (after possibly shrinking U) compactly contained in 0 and such that H y =
-
we can
the
suppose that the sequence of unit vector S Vk
vector v,,, E f. For -k E NU tool
where x,,,,
now
define
curves
Next consider the transformed
:= x.
Y(.Yk)-Y(xk)
-.
:1
Yk
-
ilk
formula
we
triangle inequality and Taylor's applying the t EI 1[ following estimate: 1,
is
subsequences
to a unit IIY(,Jk)-Y(Xk)ll converges 1, 1 [-4 R' by t " Y (Xk) + tVk)
*
curves
the
x(U)
hull K of the set
convex
After transition to
x.
H
=
-
Yk
:]
-
1, 1[-4 Rn. By
obtain for all k E N and all
-
11 k (t)
-
X(Xk)
-
tX(tJk)
-
IIY('-Jk) Ilk(t) X(Xk)
-
-
-
X(Xk) < Y(Xk) 11 tDH(Y(Xk))-Vkll
1
+
t
+
-
IIY(IJk) 1
2
Y(Xk) 11
-
11 X(IJk)
X(Xk)
-
-
DH(Y(Xk))-(Y(IJk)
-
Y(Xk))
t(t+ IIY(!Jk) -Y(Xk)ll) sup JJDZ2HII.
Note that C
zEK
:'__
JID2HII
SUPzEK
nonvanishing
and that C is
< oo
z
independent of
After
k and t.
x(vk)-(Y-k) subsequences suppose that IIY(,Jk)-Y(Xk)ll converges to vector w,,,,. The estimate (1.4.1) then entails
transition to further
we can
w,,
By hypothesis
the
on
(,(O)
=
=
Ty(,x)H(- ,,,)(0))
chart
singular
x
the relation
=
a
Ty(x)H(v,,.).
(T,,x)-'(W,,,))
E T is
true,
so
alto-
gether
(Txy)_1(Voo)
=
Jxx)_'(Wc )
E Ir El
follows. This proves the claim. proven the
By the lemma just
Whitney condition (B) is independent of by the sentence "(R, S) satisfies equivalently by "(R, S) is (B)-regular at x".
validity
of the
the chosen chart. Hence it is clear what to understand the
Whitney condition (B)
1.4.5 Lemma If the
(A)
condition
at
or
pair (R, S) is (B)-regular at x
E
R, then (R, S) satisfies Whitney's
x.
As the claim is
PROOF:
at x"
a
local one,
we can
suppose that R and S
are
submanifolds
of Euclidean space R. be
(IJk)k,N
Let
a
sequence of
tangent and Wk
v
x
Vk-X .
tk
to
=
w
such that the sequence
T.,R be a nonvanishing T,,S with smooth v. Let tk t a 11 E IIIJk XII (O) [-1, path -y(t), Then (tk)kEN converges to 0 and, after a transition to a subsequence, v
E
`=
=
-
Y(tk)-% On the other hand the sequence of the vectors Vk :7-tk Vk Wk assumption on -y. Hence the sequence (Zk) kEN with zk
some w
E R.
converges to v by 'Jk-'Y(tl) converges to t"
does
to
Let further
vector and =--
(Wk)kEN
points of S converging
converges to r c R.
of tangent spaces
-
z.
-
Z
E RI.
By Whitney (B) the
This proves the claim.
vectors
w
and
z
lie in -r, hence
so
38
Stratified Spaces and Minctional Structures Let
further notation. If the condition
(A) resp. (B) is satisfied at (R, S) satisfies the Whitney condition (A) resp. (B), or that S is (A) resp. (B)-regular over R. A stratified space with smooth structure such that for every pair (R, S) of strata Whitney's condition (A) holds is called a Whitney (A) space or an (A)-stratified space. As a Whitney space or a (B)every
us
agree
point
x
E
on some
R,
will say that the pair
we
stratified space we will denote a stratified space with smooth pair (R, S) of strata Whitney's condition (B) holds.
structure such that for
every
1.4.6 Remark If
conditions the
(A)
m E
(B).
and
N",
Whitney condition (B)
Nevertheless the condition TROTMAN
[172] (see
one can
But for the
3.4.2
under
chart transition
a
(B) a as well). is
formulate for (!" -manifolds M, R, S the Whitney proof given in Lemma 1.4.4 of the invariance of has to
one
assume
C'-invariant. A proof of this fact
that
can
m >
2.
be found in
1.4.7 Example One can construct a stratified space which is not (A)-regular by starting from Whitney's umbrella XWUznb- Intuitively we fold down one leaf of the umbrella and obtain in mathematically more precise terms the following topological space:
X
=
f (X, ig, Z)
E
W I X2 =,y2 IZI & sgn(x)
As stratification of X choose the X \So. In the ,r
origin,
the
of the tangent spaces
generated by So
one
pair (SO, Sj)
with Xk
T-,,Sl
=
(A)-regular. (0, Ilk, 0). Then
is not =
sgn(-yz) 1.
=
1(0, 0, z) I
To
see r
is
Z
E
RI
and
S,
=
consider the limit
this, given by the xy-plane,
but the
z-axis, which is the tangent space of So in the origin, is not contained in the x-y-plane. In general it is rather difficult to find examples of not (A)-regular stratifications in particular of not (A)-regular stratified varieties. A source of such examples is given by the Trotman varieties [173, 10].
Example The fast spiral X,,i,, of example 1.1.13 is a Whitney stratified space, spiral X,Pj, on the other hand not. Let us show this in some more detail. The top stratum of the fast spiral can be parametrized by -y(O) e-o'(sin 0, COS 0), 0 E R", the one of the slow spiral by -q (0) e-'(sin 0, COS 0). This gives 1.4.8
the slow
=
=
- (O) Besides that the secant
$(0)
11 (0)11
is
a
=
unit
e-02 ((COS0 ,- sin 0) tangent
connecting the origin
(0) For
Ok
=
z! 4
+ 27rk this
sequence of the
y(O).
(11,K01
lim 0--
the space Xr,,i,, satisfies the and calculate:
vector with
and
+
-
20 (sin
footpoint -y(O),
e-' ((COS 0,
spans
(sin 0, COS 0)
sin
0)
-
Let
consider the slow
spiral
0,
us now
(sin 0, COS 0)).
1(0') = (0, -1) and 71 (00 = 12 e-)k 11 0011 converges to the origin, the sequence of
implies
points 11(0k)
-
(sin 0, COS 0)
and
As
Whitney condition (B).
=
0, COS 0)).
(V2-, V2-).
Now the
tangent
spaces
1.4 Local
niviality
and the
Whitney conditions
39
subspace spanned by (0, 1), and finally the sequence of secants 71 (0 ) 0 generated by (1, 1). Hence X,,,i, cannot satisfy Whitney (B). Moreover this argument shows as well that no finer decomposition of the slow spiral exists which makes XsPi, into a Whitney stratified space. converges to the
converges to the line
Example Consider the two decompositions of the Whitney cusp Xw,,.,, given in Example 1. 1. 17. One can prove easily that the stratification induced by the decomposition into Ro 0, z 0 0} and R2 fO}, R, f (x, -y, z) E RI I x 0, ij Xwc, \ (Ro U Ri) is a Whitney stratification. One the other hand the decomposition of Xw, , into the z-axis S, and its complement S2 fulfills Whitney (A), but not Whitney (B). Let us ex(Ilk 2, 0, Ilk) E Xwc, plain this in more detail. Consider the sequence of points Wk converging to the origin. Now, if (x, -y, z) is an element of Xwc,,,, then the point (x, -1j, z) is one as well. Hence the tangent space Of S2 with footpoint Wk is spanned 1.4.9
=
=
=
=
=
=
by
the vectors
(0, 1, 0)
and
(2/k, 0, 1).
Thus for k
--
oo
the sequence of tangent
hyperplane -r spanned by the vectors (0, 1, 0) and (0, 0, 1). But the connecting secants WkW'k with w'k (0, 0, Ilk) E Xwcsp converge to the line f spanned by (1, 0, 0). As obviously f does not lie in r, Whitney (B) does not hold for the decomposition (S1, S2)spaces converges to the
=
Example Since the emergence of stratification theory one could show for more general classes of spaces that they possess Whitney stratifications. The beginnings of this go back to WHITNEY [191], who showed first that every real or complex analytic variety has a Whitney stratification. LOJASIEWICZ succeeded in [115] to prove that every semianalytic subset of a real analytic manifold possesses a Whitney stratification by analytic manifolds, and that the strata are strong analytic, i.e. they comprise analytic manifolds which are semianalytic. For subanalytic sets HARDT [77, 78] and HIRONAKA [86] could show that they are Whitney stratifiable. But it should not remain unmentioned that the first ideas for a proof of this fact goes back to THOM. In his work [169] THOM had already worked out some of the fundamental properties of subanalytic sets to which he gave the name PSA for Projection d'ensemble Semi-Analytique. In the book by SHIOTA [1581 one can find a detailed and modern account of the theory of semialgebraic and subanalytic sets. 1.4.10
more
and
1.4.11
Thom's Condition
(T)
One of the first
regularity
conditions
imposed
on
stratified space has been introduced in 1964 by THOM [168]. In the mathematical literature THOM's condition is often called condition (T). Using our notation we call a a
pair (R, S) of disjoint submanifolds (T)-regular, if every smooth function transversal to R is also transversal to S in a neighborhood of R.
g
:
R'
-4
M
The condition (T) is a relatively weak requirement to a stratified space. WHITNEY proved in 1964 in his article [1921 that his condition (A) implies THOM's transversality condition (T). The significance of (T) lies mainly in the stability theory of differentiable mappings.
1.4.12
regular
Verdier's condition at x0 E
R,
,
(W)
if there exists
a
A
pair (R, S) of submanifolds of R'
neighborhood
U of x0 and
a
is called
(W)-
constant C > 0 such
40
Stratified
that for all
X
E
Spaces
and F znctional Structures
R n U and all -Lj cz S n U
dG,(T-,R, TjS)
0,
such that for all x, -y E K -
6(x,ij)
:! ,
Cd(x,-Lj)'I'.
locally closed connected and locally finitely path connected set A C R1 is called I-regular, if each of its points has a compact I-regular neighborhood. If for every point z of A there is an I E R -' depending on z and an I-regular compact neighborhood K C Rn we will say that A is Whitney- Tougeron regular or briefly that A is regular. Finally one calls a connected stratified set (X, C') with smooth structure WhitneyTougeron regular or regular, (resp. I-regular), if there exists a covering of X by singular charts x : U -4 0 C Rn such that x(U) is a regular (resp. I-regular) set in Euclidean A
space.
By a simple calculation one shows that t-regularity is invariant under diffeomorphisms between open subsets of Rn. This implies for the stratified case that I-regularity of x(U) entails I-regularity of y(U) for any further singular chart compatible with
x.
1.6.7
Example Every subanalytic
proof
of this fact
see
set X C Rn is
KURDYKA-ORRO
[107,
Cor.
Whitney-Tougeron regular.
2].
For
a
46
Stratified
1.6.8
Spaces
and Functional Structures
Proposition Every (6t)-stratified space (X, C') is locally finitely path conI-regular in the sense of Tougeron. In particular, every Whitney stratified is 1-regular.
nected and space
For the
case
I
1 this result has been
=
also find further
can
(but
methods, how
which need not be
PROOF: R'. Let
space)
is
a
[11]. There, one composed by manifolds
space
finitely path
connected.
Without loss of E X be
x
generality we can assume that X is a connected subset of point, S the stratum of x and U an open ball around X such that U
a
only finitely
meets
stratified
a
in BEKKA-TROTMAN
proved
to check whether
many strata of X and such that for every stratum R with R n U
the relation R >
following
After
holds.
relation holds for all
z
U and
shrinking
choosing 6 (x \ s) n u
s n U and ij E
c:
> 0
appropriately
:A 0 the
611Z_.UI12-1/1
11PItJ(Z_,U)JJ
Hereby Pj is the orthogonal projection onto the tangent space T.,R of the stratum of Y. we supply S with the Riemannian metric induced by the Euclidean scalar product and consider the corresponding exponential function exp. After further shrinking U one can achieve that U n S exp B, where B, is the ball around the origin with radius 'r < I and exp, shall be injective on a neighborhood of T,. For every -U E S n U define the path -y., : [0, 11 -- S by yy (t) exp_-1 (1g) and t E [0, 11 exp (tw), where w Next
=
=
=
,,,
Then there exists C,
> 0 such that
I-Y'l where the second
inequality
smooth
curve
^?Ij (t) and
t,+J
positive escape time of -y with respect [0, q [--4 R fulfilling
to
VR,
that
:
-y-, =
VR('ylj (t)),
t E
[0, tj
is maximal with this property. As
d
dt
d
I Vyy M
-
Y-1 I'
=
dt
V (-Y" (t)
6'.Y M V ('Y' M
-
-
X, Y" M
-
-X)
=
X, -Y-Y M
-
X,
-
/" M
X)
-
X)
11
P
I I P'Yt' (t) (X holds,
one
has for 0 < t
0 such that for all F E F-'(K)
and
[170, Prop. 2.6])
TOUGERON
compact. If the Euclidean and geodesic distance
Let
oc E
Nn, 1 ,1
O.
path
Then
:! C
appropriate C
max
{ d(x, Z), d(-y, Z) 1
1/1
> 0.
estimate I > 1. is satisfied
then
+ 5 (1j,
(RAI)'
over
V n A, then after
shrinking V
the axiom
x
is
and
satisfied,
d(ljj, z)
follows
immediately,
as
Z)
C, (d(x, Z) 1/1 + d(-y, Z)
conversely (RAI)' E N with -yo
is
of
or
an
open submanifold S
z
regular
on an
smooth restricted function
>
simple argument. A, Z C 0 are regularly situated, if and a neighborhood V as well as constants
Cd(x, A n Z)c'
fis
for all
X
E V.
on
(A)-stratified space X induces over every stratum S a (S) If on the other hand a C'-function g : S --i R
E C'
then it is rather easy to
given,
T and
spaces
C' (X)
E
subsets
\
theory for Whitney functions
Extension
A function f
a
E A n Z there exists
d(x, Z)
1.7
by
sets 0
0 such that
(RS')
is
submanifold S C 0
a
53
spaces
regularly situated, if the relatively closed and 9 C X are regularly situated.
T
1.6.19 Lemma Two
C'
regular
on
stratum of X is called
a
The
T of
neighborhood
functions
-
see
on
(Ui, Xj)jEJ
As X is
X.
on
the
a
normal
whether g
can
be extended to
a
continuous
exactly then the case, if for every X E aS and every convergent sequence xk --) x of points of S the limit liMk,,,,) 9 (Xk) exists and if this limit is independent of the special choice of the sequence (Xk)kEN. Now the much further reaching question arises, in particular in view of analytic applications, whether it is possible to give reasonable criteria to a smooth g : S -4 R which guarantee that g has a smooth extension to X. Though it seems to be impossible to find such a condition as easy as in the continuous case, the theory of jets and Whitney functions will give us good tools in our hands which can help in many situations. In particular we then will be able to find criteria, when a Whitney function G : S ---) R of C' falling fast enough at the boundary of S can be extended to a Whitney function F on X. But before we come to the details let us explain what to understand by a Whitney function "flat at the boundary". To simplify notation let us fix for the rest of this section a singular atlas U
function
given (A)-stratified
space X with smooth structure COO.
1.7.1 Definition Let m E N U fool, A C relatively closed. Then an m-jet F E J'(A) Z, if the following conditions are satisfied:
(FJ1) Flz (FJ2)
=
space this is
topological
R" be on
locally closed set, and Z flat of order C E R :O
a
A is called
C A over
0.
For every
point
z E
Z and all
CX E
N, I ocl
< m
F(") (x)
lim
the
following
relation holds:
-0 _
-
1z d(x, Z)c
xEA\Z
An a
m-jet F
E
Jm(g) U
on
the closure of
closed set Z C a S, if for every
singular
a
stratum S of X is called
chart xj the
m-j et
Fj
E
flat
of order
J m (xj (9 n
Uj))
c over
is flat
Stratified
54
Spaces and 11mctional Structures
xi (z n ui) of order c. Th space of m-jets on A Z will be denoted by O',c (Z; A) (resp. gm,c (Z; U
over
(resp. 9)
which
are
flat of order
c. over
Finally
we
set
J'(Z; A)
IF E E'(A) I F() 1Z
=
IF E
U
F()
U
is the
Now
jet
we
means
0 for all
=
1Z
Note that this definition coincides with the subset of R. For such A this
0 for all
==
locl
! C2
d(x, aS)C2
B rI (ii) denotes the ball in RI around
S
(0) x B3n,3e (.,,) (x) 1, f 11 ajf (t, 1J) 111 (t, IJ) E B3'5(x) 3 f (t, ij) (t, ij) E B 16(,, )(0) x B 3e (.) (X) I,
L(x)
:=
I + sup
M(x)
:=
sup
-r(x)
F, :
such that for all
n
:=
min
3
11, 5 (x),
constants C3 E N and
To
:=
C3
'() M(X) >
01
1.
r(x)
Then
C3 d(-x, aS)C3 holds with
>
appro-
SO
U Bl(.x) (0)
X
,r
n7r-1 (x)) (Bn)(x) E(X
%ES
is
F_
regularly by a
situated
F-:= where the
J(p
norm
neighborhood
of S in R
C(TO,R7)1 y(t,IJ-)
E
is
given by I I y I Ig
F,
-4
:=
Rn.
Finally define
the Banach space
(t,,U)
E
Tol,
I I I p (t, ij) I I e-2L(iI)JtJ I(t,'IJ)
E
T'J.
E Bn
sup
x
2e
(T,(,,))(7r(ij))
for
Then the
operator t
K
:
e, (TO,
Dr),
y
"
K y (t, ij)
=
f (s, y (s, -U)) ds
ij + 0
maps the Banach space E into
itself,
because
t
JlKy(t,ij) --Ljll
0. Now let il be
a
Riemannian metric
inducing 11, and let DK like in the inequality (2.4.3). Then there exists for every weakly piecewise e,1-curve -y : [0, 11 -- K c X from x to ij a unique smallest t > 0 with -y(t) E aV. Then the restricted path -yj[0,tj lies in V. By (2.4.3) the path yj[o,t] on
0
has
geodesic length
1-YI[0'tIIR
!
IX -no,t) I
>
5,(X"j)
!
-
DK
-IX(Y(t)) DK
-
X1
>
-
FE.
DK
Consequently I E >
DK
0,
hence all the axioms of metric have been proven for considerations that the
of X.
topology generated by 5 t
it follows
6.. Moreover,
is finer than the
by these original topology
Differential Geometric
74
Objects
Singular Spaces
on
assumption that X is regular entails that the topology generated by 5,, is original topology. To check this it suffices to assume that X is a regular stratified subspace of R' and that we have given a Riemannian metric 11 on Rn such that the pullback of 11 to X is equal to t. Let x E X and K a compact, finitely path connected t-regular neighborhood of x in X, and CK like in (2.4.2). Next choose a ball B, (x) with B, (x) n X c K and a finitely path connected neighborhood V C B, (-x) n X of x in X. Then there exists for 'y E V a rectifiable curve A in V from x to -y. As V C B, (x) n X and B, (x) n X is a Whitney (A) space, there exists by Lemma 1. 6. 10 a weakly piecewise C'-curve -y in B, (x) n X connecting % with 1J and satisfying J'yJ < JAI. Together with (2.4.2) and the definition 1.6.6 of I-regularity The
weaker than the
64(X 1J)
IT14
!
! CK ITI !5 CK IAI ! C CK
11'X
-
*Y
11111)
depends only on K and 71. Hence every neighborhood of x with respect neighborhood of x with respect to the topology induced by the Euclidean
where C CK
5.
to
is
a
metric. This proves the theorem.
2.4.8 Remark Under the
Riemannian metric all
lengths 1-yl,,,
from
x
assumption that X is
X the
on
where -y
runs
In other words
to -y.
El
a
smooth manifold and [L
a
smooth
geodesic given by 6,,(x, -U) through all piecewise continuously differentiable curves 51L coincides in this case with the geodesic distance as distance
the infimum of
is
defined in Riemannian geometry.
Corollary Every regular Riemannian Whitney (A) space (X, R) with geodesic distance 51L as metric comprises an inner metric space in the sense of following definition. 2.4.9
2.4.10 Definition A
finitely path pair x, 1J E
metric space, if for every
(X, d)
connected metric space X the distance
d(x,,y)
is called
coincides with the
an
the
the
inner
geodesic
distance 5 (x, ij). If a
(X, d)
length
PROOF
is
a
complete
and
inner metric space, then
locally compact
one
calls
(X, d)
space. OF
THE
distance function
COROLLARY:
6,,.
Let p be the
Then for every two
geodesic distance with respect to the points X,-y E X and every weakly piecewise
el-curve -y connecting these points y
P(X"g)
:5
I-yI"
(2.4.4)
1 of [0, 11 the relation 6 (y (ti), -y (ti+,)) < < tk R By (2.4.4) the estimate p(x,ij) :5 5,,(x,1J) follows. One proves the inverse inequality as follows. Let & > 0 and A : [0, 11 -4 X be a curve from x to -y such that with respect to an appropriate partition of [0, 1]
as
for every
J-y1(t,,t,+,1J,
partition 0
=
holds for 0 < i
to
us
first
0 is
Before
we
start to some
2.4.11 Lemma
the
following
(cf
1-YI,
:5
P(X"Lj)
+
&-
the claim follows.
arbitrary,
provide
:5
study
El
Riemannian metrics and
fundamental BUSEMANN
E X
x
and
where
curves on
stratified spaces let
metric and inner metric spaces.
If (X,
[37, 1.1.(4)])
relations hold for all
B,,(x)=S,(x)UB,(x),
properties of
e >
d)
is
an
inner metric space, then
0
S,(x)=j-yEXjd('y,x)=ej.
point ij E S,(x) there exists a sequence By assumption on X there is a sequence of e. Choose for every k a point Ijk on 'Yk. By x to ij such that 1-ykj -1'Yk I ! d(x, IJ k) + d(IJ k) 10 ! F_ the relation V k ij follows, hence the claim follows. One
PROOF:
of points 'Ljk E curves -yk from
only
has to show that for every
B,(x) converging
to -U.
M
X in a metric space one To every rectifiable curve -y : [t-, t+1 continuous and monotone function s : [t-, t+] ---> [0, 1,Y11, t
2.4.12
the
obviously
can
--)
assign
1-YI[t-,t] 1.
[0, 1-yll -4 X with -y unique continuous curve s, and the -y- the parametrization of -y by are length. Two rectifiable curves are called equivalent, if they have the same parametrization by arc length. An equivalence class of rectifiable curves is called a geometric curve (cf. BUSEMANN [36, Sec. 5]), a representative of a geometric curve is called a parametrization. The lengths, the initial and the end points of all parametrizations of a geometric curve coincide, hence it makes sense to use these notions for geometric curves as well. Often we denote a rectifiable curve and its associated geometric curve Then there exists curve
by
;y-
the
a
is rectifiable. We call
symbol. geometric curves in a metric space is called uniformly convergent a geometric curve -y : [t-, t+1 -4 X, if there exists a sequence of parametrizations : It-) t+1 -4 X such that the sequence (Yk)kEN converges uniformly to the rectifiable same
A sequence of
to
'Yk
curve
-y.
A rectifiable curve -Y us add some notation for curves of shortest length. and if X called is a geodesic, if for every -) a t+] 1-yj [t-, segment, d(-y(t-), -y(t+)) 5 (y(s-), -y(s+)) for all times s c [t-, t+1 there exists a e > 0 such that I s- < s+ in the interval [t-, t+] n [s e, s + e] In other words geodesics are curves it follows immediately by definition that every of shortest Moreover, length. locally Note inverse is not true. the that is a geodesic. segment Let
=
=
-
.
2.4.13 Theorem
(Tk)kEN
a
(BUSEMANN [36,
sequence of geometric
5.16
curves
Thm.])
Let
(X, d)
be
a
length space, and points of the
in X such that the set of initial
76 -yk
Differential Geometric well
as
curve
on
Singular Spaces
the sequence of lengths
as
[0, t+]
:
-y
Objects
X and
-4
I-yA;I is bounded. Then there exists subsequence (-yk,) Ic:,, converging uniformly to
a
I-Y I
< liM
inf IYk I
a
rectifiable
-y such that
-
k-+oo
PROOF:
Let
proof the
us
claim under the
Later it will turn out
set is
assumption that
every d-bounded closed
the theorem of HOPF-RINOW that this
by compact. assumption is automatically fulfilled in every length space. But note that for the proof of the theorem of HOPF-RINOW the theorem of BUSEMANN for compact X is needed. Therefore We
state the theorem of BUSEMANN
we
subsequence 1-yk 1. By assumption
lim inf
the
on
curves
yA; the set of
k-+00 a
compact subset of X, hence there exists
transition to
a
subsequence lim yk, (0) I
If
already at this point. length and first choose a lengths I-YkI converges to all initial points _Yk (0) lies in
that every curve *Yk is parametrized by arc (Ykl)IEN Of (Yk)kEN such that the sequence of
assume
1-yA;, I
0 from
=
point
a
x
(=-
X such that after
a
possible
x.
00
certain index
lo on, then the sequence (Yk1)1EN obviously conx. uniformly to the constant curve 'y : 101 -4 X, -y (0) If 1-yk, I > 0 for all I E N, we define for every 1 a new parametrization A, : [0, 11 --- X Of Tki by k (t) Tki (t IYk1 1). Let L > 0 be an upper bound of the lengths I-Yk I and p > 0 such that all Yk(0) lie in the ball Bp(x). Then for every I E N and t E [0, 1] =
a
verges
=
=
I
d(x, Al (t)) that
means
that
[0, 1]
the
(At (ti), At (Q)
means
a curve
-y
=
d
:
[0, 11
a
(Ykt (tl 1Yk1 1)) Tki (t2 I'Ikl 1)) (A1)1,21q
is
lo
let 0
=
such that
one can
to
IYk1 I Itl
-
t2l ! _
L
It,
-
t2l
X. For this 'y
--
(1). Let (Xj)jErj be a Cauchy-sequence in X and K the closure of points xj. Then K is closed and d-bounded, hence lies in a compact ball B,(xo). Consequently the Cauchy-sequence (Xj)jErj has an accumulation point, hence a limit. Thus X is complete. Obviously (3) follows from (1). Moreover, under the assumption of (1) the theorem of BUSEMANN entails that for any two points there exists a connecting segment. -So it remains to prove the implication (3) => (2). We proceed like in [37, 1.1.(8)]. For an arbitrary point x consider the set R of all PROOF:
the set all
p > 0 such that
9p
:=
-gp(x)
is
compact. If p
E R then every
positive p'
< p
lies in R
Differential Geometric Objects
78 as
well. As X is assumed to be
by proving Let balls
Up
Bp B p+5 (x)
:=
Pk
Rp
E
d(x, Xk)
Lemma 2.4. 11
k
a
one
Yk from
segment with d(x, xik a
for
has
is
Bpi (xi)
is
by finitely
many
compact. Then for
U i B p, (xi) holds, consequently 1 p+6
C
U i -ffp, (xi)
c
that R is open.
means
difficult.
claim
=
=
point x'k E B p (x) with
a
X0
=
Suppose that UPI is compact for every by proving that every sequence ('Xk)kEN accumulation an point. We can assume d(Xk, X) -4 p and because in case d(x, xk) d(x, Xk+l), p one can find by
< Pk+1
point of (XkDkEN
be covered
can
for all i and such that
proof that R is closed is more positive p' < p. We will show the The
x/,
19P
Then
is compact.
such that xi E
5 > 0 the relation
points
R cannot be empty. We show R
locally compact
compact. Hence p + 5 lies in R which
of
Singular Spaces
open and closed subset of R O.
an
suppose that
us
Bp,(xi)
some
is
that R is
on
(Xk)kEN
Of
x
as
d(xk, x') k
well.
i k.
-
< d
F- >
large enough
(y, -y i)
0
an
+ d
(ii i, xkkjj )
we come
+ d
index i such that
such that d
(lu i, xA;,j)
back
again
(xkkij xk.j.) kjj ,
d(y, ij i)
e/2 for proved.
1 on
R1j such
R'O,
that
((L"nij+')* Li+1)j,,j+ for i
1,
=
thermore
-
-
-j
-
assume
xi(K'i+, \ Kj),
Ij E
1 and
appropriate
that di (y,
where di is the distance
extend the sequence of the Riemannian metric il 71 z ((Vl)
open
! i for all ij
0)
V2)) (W1) W2))
on
=
neighborhoods Uj+j C Oi of xi (Ki). FurE xi (Ki \ K'j-j) and di (ij, 0) ! i + I for all function corresponding to tj. We want to we
first define
a
by
( Lj) zj (V1) W1) + (V2) W2))
=
tj+j. To this end
tj and will construct
R'j+1
(2.4-8)
LijUj+j
Z1, V1, W1
EWj) Z2) V2) W2
E Rnj+1
-nj.
Vj+j C Oj+j be an open neighborhood of xj+l (Kj) such that Vj+j n xj+l (Kj+,) relatively compact in xj+l (Kjo+,). Finally choose smooth functions W : Rnj+1 -- [0, 11 1 on an open neighborhood with relatively compact support in Vj+j such that y Next let
is
=
Vj'+, now
c
Vj+j
of
xj+,(Kj)
and set
choose N E N such that
n Rni. As
Uj+j Vj'+j NIIij z1I ! 2 for :=
-
all
z
E
xj+,(Kj) is compact, one can xj+,(Kj) and Ij E Rnj+l \ Vj+,.
One checks easily that the Riemannian metric
Lj+j
together
with the
Wfl +
=
neighborhood Uj+j
(1
-
p)
N
has the desired
properties. This completes the
induction.
As the Riemannian metrics Lj are compatible in the above sense, we thus obtain a on X which on every one of the open sets K,' is equal to the pullback
smooth metric L
(xj) we
*
tj. We show that with this metric
prove that every
L-bounded
(X, 5 ,)
is
a
length space
indeed. More
and closed set K C X is compact. Let
precisely
us assume
this is not the case, that means that there exists a sequence of points xj E K no accumulation point. By transition to a subsequences Of (Kj)jEN and (Xj)jEN
that
having we can
achieve xj E Kj \ Kj'-,, because otherwise all xj would lie in one of the compact sets Kj and the sequence (xj)jrN would have an accumulation point. Consider the distances 5
,(xj, x).
*
to
*
point
If k >
large enough, then there exists a rectifiable curve -Yj in Kk from 2-j. We choose k as small as possible, so there exists ! Jyj I im-yj with ij E Kk \ K'k-1. Let us estimate the geodesic length of -yj from
xj with 5 -y E
,
j
is
(xj, x)
-
below: 6
,
(xj, x)
1-yj I
-
2-i
>
dk (Xk (Y)
1
0)
-
2-j
> k
-
2-3'
>
j
-
2-3.
Hence the sequence (Xj)jEN of points of K is not bounded which is in contradiction to on K. Therefore the sequence must have an accumulation point which
the assumption
by
closedness of K lies
As
we now
again
know that
immediately from
in K. This
(X, 6,J
is
a
implies the compactness of K. length space, the rest of the theorem follows
the theorem of HOPF-RINow 2.4.15 and Lemma 1.6.10.
M
80
Differential Geometric
commutative
every
[72]
GROTHENDIECK A
(with unit) one can define according Homk(A,A) of differential operators
A
k-algebra
D(A)
the space
c
to on
the set of the in the
as
cisely,
D(A)
following sense almost A-linear operators. More prespecifies recursively for every k E N a space V(A) and sets afterwards
one __`
UkENDk(A).
First let
of A.
endomorphisms
V(A)
Suppose
Dk+l(A) Thus, the
ID
=
Dk+l(A)
space
HomA(A,A) A bethe space of allA-linear Dk(A) of differential operators of order
=
that the set
at most k has been constructed for
to
Singular Spaces
on
Differential operators
2.5 For
Objects
some
I
Homk
E
[D, a]
consists of all
operators of lower order.
One
E
Dk(A)
for all
a
E
Al.
endomorphisms of A which
checks
now
Dk+l(A) by
natural k. Then define
easily by
an
are
A-linear up
induction argument that
all the spaces Dk (A) and D (A) are k-linear. Moreover, the composition D 15 of two differential operators D E Dk(A) and 5 c V(A) is an element of Dk+l(A). Hence -
D(A) is
becomes
given by
the
a
(filtered) k-algebra.
A first statement about the structure of
2.5.1 Lemma For every commutative
D1 (A) Of
PROOF:
which means
course
V(A)
let D E
A+
and set
obviously
c
b (a)
Derk (A, A)
1)(1).
=
E
a
(al)
=
15
is
A+
C
has
Derk (A, A).
D1 (A). For the proof of the inverse inclusion
We will show the inclusion
5
true, hence for all aD (b) + b c.
=
=
E
5:=
D-C E
Derk(A,A),
D1 (A). In other words this
[f), a]
=
c, In
particular the
a, b E A
aD (b) + bD (a).
derivation of A.
a
Now let M be
a
13
manifold. Then it is well-known
D k (M) :=D k(eoo(M)) of differential operators ,
all linear combinations of
endomorphisms
'C'(M) where 1 < k and
one
A there exists C(i E A with c,, is
D (ab)
Therefore
=
k-algebra
will entail the claim. First note that
that for every
relation
D(A)
following.
V1,
-
,
V,
E) f --) Df
are
=
D E
V,
on
[46,
Thm.
End(e,'(M))
...
V1f
smooth vector fields
2.3]
that the space
M of order at most k consists of
E
on
of the form
COO (M), M.
Replacing
(2.5.1) M
by
a
Whitney
(A) space X every smooth vector field V on X gives rise to a first order differential operator. Therefore the question arises, whether in analogy to the differentiable case algebra D(X) := D(e,'(X)) of differential operator is generated by the vector on X. But Example 2.2.1 and the above Lemma 2.5.1 show that this need not be the case; in general D(X) even is not generated by the derivations of 12110(X), as the following example shows.
the
fields
2.5.2
Example (cf. [46,
smooth structure
as a
3.81) Consider Neil's parabola X,,, together with its subspace of R2 (see Example 1.1.15) and parametrize
Exercise
stratified
2.5 Differential operators
X,,,i,
via the
81
R
embedding
-)
R,
t
(t2, t3)
-4
The
.
Then check that the operator at does but that the operators
12 00 (XN61)-module Derp, and a3t
actually t2 at do, and
tat and
image
not induce
of this a
embedding
derivation of
is
X ,,i,
C00(XN61),
that these two derivations span the On the other hand W-2t-lat, ta2 at t
(1200(XNeil)) 000(X, j,)).
-
t
3t-1 a2t + U-2at are differential operators on XN,,i which are not generated t2 at, hence DerR (o0o (XNefl)) e00(XNeil)) does not generate the algebra of and by tat -
differential operators
2.5.3 Lemma means
on
Every
XNeill
differential operator D
supp Df C supp f for all f E e' (X)
PROOF:
The
claim is
trivial, if
Whitney (A)
on a
space X is local that
-
D has order 0.
So let
us assume
that the claim
holds for all operators lying in DI(X). Then let D E D'+'(X). Choose for f E 12"0(X) and an open neighborhood U of supp f a function y E e' (X) with supp y c U and k 1. As Df D ((pf) y Df + 15f for a differential operator 5 E D (X), yl,upp f =
=
=
the relation supp Df C supp y C U follows. As U supp f , the claim
2.5.4
Every
was an
arbitrary neighborhood
follows.
now
open subset U C X is
operator spaces D k(U) and
of 0
again (A)-stratified,
D(U).
the sectional space of sheaves D k
Next X.
hence
we
canonically obtain
will show that the spaces D k (U)
we
Hereby, locality
are
makes it
possible that one can restrict differential operators to smaller sets. To give the restriction morphisms explicitly let fl C U C X be open and D E D(U). Then choose for every x E a a smooth function y,,. E e' (X) with supp y,, CC ft and 1 on a neighborhood ft of and define for all f x U.,, c E 000(U) and x E U an extension f-, E 0'(U) by on
p-, (ij) f (-U)
The restricted differential operator
DIff (x)
ft, 1JEU\fl.
if
0
IJ E
if
DIa
Df,.(x) for all f
=
TV(D) U
will
now
(2-5.2)
be determined
uniquely by
12'(U) and X E U. As D is local, Df,(x) does not depend on the choice of y, hence Df.,(-Lj) Df,.(ij) holds for all 1 E close to Therefore is x. smooth and sufficiently DIaf D,a is well-defined. Moreover, requiring
=
E
=
U U by definition rn Tf, o
means
D and D
k
U
are
Now it is easy to
differential operators there exists
rf, follows immediately presheaves on X.
=
see
Dj
that for any open E
D'(Uj)
with
for all open sets
covering
Djluinui
unique differential operator D
=
(Uj)jj
Dilu,,U,
1.
C
ft
C U C
of U and
a
X, that
family
of
for all indices j, i E J
D k(U) such that
Dj for all j. Djuj just defines for f E el(U) the function Df E e'(U) by (Df)luj Dj(fluj) and verifies immediately that the operator D is well-defined and has the desired properties. Hence the D kare sheaves on X indeed. On the other hand, the presheaf D is in a
E
One
general not a sheaf. The reason lies in the fact that for noncompact X one cannot "glue together" every family (Dj)jEj of pairwise compatible differential operators to a global one; namely if and only if the order of the set of orders of the Dj is unbounded.
Differential Geometric
82
2.5.5 Definition A differential operator D E
if for all S G 8 and all f E
D8(X), that
C'(X) g E C011(X)
in other words if for
means
D(X)
Objects
is called
the restriction
with gIs
=
Singular Spaces
on
stratified,
in
signs
D E
(Df)ls depends only on f1s 0 the relation (Dg)ls 0 =
holds.
Example The smooth vector fields on X are stratified differential operators of By Lemma 2.5.1 and the following proposition the smooth vector fields toget'her with the smooth functions span the stratified differential operators of first 2.5.6
first order.
order
X.
on
The
result extends
following
2.2.8 from vector fields to the
Proposition
case
of
differential operators. 2.5.7
Let X be
Proposition c COO
chart, 9
an
(0)
the space of differential operators is
canonical
a
D'(X)
D E
operator
0
on
0 C R' is
--
vanishing
on
the ideal 0 into
mapping
x(U)
a
singular
and
Dg(O)
then there
itself,
isomorphy:
family (Ds)SES
a
0
over
D(U) Moreover,
If x: U
(A)-stratified space.
the ideal of smooth functions
with
Dg(O)IOD(O).
--2 L
of differential operators
(Df)ls
for all
Dsfls
=
f
Ds
DI(S)
E
COO(X)
E
defines
and S E
a
8,
differential if and
only
if for every smooth function f : X -4 R the function X -D x -4 Ds,.fIs.,,(x) is smooth again, where S,, denotes the stratum of x. In this case D is determined uniquely
and necessarily stratified. Vice versa, any stratified differential operator D E DII(X) 8 originates in this way from a family (13s) SEs of differential operators DS E DI (S). As all sheaves involved
PROOF:
that U X
=
(Xi,
that X is closed in
X,
=
.
.
.
,
Xn)
is
given by the morphism
fine,
are
identical
suppose without loss of
we can
that 0 is
0,
a
ball around the X
embedding
origin of
R". Then
-4
we
generality
R' and that
consider for
k E N the canonical
TTk
:
D k (0) 0
--
D k (X) ,
D
(f
-4
f+0
=
and first show that TTk is surJective. To this end let I
V)
(7t -y)
E C-
-
projection
(Xi
the smooth function
by (x
(0)
the function
(7r,
-
y 1) 11
-y
.
.
onto the i-th coordinate and -yi
oci
us
.
.
.
.
.
(7rn 7ri(-Lj). C-(X), ..
.
Df +
-4
-
0),
f E
denote for
C' (X), and
Cz
-
one
Nn and -y E Rn
oc E
(Xn 1Jn) IXn Ij n) 0-n, where
Then
C'(0),
proves
by
the order k that for every D E D k(X), g every multiindex and every point -Lj E X the following relation is true: D (g
Now fix D E
D'(X)
and choose for
-
(X
-
1J)') (IJ)
locl :5
=
by
7ri :Rn __4 R is the
oc
induction with
I ocl
on
> k
(2.5.4)
0.
COO(O)
k functions do, E
such that
1
d, , (ij)
Setting
D
every f E
=
Elocl
-1,
Schouten-Nijenhuis polydifferential
-4
E Djofo
bracket
the space D k(M) of
Section 6.3 for the definition of Hochschild
(see
cochains
(9
0 and is
On the other
bracket.
:=
fo)
Djkfk)
fk
E
cochains)
1200 (M),
Lie
consists of all
Djk
Djo,
as
Hochschild k-
E D
(M),
iEJ
where J denotes
a
in the Hochschild
There exists Lie
a
finite index set. As differential
complex, canonical
algebras: Fl: T*(M)
-4
VA;)
-4
the Lie bracket is
one takes the ordinary differential given by the Gerstenhaber bracket.
mapping between the just
(VoA
where the Vi
...
are
A
graded
D*(M) 1
Fi:
introduced differential
fo
(9
...
vector fields and the
0
fi
fk
are
'-4
-
(k
+
1)1
E
sgn (a)
smooth functions
11 V qi)fi i=O
,Esk+l
on
M.
version of the theorem of HOCHSCHILD-KoSTANT-RoSENBERG
)
By the topological [881 F, is a quasi
isomorphism. (see as well Theorem 6.4.5 and PFLAUM [143]), hence HKR quasi isomorphism. Now we can formulate:
we
call it the
2.6.11 Formality Theorem (KONTSEVICH [103, 102]) There exists an L,,morphism F from T*(M) to D*(M) with first term given by the HKR quasi isomorphism. Fl.
2.6 Poisson structures
89
Now the
question is what to understand by an L,,.-morphism and what is the theory resp. deformation quantization. First let us explain L,,,,-morphisms. Consider a graded vector space. g and associate to g the following graded coalgebra: Ak (0) [k], SyMk(9[1]) C(g) connection to deformation
kEN
where
Sym
k
kEN
is the functor of the k-times
C(g) possesses additionally (C(q), Q) a L,,,,-algebra, and
symmetric tensor product. If the coalgebra Q of degree +1, then one calls the pair L,,,,-morphism is nothing else than a morphism in the
differential
a an
category of L,,.-algebras. For the
case
the differential d and the Lie bracket on
C(g)
Q, =d, Q2= [.,.] and Q3=Q4=''*=Oto deformation theory. One assigns to every differential graded the moduli space M(g) of solutions of the Maurer-Cartan equations:
with
Next
algebra
that 9 is even a differential graded Lie algebra [-, -] of g induce automatically a differential Q
we come
g
NE(g)
f
:=
gj
E
d& +
&I ![&, 2
=
Lie
01/Go,
where Go denotes the Lie group corresponding to the Lie algebra go. If now m is a commutative algebra without unit, where for our intended applications in deformation
quantization we will always have of g is given by
m
=
AR[[X11,
Def,&) Hereby
one can
tal statement
interpret
now
Def,,,(m).
to deformation
Now
a
theory by
and remark that
of deformations
M(g 0 m).
the parameter space of the deformation. The fundamenL,,.-morphisrn F between C(gi) and C(02) such that
quasi isomorphism induces
This is the
we come
Def,
is that every
the first term F, is and
m as
=
then the functor
back to
(with
m
an isomorphism between Def", (m) quintessence which essentially goes back to the approach
SCHLESSINGER-STASHEFF our
original
=;kR[[A11)
the formal Poisson structures
[147].
matter of concern, deformation
quantization,
the deformations
DefT-(m)(m) consist of exactly and that the deformations DefT.(M) (m) are precisely the
products on M. Consequently, an L,,.-morphism like in the formality theorem provides a natural isomorphy between equivalence classes of formal Poisson structures and star products on M. Let us transfer these considerations to the stratified case and let X be a symplectic stratified space with Poisson bivector A. Assume for simplicity that X is a closed
star
subset of
an
open set 0 c RI and inherits from RI
a
smooth structure.
Let
W be
antisymmetric bivector field on 0 such that W coincides over X with A. Then the bivector field W induces over 0 a not necessarily associative "star product" :; , as for W an
identity need not hold. At this point we consider the strata S and stress that embeddings LS : S --> X are Poisson maps. If now the construction of the L,,-morphism F from C(T*(M)) to C(D*(M)) is in a certain sense natural or functorial with respect to Poisson maps, then for every f in the vanishing ideal 0 of X and for 0 holds. Consequently T- can b e pushed gTf Ix every g E Coo (0) the relation f-Tg Ix down to a R[[A]]-bilinear map *: A x A -) A with A C00(X)[[A11 the Jacobi
the canonical
=
=
=
Applying
the
naturality of the
construction
again
=
it stratawise becomes clear that the
Differential Geometric
90 restriction of * to each of the S
X. Let
on
2.6.12 orem
us
gives
a
star
product,
on
Singular Spaces
hence this must hold also
globally
summarize:
Result Under the assumption that the
can
Objects
be constructed in
stratified space
a
star
a
L',,-morphism
in the
formality
natural way, then there exists for every
the-
symplectic
product.
immediately clear whether indeed, but this seems to be plausible. Therefore let us formulate the following conjecture, where we do not want to conceal the fact that the notion of "naturality" in the context of the formality conjecture needs further explanation. By KONTSEVICH'S proof of the formality the construction of F is natural in
Functorial
theorem it is not
functorial
formality conjecture L,,.-morphism according to KONTSEVICH
2.6.13
the
a
sense
The
graph
theoretical construction of
is natural in
an
appropriate
sense.
Chapter
3
Control
Theory
Tubular
3.1
In this section
we
will introduce the notion of
the classical tubular some
neighborhoods
neighborhood
a
tubular
theorem. But before
neighborhood
we come
and will
to this let
us
proof provide
useful notation.
Let X be a locally compact Hausdorff space with countable topology, S c X locally closed subset, and TS an open neighborhood of S in X. Moreover, let 7rS : Ts --> S be a continuous retraction (7rs)ls ids, and ps : Ts -- R :O a continuous function with ps-1 (0) S. One finds such a situation for example, if X is given by a metric vector bundle E over a manifold S; then one can choose Ts E, 7ts as the projection of E, and ps : E -i R as the distance function v 1-4 1 Iv I I' (v, v), where il denotes the scalar product on E. Assume to be given two (in most cases continuous) functions F-, 6 : S --4 R R U JooJ with e < 6, i.e. E(x) < 6(x) for all x E S. Then we set 3.1.1 a
=
=
=
[r-1
=
[F-, 5[
=
T's
=
t's
=
TsF-
=
I(x,t) E S x RI t e(x)J, J(X, t) E S x RI e(x) : t < 6(x)J, JX E Ts I ps (x) < z (7rs (x)) 1, =
T's
\ S,
fx E
Ts I
ps (x)
neighborhood
W>0 with S
There exists
(7ts, ps)
possibly shrinking Ts,
:
T's
a
--
c
Vs
c
following statements
W of S in X there exists
a
hold:
continuous function
W.
continuous function
[0, e[
7rs and ps the
is proper and
M.J. Pflaum: LNM 1768, pp. 91 - 149, 2001 © Springer-Verlag Berlin Heidelberg 2001
E
:
S
---)
surJective.
R` such that the restricted
map
92
Control Choose for every
PROOF:
Vx-
closure a
basis of
hoods of with
neighborhoods in S and
x
limn,,,. 5,.,n
yn lie in the
set
compact
a
x.
element of the
no
Now let such that
the claim is not true.
V-,-
Ts
C
=
basis
x
of
C
point
Ajn
Ts. By definition of the ps (-y n)
=
W-,,,, \ a
W-,,n
sets
Thus the
open
As the
subsequence the relations
hold,
V of -Y
Wx,,,
an
V.
0 then would
=
neighborhood
Odn)nEN.
sequence
E
form
hence
x
=
there
basis of
a
x.
a locally finite covering (Vn)nEN of S by V., for appropriate xn E S. Then define TS
sets
Vn
_
open in X
TS
C
WEN Vn
=
and note that
open
S
G
x
an
'C W. As S is paracompact, we well as open neighborhoods U,,,
fV is
an
open
covering of S and Um
neighborhood
a
continuous function 6 E
U,,,. But this
R
:
means
-)
R>0 with
'NS
that
fV
C
S
R>0
-
a
p (x)
Then the
mapping
we can
CK,5
p
:
choose
i s-' (K)
is closed in restriction
number
6.,,o
5X'i
=
J c
>
Ux,.,, of xn in S such that (Um)mEJ is U-,,_,O. If one now sets W UMEJ W-,'-,,,
fs
n
=
IV
C
5(x)
W. As S is paracompact, there exists
0
[0, d
proof
of
with 6
(2) for this e < F-(x) for all
is proper, and
by
UVnnKOOVn
the choice of
C
ts.
e
ef,s(v)
0,
and
ef (v) > 0 and Ti (1j) > 0 have to be true.
e,,s(v)
:,
ef,j-,j(v)-
We will prove
only analogously with
of the first
one can
Chap. 11].
It suffices to consider the nontrivial
be carried out
real number with tf < tj
0. Let further t, be we
had t,
path -yl[o,,cj
a
>
real number between 0 and
t,,,,. then
would have
we a
would have s,
shorter
length
ec(v)
kn. Hence there exists for
every k >
ko
some
recall that 'Yk minimizes the exp(twk) for Wk E Sj,M with 'Yk(t) As follows. N distance from 'Yk (1) to S. Thus Wk E S,, IJ k ::: Yk (0) is an element of S for sufficiently large k and iMl'k C Un7 the sequence (Uk)kEN Of fOOtPOints of the 0 < t < Sk. Now
=
Wk converges to assume
x.
appropriate subsequence we can therefore (Wk)kEN converges to some vector w (-= S,,N. At this point
After the choice of
that the sequence
an
recall that
eXP((tk
+
5k)Vk) =Yk(Sk)
passing to the limit k -4 oo consider the following two cases.
After
the relation
exp(skwk).
exp(t,,,,v)
=
exp(s(,ow)
follows. We
now
points x and exp(t(,.v) in two different ways by geodesics, hence the geodesic (exp [0, t,+ [ v) does not anymore minimize the distance to x beyond t,,., which is impossible. 2. CASE W v. Then by t,,. < e,(v) the relation t,,,, s(,o follows. Hence for W'. By sk < tk + 6k lie both in sufficiently large k the vectors SkWk and (tk + 6k)Vk 1. CASE
W
v.
Then
one can
=
=
the vectors Skwk and
Tt((tk
+
6k)Vk) holds,
connect the
(tk
+
5k)Vk
have to be different. On the other hand
n(SkWk)
which contradicts the fact that n1w, is injective. ec(v) does not hold, so e, is lower semicontinuous.
The lower Altogether too < semicontinuity of Tj follows immediately by the one of e,. The proof of the continuity of ef, e, and Tj for the case that M is geodesically complete will not be performed here, as one can find the corresponding proofs in the literature [16, 101]. Moreover, the continuity results will not be needed further in this work.
n
100
Control
Now will
define
we can
special
a
yield the desired
neighborhood of the neighborhood.
open
zero
Theory
section of N which later
maximal tubular
3.2.4 Lemma Set
T' ScN Then expITm- is SCN PROOF:
an
:==
ftv E
open
S,
E
x
We show first that expjTmfi' is SCN real number t
0 with
minimizing that
the distance from ij
6,(y, S)
=
S,,N
is maximal among the tubular
were
neighborhoods
1-yi,
=
s.
not maximal in the claimed
T of the
that the conditions for T in the claim exist
f e,(v) I V E
result of this section:
of S in
R.
Let
there exists
lies in T.
zero
sense.
section of N with Tm'
SCN
Then
C T such -
satisfied. Under these assumptions there + 5)v E T. Let -y : [0, s] --4 M be a geodesic are
(e,(v) exp((e,(v) =
By the theorem
+
6)v)
to S. In other words this
of HOPF-RINow 2.4.15 such
a curve
means
-y exists
101
3.3 Curvature moderate submanifolds
indeed, and y(O) has exists
to lie in
S-y(o) N with -y (t)
W E
the relation e, (w)
S,
S is closed. Then -y is normal to S, hence there s. As y is distance minimizing,
as
exp, (tw) for 0 < t
s is true.
[0, s [w
hence in T. lies in Tmax SCM ,
5)v and [0, s[w would have disjoint neighborhoods Uv and Uw in T. But by -y E exp(Uv) n exp(U,) the map explT could not be an So s < e,(v) follows, hence v. open embedding anymore. Therefore we have w T hand other E the On Tm '_, which by the fact that + \ 6)v E (v) (e,, ). (Tmax 'Li exp, SCN ScN is This proves that Tmal contradiction the entails is exp(Tml). -y ScN open ScN explT If
=14-
w
v, then the sets
(e,(v)
+
=
maximal
as
claimed.
3.2.7 Remark In
n .
case
(M, R)
is not
complete
or
S is not closed in M the tubular
need not be maximal anymore in the sense of the last proposineighborhood Tm' SCM tion. Nevertheless we can define in this case em' and Tma' like in the Corollary. For our
purposes
we
do not need any tubular neighborhoods language and call Tm' in every
abuse the
fore
we slightly neighborhood of
S in M induced
by
larger case
than Tm.
There-
the maximal tubular
R.
complete and S is closed there might exist other maximal tubular neighborhoods of S induced by R besides Tmax. Therefore the claim appearing occasionally in the mathematical literature that 3.2.8 Remark
T'nax is the
largest
let
us
remark that
among the tubular
even
in
case
neighborhoods
M is
of S induced
by
t is wrong.
Curvature moderate submanifolds
3.3 In the
Finally
following
we
will introduce
a
notion which describes how
a
stratum
or a
sub-
manifold curves within the ambient stratified space respectively ambient manifold when approaching the boundary of the stratum or submanifold. Take for instance the standard
cone.
Then it is
intuitively
clear that the behavior of the curvature of
the top stratum near the cusp does not change much. More generally consider a real or complex algebraic variety with its natural Whitney stratification. Then the curva-
speaking again intuitively bounded by a rational function, so cannot grow "too" fast while approaching a lower stratum. But the situation is different when considering the slow or fast spiral. Here the curvature of the top stratum grows exponentially with the distance to the origin. The notions introduced in this section will help to separate the first two cases, which in the following we will regard as curvature ture is
-
-
moderate, from the latter 3.3.1
S is
To
cases.
notation let
us
submanifold of R'
or
simplify
alway neighborhood of S in R' be given by the total space of
open
a
or
agree for this section that ME N>0 U too} and that a manifold M. Moreover denote by T always an
of
M such that S is closed in T. In most
cases
T will
neighborhood of S. We consider first a submanifold S C R7. According to the classical tubular neighborhood theorem and Section 3.2 the Euclidean scalar product induces a maximal will be of S in Rn; the projection corresponding to T" tubular neighborhood T'ax S S Rn the T,,Rn denoted by 7rS or shortly by 7r. For every point x E T1 tangent space S a
tubular
ax
-==
-
Control
102
Theory
T,,S ED ker T,,7r. Hereby T,'S origunique orthogonal decomposition T,,R' T,,(-,)S by parallel transport along the line connecting 71(x) and x. Now, denote by PS,-, : TR' R -i T-,S the corresponding orthogonal projection and a
possesses
=
inates from we
=
write, if PS
:
misunderstandings
any
Tm' S
becomes
End(Rn)
--
3.3.2 Definition Let
End(RI)
not possible, simply P.,, instead of Ps,.,. Hence projection valued section in the sense of 1.4.14.
are
a
oo, S C R' be
m
0 such that for all oc (=- Nn with locl < m the following estimate
regularly
a
partial derivatives of
is satisfied for the
distance
neighborhood
P in
dependence
on
the Euclidean
d(x, aS):
Pap-il
i
E J the
following
satisfied:
:5 C-1
(1
I +
d(x, aS)c-1011
)
'
xEBnT, locl<m, MENn.
109
3.3 Curvature moderate submanifolds
(3)
For every over
(4)
is
orthogonal
an
>
0 such that for all j E J and
1 +
cin
frame with respect to R
n T.
There exist constants c,,, E N and C,' the following estimate is satisfied: a 'fjI (Y-)
(5)
fjn)
(fji,
j the n-tuple (Ok n B
the set SUPP
E supp
X
d(x, aS)c-'
4)i
11
1,
T, locl
n B n
n,
< in.
neighborhood of S in M induced by L, let 7& : T" --) S be the corresponding projection and P" : TR -- End(Rn) the projection valued mapping which assigns to every x E T the orthogonal projection with respect to ti ontothe horizontal space of the tubular neighborhood at the point x. Then P ' has over B n T the following representation: Let T '
(E,
=
be the tubular
p)
Fs,
dimS
P' 'Iv
=E E _a
1=1
iEJ
Moreover, PROOF:
that all
(V,fjl(x)) fjl(x) L-(fjI(-X)' fjl(x))
(mx)
P ' is curvature moderate of order
xEBnT, VEW.
(3.3.9)
m.
As P := Ps is curvature moderate, there exist d,,, E partial derivatives aPel at most of order in satisfy the
N and D"L > 0 such estimate
1 sup
JJD'PeI(x)JJ d(x, aS)d-
,:
V _n_
xEBnT
t there exist constants
Moreover, by assumption on all V E Rn and x E B2,nT
d(x, aS)c
a
locally
5,
C'M
=
-
maxf 1, C2 I
lam- j(X)1:5c' where Cm
>
0
depends only
one can assume
1+ on
>
n
those indices
T n B and choose xj E supp
j such that
qbj
)
-
=
C
c'M +
of Rn
0 such that for
>
\
11VII.
c
according
to Lemma 1.7.9
K such that
(3-3-11)
(j
(3-3-12)
XERn\K, lal: m,
I
and
that all the supports supp
exactly
(3.3.10)
T.
2n6m diam (supp ( j),
d(x,K)IInIII
6m,
c-
N and C
c E
f-(X'aS)
OMiEJ
1
cm,
:! -
and c,
finite smooth partition of unity
d(supp ( j, K)c-
sets J to
11VII 1-
-
< 1 +
P,,v for
norm
E supp
X
'
2n
estimate the
we
help of (3.3.11)
11PXV11
11filNil
(3.3.13)
-
JIT,,Pvll d(x, aS)"
sup
__
Tn C,
(3-3.14)
1
:
IlPxjvll
-
'JE[%,,xl
_11V11, 2n
where it has been used that the segment 1
-
-L 2n
llfjl(x)ll
IlPxOjez,+lll
< 1. Let
suppose that
us
well:
llfj(,.+,)(x)ll
1
E
-
k=1
10
1
1:
_
-
-
2n
k=1
fjk(x)) fik (X) lix(fik (X), fik (X))
p, (PxOjel,,+,,
11 PxOj e4,+, 11 Jjfjk(X)jj[Lx
,
I
I Ifjk (X) I I
10
E
2n
C2
JIT,,POjez,+lll llx-xjll
sup ij E [xj,-Xl
k=1
d(x, aS)c
10 >
1: TnTC,
-
-
2
k=1
sup
JjT,,POjej6+jjjd(x,aS)'-
IJE[xj,xl
+ >
2n
llfj(lo+,)(x)ll
:: ,
11Px0je,0+111
+
e10+1 fik (X)) Elix fik (X) 1 (PxOj '. (fjk (X)) fik (X)) I
k=1
10
1
:51+LC'
sup 'Y E
k=1
JjTjPOjejO+jjjjjx-xjjj
[xj,,Xl
d(x, aS)c
1+
2n' thus obtain
(3.3.14)
for all 1, hence fj I (X)) fjn (X) comprises for 4 j n B n T a tx-orthogonal basis of Rn. Now it is not difficult to check that the thus defined fj, can be extended to smooth vector fields over Rn \ K such that every fj, has compact support and such that I I fj, (x) 11 :5 2 holds for every x c- Rn \ K.
Inductively every
X
We
we
E supp
already know by
of the functions
C,,,,
> 0
POjej
the fact that S is curvature moderate that the components M'(aS;T), hence there exist constants c,j G N and
lie in
such that for all
a
with
sup XESupP.i)jnBnT
I al
< m
jja'fjj(x)jj d(x, M)c-'
& >
ell Qs,.uwll (4)
Given
End (R) the projection
and
at
=
X
X
x.
dG, (T-,,R, T',S)
a
neighborhood
U of x such that
a
neighborhood
U C TR of x, such that
IIQV'Qs,,Jwll
U C TR of x, such that
JJQS Q,'wll
< e
for
S n U.
E
Given
0 there exists
-4
Qs Ts -4 End(Rn) End(Rn) idRn given by Q,' R', and Qs,. the following four statements are equivalent: By QI: TR
pair (R, S) satisfies Whitney (A)
Given
all'y
by Ps : Ts
-4
valued sections
Then for everyx E R
Ps,,,.
The
projection
115
0 there exists
for all y E S n U and
e >
ell Qll, wll
0 there exists
E Rn
W
equivalence of (1)
:5
Rn.
neighborhood
for all -y E S n U and The
PROOF:
a
w E
and
(2)
follows
immediately from the definition Appendix A. 1 and
the definition of the vector space distance dG, in
Whitney (A), Proposition A.1.1 (2).
of
points of S converging to x and (TJkS)kEN be Q' orthogonal projection onto TR, and Qs,,J, the convergent to r r' C JxR)', hence TxR C -C. Therefore Whitney one onto (T.,,S), (3) implies that (A) follows. Property (4) entails immediately T,,R C r, hence Whitney (A) follows Now let
('Jk)kEN
be
a
sequence of is the
C Rn. As
X
again. Next let us suppose that (1) holds but not (3). Then there exists e > 0, a sequence (IJk)kEN of points of S with limit x and a sequence of unit vectors Vk E (T,,S)' with IIQ7' VklJ > e. By transition to subsequences one can achieve that (Vk)kEN converges 'Yk v E Rn. As the projections Q,',' converge to the orthogonal projection TR, the vector v has nonvanishing projection to TxR, which contradicts Whitney (A). Analogously one proves (1)=>(4).
to
a
unit vector
onto
Proposition Let sisting of two strata S 3.4.6
X C M be =
an
X' and aS.
(A)+ (5) -stratified Let further
closed
L be
a
subspace of M
con-
Riemannian metric
on
M and for every stratum R of X let TR be the maximal tubular neighborhood of R in M induced by L, nR the projection and PR the tubular function. If X is curvature moderate of order m, then after
(7ras, pas) : TsnTas PROOF:
-4
aS
x
As the claim is that M is
R" is
appropriately restricting Tas the submersion even strongly curvature moderate of order Tn.
essentially
a
local one,
we
can
assume
without loss of
open subset of Rn.
an Moreover, we can suppose that R is given generality Euclidean scalar the the case of arbitrary curvature moderate [t is by product (., .); technical. somewhat more Finally we abbreviate p := pas, proved analogously, only
7r:=
7ras and set R
-4
aS.
by Ps : Ts --> End(R) the projection valued section onto the horizontal Ts along the vertical bundle. Further projection valued sections PP : TP, \ End(R) and P' : TR -4 End(R) are given by the orthogonal projection onto
Denote
bundle of R
:=
116
Control
the kernel bundle of
T(PITR\R)
resp. onto the kernel bundle of T7r. More
Theory
explicitly
PP X
has the form P PW
=
W
-
V
(W, IJ
As the claim has to be
shrinking
M and thus
-
1J
7r(Ij))
I JIJ
-
-
7r(IJ) 7r(y) 11 2
TR \ R,
-y E
'
proved only locally around a point shrinking R) that P' has the form
of R
W E
R.
we can
(3.4.1)
suppose
(after
dimR
P!YW
W
(W, fi (7r(Ij))) fi (7r(L-J
-
TR,
-LJ E
W
E
Rn)
(3.4.2)
: R -4 R' denote vector fields spanning an orthonormal frame of R around projection valued sections PP and P' are obviously curvature moderate. Moreover P,,P and P,', commute, hence PPP' comprises the projection onto the kernel bundle of T (7r, p) over TR \ R. Furthermore, as P P and P' are curvature moderate, this holds for PPP' as well. For the proof of the claim we thus only have to show that the projection valued section onto the kernel bundle of Ps PPP' is curvature moderate.
where the f i x.
The two
-
To this end
will
As PS
Lemma 3.3.6.
apply
PPP' is
selfadjoint, it neighborhood U and A > 0 such that for every -y E S n U and every eigenvector V E (ker(Ps the corresponding eigenvalue Av has absolute value JAvI > A. We will show following. Let us calculate: we
Lemma 3.3.6 to show that for
ker (Ps,,j
P,,PP,',))
-
where the vector space
-L
-
E R there exists
X
a
(im Ps, n im R.P11.1)
C
ker
is not direct in
sum
general;
in the
considerations it will turn out that instead of the relation C estimate the
now
11 (Ps,.u
norms
from below. First let
w E
-
for
P,,PP,',)w
vector
a
+ ker
Ps,.y
-Y
E ker
w
by
constant
PPPJ)) 71
J-
this in the
P,,P + ker P,', of the foll owing
course even
suffices a
equality
Ps,,u
+ ker
holds. We
PP + ker P,','
ker R.P. Then
W
(W,g 7t(Ij)) (IJ I JY 7r(IJ) 11 2
-
7t(Ij)),
-
hence
by (5) for appropriate
(PS'-Y Next let
E
PS'-YW I I
P-"Ppi")w
ker
Ps,j.
=IIPS'Ju 1 1-Y
Tr(-Y))Il IIWII 7r(IJ) I I
-
-
We shrink U such that
according
>
5
IIWII-
ker
Ps,,j.
Then
(3.4.3)
to Lemma 3.4.5
I I Q,7JIvJJ :! F-JJvJJ holds for all -y E one calculates with Eq. (3.4. 1) and relation P"(-y 7r(-Lj))
0 to be determined later the relation
r > V E
w
-
U and 6 > 0
-
(3)
for
a
S n U and 1J -7r(IJ)
that
(W, IJ 7C(IJ)) IIIJ 7t('U) 11 1 (W, Q S,.y (IJ 7r(Ij))) I I IIJ 7r(IJ) 11 IIQ Wlr IIQS,.Y(lj 7t(IJ))JJ 11WI, IQ,UWJr -
11(pS"l
pPP7t)WJJ
=
JJPLYPP7rWJJ Y 't
r
11p.7.W11 U
-
-
=
11P,7!U W11
-
11W IF
-
(8/1--F2
-
7r
-,
52)
1-
IIWII
57r
JJW112.
(3.4.4)
3.5 Existence and
117
uniqueness theorems
point we determine F_ > 0 such that 5' := V1_--F-2 V11_-62 > 0. Finally let w E ker P.,'. By Lemma 3.4.5 (4) we can shrink U such that for orthonormal frame (f 1, fdim R) of R around x the estimate At this
-
-
the
-
)
1
JJQS,-Yfi(7T(1J))JJ
:5
SnU,
-y E
,
2n
dimR
by Eq. (3.4.2)
holds. On the other hand
(w, fi (7r(v))) fi (7r(-U)),
w
hence
dim R
11 (PS"Y
(W, f,(7r(1j))) 111 QS,yf, (7r('Y))
JJWJJ
JJPS,_YWJJ
P,,p,7r)WJJ
_
(3.4.5)
dimR
JJQS,Ufi(7t(1J))JJ
JJWJJ
2
JJWJJ. .
We
now
minJ5, P, 11. 2
set A
for the
eigenvector V E This finishes the poof.
(
Our considerations
ker (Ps,,j
-
P,,PP.',))
-L
3.4.7 Definition An
far
imply that
the
eigenvalue
I;kv I
> A > 0.
definition.
space X is called
(A)-stratified
now
must have absolute value
preceding proposition suggests the following
The
so
strongly curvature, moderate
of
order m, if every stratum is curvature moderate of order m, and if for every pair R < S of strata (after possibly shrinking the tubular neighborhoods) the submersion
(7tR) PR)isnTp
:
S n TR
R
--
x
R>1 is strongly
curvature moderate of order
Example Subanalytic sets with their strongly curvature moderate of any order. 3.4.8
Existence and
3.5
coarsest
Whitney
M.
stratification
are
uniqueness theorems
generalized the classical tubular neighborhood theorem in his notes particular he proved far reaching theorems about the existence and unique[122]. ness of tubular neighborhoods. In this section we will explain and proof the results of J. MATHER. Moreover, we will supplement these results by "curvature moderate versions". But before we will come to this let us provide some necessary terminology, J. MATHER has In
which has been used in
3.5.1 section:
always
As
[122]
prerequisites assume to be given the same objects like in the preceding E N U fool, S is a submanifold of M; moreover let T, To, T, and so on tubular neighborhoods of S in M.
be
For every subset U C S
we
well.
m
hood T to U that If T
as
=
(E,
F-,
understand
means
y) by
Z, p)
the restriction of the tubular
neighbor-
01F-junTI,,)S
neighborhoods of S in M, then morphism of tubular neighborhoods from T to f (of class C )
and a
denote by Tlu triple (EJU, EJU,
we
the
are
two tubular
..
118
Control Theory
pair (*, 5), where
the
C'),
5
S
:
--)
RIO
a
E
is
isometric
an
V' -function such that 5
0
D, D,
d(x, aS)d
such that for all
j
J,
E
i
by (CM7),
=
there exist constants
< D
1+
supp, j
E
X
d(x, aS)d
n B n
one can
constants
D,
representation (3.5.10) there exist
on
the other hand constants d' E N
0 with
d(x, aS)d' D'
S
open
the relation
(TB2)
space, and S E 8 a
--
S is
=
one
of its strata.
(Ts, 7rS, PS) satisfying
for every other stratum R E 8
-
a
continuous retraction of S such that for every stratum R > S
the restriction 7rs,R
:=
7rSITs,R
:
Ts,R
---
S is smooth.
126
Control
(TB3)
ps
:
Ts
R :' is
--i
continuous
a
mapping
such that
ps-1 (0)
such that for every stratum R > S the restriction PS,R is smooth.
(TB4)
The
mapping (7tS,R, PS,R)
:
Ts,R
S
--
R>0 is
X
a
:=
=
Theory
S is satisfied and
PS]Ts,R
:
Ts,R
-
R o
submersion for every pair of
strata R > S.
3.6.2 Lemma If the stratum S possesses
S is
surjective
PROOF:
That 7rS,R is
tube
compatible with 8,
surjective follows easily from the fact that
The statement about the dimension is
3.6.3
a
then 7rsp,
:
TS,R
--4
and dim R > dim S for all R > S.
Example
Let X be
that X inherits from M
a
locally compact
stratified
subspace
C"' -structure and let TS be
a
stratum S of class E!'. In
X is
a
of
K and (TBI).
S C
immediate consequence of
an
a
(T134).
E]
manifold M such
tubular
neighborhood of the triple (TS, 7rS, ps) defines of S in X, which we denote
Whitney according to Proposition 3.4.1, after shrinking TS, a tube by the letter T. In this case we say that the tube is of class C' and that it is induced by a tubular neighborhood or that it is normal of class C'. Using the symbol T both for the tubular neighborhood of S in M and the tube of S in X does in general not lead to any confusion but rather simplifies notation. case
If the stratified space
Ts
(Ts, 7rs, ps)
=
(T135)
that
carries
normal of class
There exists and ps
(X, 8)
stratified the
12',
that
of order m, if the
TS
is induced
is
a
a
by
X
=
X
-
X
=
PSIU2
are
-
t1u,
(TB7)
if
we
call
a
tube
7rS
Tx.
we
call
TS
curvature moderate
holds:
covering of S by singular charts x : U --4 Rn of X such that TS neighborhoods T' of x(S n U) in Rn and such that every
tubular
TS and
ts
thefollowing
There exists
7tSIU
P-1-mappings.
Tx is curvature moderate of order Two tubes
o
normal tube of S of class C!'
following
There exists
o
projection and p' the tubular function of
the functions 7rS and ps
Assuming
Tlu
then
means
where 7e' is the
(TB6)
(!',
axiom is satisfied:
covering of S by singular charts x : U -- Rn of X such that by tubular neighborhoods T' of x(S n U) in Rn of class
a
PX
case
following
induced
are
e
In this
smooth structure
a
if the
a
of S
are
m.
called
equivalent
over
the set U C
holds:
neighborhood TU
C
Ts
n
7rS ITL'i
=
PS ITLI
=
rts
of
ftS JTU'
PS ITU
2
-
U, such that
S,
in
symbols
127
3.6 Ilzbes and control data
One checks
easily that
the
of tubes is
equivalence
equivalence relation
an
on
the set
of all tubes of S in X indeed.
Ts
stratified space (X, 8) consist of a family js)scs of tubes such that for every pair of strata R > S and all X E Ts n TR with
Control data for
3.6.4
(Ts, 7rs, Os) P
=
7CR(x)
E
a
Ts the following control conditions
(M)
7tS
(CT2)
PS
are
satisfied. A stratified space
7rR(X)
-
o
which
on
=
nR (.X) some
=
7tS(X), PS N
control data exist is called
a
controllable
space.
(X, 8)
If
carries
additionally
normal, then (Ts)SEs data
are
called normal control data of class C'. Some normal control
called curvature moderate of order m, if any two strata of X are situated and if for every stratum S the corresponding tube TS is curvature
(TS)SEs
regularly
are
C" -structure and if all tubes of the control data
a
are
moderate of order
Tn.
(TS)SEs
Two families of control data
and
(ts)sEs
of
a
stratified space
(X, 8)
are
ts are equivalent over if for every stratum S E 8 the tubes TS S. A stratified space (X, 8) together with an equivalence class of control data is called
called
a
and
equivalent,
controlled
(stratified)
space, the
corresponding equivalence
class
a
control structure
for X. A
morphism
map that
between controlled spaces X and Y is given continuous mapping f : X -4 Y which is
means a
spaces and for which control data
(Ts)SEs
and
(TR)RE9Z
of
by a so-called controlled morphism of stratified
a
(X, 8)
resp.
(Y, JZ)
exist with
following properties: For every connected component So of a stratum S E 8 the relation f (Tsj C TR,0 holds, where TS0 := 7ts-1 (So) and Rso is the stratum of (Y, 9Z) with f (So) C Rs,,, and for all x E Ts,, the following conditions are satisfied: the
f
(CT3)
o
7tS(X)
Ps('X)
(CT4)
=
=
7%
PRS
"
o
f(X),
f(X)-
appropriate control data fulfill only condition (CT3), then f is called weakly controlled. If each of the restrictions f1so : So -4 RS0 of a controlled mapping f : X --
If f and
Y is submersive
(resp.
controlled
(resp. immersive), immersion).
then
one
says that f is
a
controlled submersion
Controlled spaces together with the controlled maps as morphisms form a category (ESP,j,. By associating to every manifold M the trivial control data consisting of the single tube Tm (M, idm, 0), smooth manifolds and e'-mappings form a full subcategory of (Esp,t,. If f : X --- M comprises a stratified mapping from X into the manifold M with the canonical stratification then some control data (Ts)scs are called compatible with f, =
if'for every stratum S and all
(CT5)
x
E
TS f
-
7tS (X)
=
f(X).
128
Control Theory
Hence the
mapping f
is
compatible
with
(Ts)sc:8,
if and
only
if it is
a
morphism of
controlled spaces from X to M. 3.6.5
Let M be
Example
Choose
a
smooth collar k: R
a
smooth bounded manifold and R
[0, 1[--+
x
for M with its natural stratification into the sets M' and define 7rR and PR
as
the
with Tm-
Together
M',
=
7rm.
=
-
boundary.
M \ R and R. Just set TR
=
uniquely determined functions k
aM its
=
U C M for M. Then k induces control data
(7 R) PR)
=
idm. and
=
U
satisfying
idu.
pm.
=
0
we
thus obtain normal control
data of class C' for M. 3.6.6
Example
into
manifold
a
from X in
a
Let f
M,
:
X
M be
--
and let ij be
a
controlled submersion of
a
a
controlled space X
point of M. Then the fiber X,,
natural way the structure of
=
f-'(1j)
inherits
controlled space. By assumption f is a hence for every stratum S of X the intersection Si. f-1 (-Y) n S
controlled
a
submersion, S, so the family (SY)SE8 comprises a decomposition and thus a stratification of X.- Moreover, after choosing control data (Ts)sEs of X compatible with f the family (Ts,,)SES of restricted tubes Ts.,, (Ts n XIj) 7 SIT,nx,, PSITsmy) has to comprise control data for X... is
=
submanifold of
a
=
3.6.7
Proposition
(Ts)sc,s
(1) IfR,S
(2)
a
controllable space.
point
7til(U)
x
of a stratum S there exists
For every
(4)
For every stratum S there exists =
Moreover,
(5)
pair of strata R
S the relation
TS and such that (7rs, ps) such control data
connected. If S 1 is
(6)
>
satisfy
:
7cs-'(SI)
are
a
x
an open neighborhood U C only finitely many tubes TR,
E
TR
n
Ts implies
smooth function es
a
Ts
the
For every connected component
and
R=S orR>S holds.
has nonempty intersection with
(3)
T's S
Then there exist control data
following properties:
andTsnTR:AO2 thenR<S,
ES
For every
that
Let X be
of X with the
[0,
--
es [ is
following
So of a
a
Ts
--4
R > S.
Ts.
R>' such that
surJective mapping.
relations:
preimage 7ts-1 (So) is path disjoint to So, then 7tS-1 (SO)
stratum S the
second connected component of S
disjoint
:
proper
7rR (x) E
S such
well.
as
For every pair ofstrata R > S the map
(7 S,R, PS,R) : Ts,R --410, F-s [ is a differentiable
fibration. Control data
satisfying conditions (1)
to
(4)
in the
proposition
are
called proper control
data. PROOF:
We first show
of the strata of
X,
and
(1) by an induction argument.
(T-1 S )SE8
k > -1 has been constructed
Let
(Sk)kEN be a denumeration
k with Suppose that (T S)SES by appropriately shrinking (T S1)SE,3 and that for all some
control data for X.
129
3.6 7bbes and control data k
with S 1 the relation
comparable
strata S 1 with I < k and all strata R not
k
Ts, n T.
0
locally compact and paracompact there exists a locally finite covering Of Sk+i by sets open in X such that U,, is compact and has only nonempty (Un)nEN k intersection with strata R > Sk+j and R < Sk+1. Then Tk", Ts,+1 n UnEN U" is Sk+
holds. As X is
=
\ A contains UnEN Un neighborhood of Sk+,. Moreover, k+1 Tk for I < k and T set let us all strata not comparable with Sk+1. Therefore St S1 and Tk+1 pkS appropriately we thus T Ski \ A for 1 > k. Restricting the functions 7tsk S1 k+1 obtain new control data (TS )SES which satisfy the above induction assumptions for an
A
open
is closed and X
=
=
=
family (Ts)sEg where Ts
The
k + 1.
satisfy (1). By further restricting the
tubes
=
TkS for S
Sk then
=
to Lemma 3.1.2
according
control data which
are
immediately
we
obtain
following we will further shrink procedure properties (1) and (4)
In the
control data (Ts)s,,-8 which fulfill (4). (TS)SE,S by shrinking the functions es. By
some
this
remain true.
Now let
that
(2)
exists the
a
(2).
to
us come
there exists
topology
As X is locally compact and has a countable basis of its on X compatible with the topology. Let us assume
metric d
a
does not hold for any control data obtained by shrinking (TS)SE'S. Then there S, a point x E S and a sequence of pairwise different strata R,' with
stratum
neighborhoods (Un)"EN
For every basis of
following property:
of
in S and all
X
R>0 and 5,, : S --) R>0 there exists a point sequences of smooth functions " : R,, -4 that all the sets U,' are relatively compact We assume can n E (u,,) n7ts-1 T6'Lj,, TrS -
Rn
in S and
thatnnEN Un
=
Jx).
Then
according
to Lemma 3.1.2
choose the functions
we
F-n and 6n such that
1 7r F1 (R n):=fIJET Rn I d(ij, Rn (10)
c B
TRI:n
By definition TIn S n7ts-I (u-,,) converges to
of the 5n and Un and form
a
B,(x)
Choose N E N
metric d.
2n
to the
proof
basis of neighborhoods of x in X that
Now let
x.
according
so
the ball of radius that
large
1
< 1 2
2n
T >
0 around
and 1Jn E
to
control data which
some
(3)
we
again
we
denote
(Ij n) "EN
with respect to the for all
>
n
N.
By
2
n >
N that
Hence
by (TS) SE'S
first suppose that the control data
(S).
the sequence x
Bijx)
definition Of 1Jn and En then 7rR.(lJn) E B,(x) meets infinitely many different strata which cannot be true.
To prove
1
In-
of Lemma 3.1.2 the sets
means
follows for
(TS) SE8
T In C B S
and
S there exist smooth functions
R` such that the relation
help of
an
inductive argument
KR
E TR
x
n
Th S
entails
construct from
we
5R
nR(x)
(Ts)SES
S
:
E
-4
R>0 and
Ts.
control data satis-
the union of all k-dimensional strata of X and
by Sk fying (3). call by slight abuse of language Sk a stratum of X as well. We now suppose that after shrinking (Ts) SES property (3) holds for all strata S, with 1 < k and every stratum R > S1. Then we choose a locally finite open covering (Un)nEN Of S Sk+j by in S for that such sets CC and finitely Un only S, many U. the relatively compact open To this end
we
denote
=
intersection tition of
7rs-1 (Un)
unity
n
7 s_l (Un)
is nonempty. Let further
of S subordinate to
(Un)nEN
.
(W n) nEN
be
.
a
smooth par-
Then choose for R > S smooth functions
130
5R
Control S
:
R" and
--4
the intersection
R
:
KR
Ts
RIO according to
--
TRn7rs-I (U,,)
n
there exists smooth functions
Ts
5,, =,A 0. Then let
TRn7t,-'(U,,)
n
d,,
f 5,, (X) I
inf
=
S
:
i
-
After
possibly shrinking the U,' only for finitely many strata R, hence
is nonempty
WO with 0
0
is true for all
Hence 5
n.
0
>
follows,
1
and R
with7ts- (U,,)
and
by
definition
TR :A 0 and U,,
6IUn
(5-)]Un
S
exist, every tube Ts will be restricted only finitely many times that means we finally obtain by this procedure control data satisfying (3). But it remains to show (3)'. More precisely we will prove that for some control data (TS)SE8 fulfilling (1) and (4) the relation (3)' already holds. Let R > S be two strata X. Then we choose a smooth function SR : S -- RIO with6R O
-)
of
TS
(6)
E
x
,
a
note that W is
U is
a
KII TR nT 6R S
will not
point
x
of
UjER 7rR-'(Uu)
ETs, which
P-
m+
to
S and which is curvature moderate of order
the curvature moderate
a
case.
curvature moderate
This finishes the induction step with tubular neighborhood To of So, which satisfies (CTo) for
all R
so
(dim(S) )2 +2(1+1)-1,
+ 21.
respect
0 a vector field 0, X' with the desired properties. By possibly shrinking the tubes of the strata =
the claim is trivial. So let
S of dimension < k
achieve that the control conditions
one can
Vk and all pairs of strata S
< R
with dim R < k
are
(CT8)
satisfied. After
and
(CT9)
for
possibly shrinking
136
Control
the tubes
again
achieve
we can
according
Proposition
to
3.6.7 that for all
Theory pairs of
strata R < S the relation
7TR(Ts is true, and that for any two not
disjoint. Moreover,
are
es
S
:
--)
TR)
n
comparable
TR
C
strata S and
shrink the tubes such that for
we
R" the mappings
(7rs, ps)
:
T's s
[0,
--
es [
of X the tubes
appropriate
are
Ts and T
smooth functions
proper, and
Ts
T's holds s
=
true.
Let
suppose that for every
us
vector field
Vs : S
-)
control conditions
are
(k + I)-dimensional stratum
such that for all R
X
-oo)
the restricted map
(bzw. yjj.,,jx ]t.,01
"
integral flows by
fxj x I tx-, 01
:
-4
X)
proper.
First fix
conditions
X
(resp. t,-
< oo
a
x
uniquely determined maximal stratified integral
a
flow -y : j -- X of V. Moreover, -y is determined among all stratified the following property:
(IF)m,x
calls
V, if for every stratum S the curves 'Y,,, (S x R) the condition (IF) is satisfied.
[122, Prop. 10.11)
MATHER
X, then there
stratified space X is called
on a
element of the stratum S
an
flow -y j -4 X an integral flow of class C' and if for all (x, t) E j n
3.7.6 Theorem
X
-4
is
x
Theory
(CT8)
proper control data
some
(CT9)
and
are
satisfied.
X and for V such that the control
on
By assumption on V the restriction Vjs By the classical existence and unique-
smooth vector field for every stratum S. theorem
maximal
on
the
integral
integrability : Js -- S
of vector fields there exists
flow ys
U
j
of
VIS.
and
Js
We
j
-y:
-4
a
uniquely determined
set
now
X,
-ylj, :=,ys,
S ES
and claim that -y is that
(IF)max
maximal
a
holds. Under the
integral flow of V, that it is determined uniquely and assumption that J is open and that the continuity of -Y
has been
proved, it follows by definition that -y comprises a stratified integral flow integral flow is maximal, as for every integral flow - > -y
V. This
;Y_ijn(sxR) !'Yijn(sxR) which
, y-
=
by
the
maximality of -ys implies analogous argument
-y follows. An
To prove
(IF)max,
that
3.7.7 Lemma Let
J is
be
x
a
point of the
es
(-yj (t))
Then the
(CT11)
tj
for Ij E K and t E
mapping -y
max(t , ti)
S. As R, there exists a sufficiently small to > 0, such that (CT11) is satisfied with Itl < to. By the control conditions (CT8) and (CT9) and after
OF THE
flow
a
on
for all t E R
a possibly smaller to the conditions (CT12) and (CT13) are satisfied for T's --4 [0, es [ is Consequently -yj (to) has to be in TS, as (7rs, p.,) : Ts S But this implies that (CT12) and (CT13) have to be true for t to as well. flow property Of 'YR the relation y-u (to + S) E Ts then holds for sufficiently > 0. Moreover,
transition to
Itl
to.
2. geometrically flat with respect to a smooth Riemannian metric L, then f is geometrically flat with respect to every other smooth Riemannian metric 71 on X. The reason is that L and -q are locally equivalent and are both smooth. Hence the notion of geometrically flatness at the boundary is independent of the special choice of a 3.8.2 Remark Recall that
=
=
=
If f is
smooth metric Now
on
to
we come
X.
Extension Theorem Let ME
3.8.3
Riemannian metric erate of order
by
main theorem.
our
m
If then
i.
Cm(S) Cm(X) such
X. Let S be
an
N, X an (A) -stratified space, (T, I) -regular stratum which is
CM(T, t) S
>
tion g E
which is
f E
that
is sufficiently large, then one geometrically flat at the boundary
fIT5
g
=
S
6
Hereby
:
S
-4
R` is
We
PROOF:
and 1i
a
smooth
curvature mod-
3, and let TS be the orthogonal projection of the tube of S induced
+
c
on
an
extend every func-
can
of order
c
to
a
function
7tSIT5S
o
appropriate (smooth) function.
suppose that X is closed in
an open bounded set 0 C R; the given in the following with the help of a smooth partition of unity. According to Proposition 3.3.14 and the assumptions on S the components of the projection 7r := 7rS are tempered of class em. Finally recall that the restriction of L defines a Riemannian metric on S; the Porresponding LeviCivita connection will be denoted by V. Under these assumptions let us consider the jet G on 9 which over S is induced by the smooth function g 7r and which vanishes on the boundary aS. To prove that G is even a Whitney function we only have to show according to the generalized lemma of Hest6n6s 1.7.2 that G is flat of order cm(T, be arbitrary Vk, t) over the boundary B. To this end let V1, V2, S constant vector fields Vk : T -4 R, in other words this means that all derivatives DVk vanish. As the functions T7r.Vk comprise vector fields along 7C, the covariant derivative VV,T7r.Vk along 7t in direction Vj is defined (see KLINGENBERG [101, Prop. 1.5.5]). Using local coordinates on S and the corresponding Christoffel symbols one checks that the following equalities hold:
general
can
case can
be reduced to the statement
-
* (g
-
7r).Vi
=
(dg, TmVi),
* 2(g
o
7r
*3 (g
.
7r). (V1' V2, V3)
).(V1 'V2 )
+
Analogously
D (dg,
=
=
2
concludes D
k
1
2)
(D T7r.V
+
(dg, VVIT7t-V2)
(V3 g' T7r.V1(2) T7r.V2(g T7r.V3)+ (dg, VV,VVT7t-V3)
(V2g, VV,T7.r.V
one now
(,V2g, T7ry
T7r.V2).Vi
(g T7r.V3 + T7T.V2 (9
by
an
(g 7r). (V1 .
VV1T7r'V3 + T7r'Vl
(D
induction argument that for k
R >o such
that t I
T'('U)
,r(x) >
From
now on
Out of H " over
the function obtain
we
the open set U
P
-4
=
H(y, t)
with
(1j, t)
E
Q'x 10, 3/21,
if
x
=
H'(-Lj, t)
with
(1j, t)
E
Q
if
x
Q'x [0, 2[-- X, (ij, t)
H " (Q'x
x
13/2, 2
H'(Q'x 10, 2[),
vector field W
a
:=
I
U
:
10, 3/2 [).
H'(1J-, tT'(lj)) H (y, t) TX, x -4
"
-
=
The
integral
will be denoted i-->
i)s
H
"
by H".
(1j, t + s) I s=0,
flow of W will be denoted
J(x,s) EUxRj x=H"(-y,t)
and O<s+t
6
1-dimensional submanifold of
Q i n Qj, i, j
fact that
1,
i[j,
=
1, 2, 3
H"(Q'x 1) 3
all open in
are
the definition of the functions
finally the
6
T
and
5,
the
and S' intersect
X',
Qj.
as
its
For the
one
of the
transversally
in B.
6. STEP: End of the induction step, construction of F.
Hi : Qix]0,11 -- X, i 1,2,3 with the desired EQ, set Hl(,Lj, t) Qu, 13 1). If -Li H"(G(b, C2(S)), Cl(S)) with b E B and s E10, 1[ letH2 (IJ, t) H"(G(b,W2(S,t)),Wj(s,t)). Finally define H3 (1J) t)= -y(ij, 13 1) for'y E Q3- One now checks easily that the functions Hi and Hj coincide on the intersection (Q i n Qj) x 10, 11. Hereby one has to use the fact that the integral flows -y and commute. Altogether we thus obtain a smooth function H : Q x ] 0, 1] -- X the restriction of which to Q x] 0, 1 [ has to be a diffeomorphism onto its image. Using the commutativity of the flows -y and again, one realizes that H can be extended to a continuous function H : Q x [0, 11 -1 X and that then the 0 properties (1) to (4) hold true.
We will
provide
properties.
three functions
For -Lj
=
=
=
-
-
=
3.10 Cone spaces
147
Applying the theorem the following result.
and
using
a
simple gluing argument
one
checks
immediately
Corollary Let X be a controlled stratified space and S a stratum of dimension :A 0. Then there exist a d-dimensional manifold-with-boundary M and a continuous mapping f : M --i X such that the following properties hold proper 3.9.6
d with aS
equality f (M')
(1)
The
(2)
f (M)
A proper continuous
3.9.7
Let
a
S holds true, and
more
we
call
be called
generally a
proper
tempered resolution
(RTI) f1m.
:
f1m.
is
a
smooth
embedding.
aS.
=
corollary will Then
=
M'
-4
a
mapping f
:
M
-4
X for
a
stratum S of X like in the
resolution of S.
X be
an
(A)-stratified
space of class C,' and S C X
a
stratum.
C-mapping f : M --19, where M is a manifold-with-boundary, of class C', if the following properties hold:
S is
a
(!I-diffeomorphism.
i be a smooth Riemannian metric on X and L : M -4 R' a proper T,,, of the composition T := L. f1m. -1 : embedding. Then the components T1, S -4 RI are geometrically tempered of class C!' that means for every Ij (=- aS there exist a neighborhood V and constants C E N and C > 0 such that for all
(M)
Let
-
-
-
,
k
singular chart Whitney's condition (B).
We consider the
space L c
CL is
every manifold-with-
as
can
technically more can prove that every polyhedron X with the smooth triangulation h: X - Rn is a cone space of class 'C"O. a
well
show without any difficulties. involved but nevertheless canonical consideration as
a
a
following
a
situation: Assume to be
stratum S' c
cone
2 is
space of
L, an depth
a
fixed
Whitney space hence
given
a
open ball B C Rn and
d + 1.
by
By one
Moreover,
compact a
B =2=
cone
point x E B. B x 101 and
149
3.10 Cone spaces
S B.
B
:=
(10, 1 [. S')
x
According
are
R'+I+' is
E
point of X
a
So let
points of S with liMk-4oo'lJk converges in projective space of
in the form ij 1,
uniquely
=
x, and
to
R'+'+'. Now
with x'k E
1)
k
we can assume
Whitney's,
(IJ k) kEN
X5
a
that the sequence of secants
assume
line f C
a
(Xk) tk
=
subsequences
transition to
in the stratum
to Lemma 1.4.4 and Remark 1.4.6 it suffices to prove that
(B) holds for the pair (B, S) at the point x E B. (Xk) kEN be a sequence of points of B with liMk--)oo Xk
condition
some'y'
(x, 0)
strata of X and
we can
B, tk
10, 1 [
E
sequence
fk
=
Xk-yk
represent every 1Jk
and
that the sequence of the
-Y'k E S'. After -y.' converges to
S' and that the sequence of normed difference vectors
E
(Xk
I
-
I
Xk) -tk
*
Vk )
V_IX k _,X1k 112 + t2k converges to
a
,r
tk
=
Rn
converges to
a
Tj, S'). By
case w
=
0
a
subspaces
further transition to
linlkloq follows,
Cone metries In the
given
to metric
11Xk-k,11
holds true
=oo
in the second
w
to
one can
or T :=
tk
11(t,x)
provides is called
geometric analysis
a
and the
f,
subspace achieve, as
a
liMk--)oo
I 1-k -k, I I tk
Hence in both
T+1
E
cases
=
dt2
ED
t2
canonical Riemannian metric
a
over
cone
and 71 its
-q and
so on.
such
be constructed
cone
space that
(t,x)
jx,
metric
a cone
singular spaces particular attenfollowing shape. Let us be given
Riemannian metric R. Then
a
the definition of
of
These spaces have the
cones.
compact manifold M with
metric
spans the line
(v, w)
Then
E r resp. f C r is true. This proves the claim.
3.10.5 tion is
+
that either
0,
R -'. In the first
(v, w)
R1+1.
E Rn x
tangent spaces T,,,S converges in the GraBmannian
(span -Lj'
x
(v, W)
vector
sequence of the
the stratified
on
metric
means we
build
A Riemannian metric
on
we
continue
according
the
cone
iterative processes will be called
pair (C M, -q) analogously to
C M. The
cone
R. Now
over
(3.10.1)
CM\fo}
E
to
Eq.
3.10.1 the
space X which
cone
locally
can
metric for X.
an by Considering the interesting results already obtained for metric cones the study of more general cone metrics appears promising, though one can expect it to be rather involving. We close this section with several historical remarks which essentially are taken form LESCH [111]. The study of metric cones was initiated by CHEEGER [38, 39, 41],
differential operator of order 1 and 2
on
a cone
such spaces have been considered among
by BRUNING-SEELEY [32, 33, 34]. MELROSE [126] and SCHULZE [148, 149] have introduced independently an important class of differential operators on metric others
cones:
the so-called operators of Fuchs type. These
are
differential operators of the
form a
t-n
E Ak (t)
(_ ) t
at
,
k=O
where m,
n
G
Mo and the Ak
are
smooth families of differential operators
on
M. The
detailed exposition of the theory of Fuchs type operators together with further information on metric cones and many references in LESCH [111]. reader
can
find
a
Chapter Orbit
As
already
4
Spaces
mentioned in the
introduction, orbit spaces of certain, or more precisely give nice examples of stratified spaces with smooth structure. Moreover, play an important role for many considerations in mathematics and mathematical physics. For this reason, they will be treated here in rather detail, where in accordance with the scope of this monograph attention is given primarily on the canonical stratification by orbit types and the construction of the smooth structure. The results of the first three sections of this chapter are standard, at least for the compact case. Thus, we have formulated them from the beginning in the greatest possible generality not only for the case of compact G's but also for the case of proper G-actions. As references for Sections 4.1 to 4.3 serve in particular J,KNICH [95], BREDON [25] and LESCH [110]. Concerning the canonical stratification of an orbit space original references are BIERSTONE [13, 14], SJAMAAR-LERMAN proper G-actions of
[162],
a
Lie group G orbit spaces
DOVERMANN-SCHULTZ
[52,
and FERRAROTTI
[56].
Differentiable G-Manifolds
4.1
Let M be
4.1.1
smooth
a
manifold and G
such that for all e
a
Lie group.
By
a
(left)
action of G
we mean a
mapping
0:GxM- M,
hold,
671
p.
being
(g,x) -4(D(g,x)=(Dg(x)=gx
g,h E G and x E Mthe relations (D,((Dh(,x)) (Dgh(x) and (D,(x) x identity element of G. By a right action of G we mean a smooth =
=
the
mapping 'IF: M
x
G
--)
M,
(x, g) "Y(x, g)
=
IF, (x)
=
xg,
M, (g, x) --4 'F(x, g-1) describes a left action of G. together with a G-action (D : G x M -- M a differentiable G-manifold or shorter a G-space. A left or right action of G on M is said to be transitive provided that for all pairs (%, 1j) of points of M there exists a g E G with gx -y and xg -Lj, respectively. The G-action is called effective or faithful, if the relation 0. idm respectively 'Tg idm is fulfilled, if and only if g e. In other words, this means that the canonical homomorphism of G into the group of diffeomorphisms Diff (M) is injective. such that (D
:
G
x
We often call
M
a
-
manifold M
=
=
=
M.J. Pflaum: LNM 1768, pp. 151 - 168, 2001 © Springer-Verlag Berlin Heidelberg 2001
=
=
152
Orbit
A
morphism of
Spaces
G-actions
or a G-equivariant mapping is a differentiable mapping G-spaces M and N such that for all g E G and x E M the equation f (gx) gf (x) is satisfied. Now, if -y : G -4 H denotes a smooth homomorphism of Lie groups, we call a smooth mapping f : M --i N from a G-space M into an H-space N -y-equivariant, if the diagram
f
:
N between
M
f
G
x
M-Yx--H
1
x
N
(4.1.1)
I f
M-N commutes. The
G-equivariance
is therefore
equivalent
to the
idG-equivariance.
apointxE Mtheset Gx=fgxE MJg E Glissaidto bethe orbitof xin partition of M into its various orbits then describes an equivalence relation on M; we call the corresponding quotient space of equivalence classes the orbit space of M, denoting it by G\M. In an analogous way one defines for a manifold N with a right action of G the orbits qG with q E N and the orbit space N/G. Next, we equip G\M (resp. N/G) with the quotient topology with respect to the canonical projection 7r : M -4 G\M (resp. 7r : N -- N/G). This makes 7r into a continuous 4.1.2
For
M. The
and open
mapping,
for all U C M open
as
7r-1(7c(U))
=
Ug(=-G 9U
is open in M.
Usually, the orbit space G\M is not a differentiable manifold, sometimes not even Hausdorff. For a relatively large and most applications sufficient class of G-manifolds -
namely
those with
so-called proper G-action
a
-
the orbit space
G\M
possesses the
structure of
a Whitney space. In the next sections we will explain this in more detail and introduce in this paragraph the new notions necessary for this purpose. In the following, if not otherwise mentioned, definitions and results will be given explicitly
only for
the
case
of left
actions, tacitly assuming that these hold in the "right"
case,
too.
4.1.3
by G,
For each =
Ig
C-
point
GJgx
E M
x
=
xJ.
for all g E G the relation groups of two
define its
One
Gg.,,
=
points of
easily
isotropy
group
checks that
gG,.g-1 orbit
or
stabilizer
G-, is
holds. In other
or
subgroup words, this a
symmetry of
G,
means
group
and that that the
conjugate to each other. Consequently, to each orbit there is a uniquely assigned conjugacy class, namely the conjugacy class (G,,) of the isotropy group Gj of an arbitrary point'y E Gx- In the following (G1.) will be called the type of the orbit Gx. A G-action of M is said to be free, if all the isotropy groups G.,, are trivial in the sense of being equal to {e}. Every free group action is effective, for using the fact that (D is free it follows immediately from 09 idm that e. Conversely, not every effective G-action needs to be free. g To every closed subset H C G one assigns the following three subspaces of M: isotropy
an
are
=
=
Mij
MH
M(H)
:=
JX E MI fx E MI JX E MI
G.,,
=
Gx
D
G,
-
M" then describes nothing else but the fixed point
HJ, HJ, HJ. set of H in M.
4.2
153
Proper Group Actions
Differentiating
4.1.4
obtains
one
a
G-action (D
a
G-action
:
G
(g,v)t-4gv=T(D,(v).
(D with respect to the first variable
Conversely, differentiating element & E g of the Lie
fundamental
The G-action (D bundle
also
-
bundles and
G
x
T*M
&m(x)
:=
a
at
(1) (exp
(g, txx)
-4
t&, x) -
bundle,
is
on
obtains for every
M,
x
E M.
apart from the G-action on
D V F-4
the so-called
given by
L01
tensor and exterior
completeness
(TgxM
one
canonical vector field
functorial way
For the sake of
T*M,
a
Explicitly, &m
the cotangent
one on
so on.
-)
of
in
of G
algebra
field &M
yields
M with respect to the second variable
-4
the tangent bundle of M:
on
GxTM--iTM,
vector
M
x
we
give
on
products of
here the action
(0(%, g-lv)),
the tangent
oc,, E
on
these
T*M:
T-*M,
X
E M.
a G-manifold, hence it makes sense to speak M; this is then a differential form a E Q"(M) that such that oc,,, g oc, for all x E M and g E G. If one finally requires additionally 0 the contraction of oc by each fundamental vector field &M vanishes, i.e. that i&m OC holds for every & E g, then oc is said to be a basic differential form. The space of basic
Now the bundle of exterior forms becomes
of
a
G-invariant
differential form
on
=
=
k-forms
on
The basic differential by Q',,ijG\M). b computation of the cohomology of G\M (see 5.3).
M is denoted
used for the
forms
on
M
can
be
Proper Group Actions
4.2
4.2.1 Definition A G-action 4)
(De.t
G
:
:
M
x
G
--i
x
M
M
x
--
M is called proper if the
M,
(g, x)
1-4
mapping
(gx, x)
is proper
4.2.2
Example
For
a
compact Lie group G all G-actions
are
obviously
proper.
Example One might think that all free G-actions are proper. This is, however, S' x S' with the case as shown by the following action of R on the torus irrational angle 0C E R/27rZ:
4.2.3
not the
RxS'xS'-4S'xS',
(r,e i27rseU?rt)
_4
(e i27t(s+T
cos
'), ei27r(t+T sin oc)
following theorem is aside from the slice theorem proved later point of all further investigations concerning proper group actions. The
4.2.4 Theorem Let (D
holds:
-
:
G
x
M
--
M be
a
on
-
the
proper group action. Then the
starting
following
154
(1)
Orbit Spaces Each orbit canonical
Gx, x E mapping
M describes
a
closed submanifold of M.
(Dx: G/G-,
yields
a
difteomorphism
from
M,
G/Gx
g
Gx
-)
Moreover,
the
gx
onto the orbit Gx.
(2)
The
(3)
The canonical projection 7t : M -4 G\M is closed. The orbit space G\M Hausdorff, locally compact and endowed with a countable topology.
(4)
isotropy
x
E M is
compact.
To any
covering of M by G -invariant open sets there unity by G-invariant smooth functions.
of
(5)
The
(6)
M admits
algebra
PROOF:
Gx
Gx of any point
group
=
fg
begin with, we first GI gx x1 in the form
show
(2).
It is
possible
to write the
isotropy
group
=
it is therefore
Let
subordinate partition
G-invariant Riemannian metric.
a
Gx
the proper
a
Coo (M)G of G-invariant smooth functions separates the points of M.
To
E
exists
is
=
pr, (0-1 ext (x,
x));
compact since it is the inverse image of the compact
set
(x, x)
under
mapping (D,,,t.
prove (1). Since G, is compact, hence a Lie subgroup of G, G/G.' needs (real analytic) manifold. We first show 0, to be an injective immersion. The hx it follows immediately g-1h E G-x, hence injectivity is obvious, since from gx to be
us now
a
=
hGx. To show that (D-, is immersive it suffices to prove that the differential gGx T,Gx(Dx is injective, since (Dx is equivariant with respect to the G-action G x G/G." G/Gx. So, let V E TGxG/G,, be a tangent vector with TGx(Dx.v 0. Because of =
=
the fact that the canonical E g
=
TG with T,7r.F,
=
projection
v.
curve
a -
at
(D (-y (t),
x)
-y(t)
exp
=
It=.'
at
-)
TeG,,(Dx-Te7r-&
=
G/Gx
is
submersive,
there exists
a
=
TeG,.(Dx-V
=
0)
t&
a -
G
Then it follows
Te(D(-tX)-& and for the
7r :
(D (-y(t +
s), x)
I
(T.(D (-y(s), -)
-
TA (-, x))
(&)
=
0,
t=O
using the fact that 4)(-y(t + s),x) (D('y(s),-y(t)x). The result is (D(-y(t),x) x for R, or in other words y(t) E Gx- This implies E T, Gx, hence v 0. T,7t.& Consequently, (D., is imme'rsive. Since the mapping G x fxJ -- M x JxJ is proper, this also holds for (Dx. Regarded as an injective immersion, (D-, is therefore an embedding, =
=
all t E
hence
=
a diffeomorphism Next, we show (3). (Dext (G, A). needs to be
GA
=
orbits
onto its
image
=
Gx.
Let A C M be closed.
Since (Det is proper, GA x A M, hence GA is closed in M. Because of 7r-1(7r(A)), 7r(A) is closed in G\M, i.e. 7t is closed. Consider now two different Gx and Gij. M is normal, Gij closed, and therefore there exist two disjoint open closed in M
x
4.2
Proper Group
neighborhoods
7r(ig) open
Actions
U of
x
155
and V of
7r(U).
7r(x)
of
neighborhoods
local compactness and
In
G U.
Due to the fact that
7r
particular,
is closed
7r(U)
this
and
means
that
7r(ij), respectively. Thus, G\M
and
of
separability
follows
G\M
Un Gy
(G\M)\7t(U)
directly
then
=
are
0 and
disjoint
is Hausdorff.
from the
The
corresponding
properties of M.
by (3) the orbit space G\M is paracompact. we assume covering of M by G-invariant open sets U,. By (UL)LEJ the paracompactness of G\M there exists a locally finite covering of G\M by open sets V, such that n-'(V,) C U,. Moreover, there is a locally finite smooth partition of unity (*j)j EN on M and a mapping L : N --) J in such a way that supp *j is compact for all j E N and such that suppipj C 7r-'(V,(j)). Choose now a right invariant Haar measure L on G. By virtue of the hypothesis that the supports supp % are compact, there exists for x E M and j E N the integral proving (4), first
Before
that U
Then
note that
is
=
*39 N
a
=
L *(gx) d t(g).
An easy argument shows that all the ip'P describe smooth functions on M and that G holds. On the other hand, the family of supports (supp EN SUPP *J C 7T-1
*jG)i
(V,(j))
need not be
locally
finite any more,
tion of the functions seminorms
for all
j
11 JJj -
on
ipjG.
a
lack
CI(M)
Since
C-(M) defining
we
is
a
intend to
remedy by
a
suitable
Fr6chet space, there exists
the Fr6chet
topology
such that
a
sequence of
11 JJj -
summa-
-::
11 11j+1 -
E N. We define
E
Cp':=
2i
jEN
L(j)=t
Then
even
7t-'(V,).
the functions
p,
well, G-invariant and satisfy supp Cp, c the covering (VL)LEJ is locally finite the family of
smooth
are
Because of the fact that
supports supp Cp, is locally finite
on
as
its own, hence for all
I
YL(X)
is well defined. Now the
unity subordinate
=
CP (Y-)
CP (X)
with
L
Cp (x)
=
E M
X
E
y, (x)
LEJ
family (Y,)IEJ
is
a
locally finite and G-invariant partition of
to U.
On (5): Let Gx and G-y be two disjoint orbits. Since G\M is paracompact, hence in particular normal, we can choose two open neighborhoods V, and V2 Of 7T(X) and 7r(.Lj), n-'(Vi) for i 1, 2 and U3 M\(GxUGli), we obtain respectively. Setting now Ui A) U2) U3) as a G-invariant open covering of M, with a G-invariant partition of unity ( 01) (P2) Y3) subordinate to it, existing by virtue of the statements just proven. Then 0 holds, meaning that C'(M)G separates I and 02(y-) pi(y) Y1(X) (P2('IJ) the points of M. Finally, we would like to prove (6). To this end, we first choose an arbitrary =
=
=
=
choose for every
suppXj c
Kj'+,
=
=
Riemannian metric 11
on
M and
=
a
compact exhaustion
(Kj)jEN
of M.
Afterwards
smooth cut-off function Xj : M -4 [0, 1] in such a way that j I for all x E Kj. By means of the Haar measure on G and xj(x) E N
a
=
156
Orbit
already
used above
lij W
(V, W)
we
=
define G-invariant smooth sections -qj
L
xj (gx) -q gx (gv,
gw) d. L(g),
X E
:
M
M,
--
Spaces
T*M 0, T*M
by
V,W E TM.
By the assumptions concerning il and the xj all the forms ilj(x) are symmetric and positive semidefinite. If X E GKj, ilj(x) is even positive definite. Since the family
(Uj)jerq (4)
a
with
Uj
=
GKj' describes a G-invariant open covering of M, there exists by ((Pj)jEN subordinate to (Uj)jEN. Define R: M -- T*M 0.,, T*M
partition of unity
by
L(X)
=
E
oj N 11j M,
X E
M.
jEN
This
gives
a
G-invariant Riemannian metric
on
M,
thus proves the last claim of the
theorem.
13
In order to prepare the slice theorem, consider for x E M the normal space TM/T- Gx of the orbit Gx at x, the so-called slice of x. For each element g of isotropy group G,, the differential T(Dg maps the tangent space T-,,Gx of the orbit again into T,,Gx, hence induces an automorphism of Vx. Consequently we obtain
4.2.5
V,,
=
the
Gx
the so-called slice representation
Sx: Gx Since the
homogenous
obtain
associated bundle Nx
space G
--i
GL(V-,).
G/Gx describes a Gx-principal fiber bundle, we Vx, the slice bundle of x. As Gx is compact, there exists on V-, a Gx-invariant metric, with respect to which one can define the sphere SVx fv E Vxj jjvjj 11 and the sphere bundle SNx G XGx SV.,. Then the group G,, acts in a natural way on SV-, such that SN., is well-defined and becomes a differentiable G-space. The slice theorem now states only one thing, namely that every G-manifold with a proper G-action locally looks like a neighborhood of the zero section in the slice an
=
--
=
G XG.,.
=
=
bundle. 4.2.6
be
Slice Theorem
(KoSZUL [104,
p.
139],
PALAIS
[138])
Let (D
:
G
x
M
-)
M
point of M and V, T,,M/T-,Gx the normal space to the orbit of x. Then there exists a G-equivariant diffeomorphism from a G-invariant neighborhood of the zero section of G X G- Vx onto a G-invariant neighborhood of Gx such that the zero section is mapped onto Gx in a canonical way. a
action,
proper group
PROOF:
Since the
x a
=
exponential function of a G-invariant metric
the slice theorem follows
is again G-invariant, immediately from the classical tubular neighborhood theorem
3.1.6.
M
4.2.7 Remark In the literature
borhood V C V,, with
p(U)
element of G and
-
as
o
:
U
in the slice theorem.
G
n
one
(Jej
X G.
x
often calls the
V,j
Vic the
uniquely determined zero neigh{ej x V the slice of x, e being the identity G-equivariant diffeomorphism which emerges =
157
Proper Group Actions
4.2
Corollary For
4.2.8
every
compact subgroup H
E-submanifolds of M. In other words, this
are
MH; M(H) and MH following relation:
is
Of
that each connected component
submanifold of M. Moreover, these three sets fulfill the
a
MH
M(H)
--`
n
MH.
Due to the fact that the statement is
PROOF:
MH, M(H) and MH
G the stes
C
means
(4.2.1) local
a
one
it suffices
by
the slice
.
theorem to consider the
case
that M
G XH
=
Y, where H C point [(g,v)] E
G is compact and Y
G XH Y is G[(g,,)] K-module. Then the isotropy group of a V of v. Indeed, H-manifold the of the H denotes where C isotropy Hv group gH,g-1,
is
an
6[(g,v)] hV
=
=
[(g,v)]
holds if and
only
if there exists
an
h E H with
ggh-'
=
g and
V.
By virtue of the lemma below Hv and consequently G[(g,v)] are conjugate to H, if H, i.e. if v lies in the fixed point space VH C V of H. Using this, only if Hv. G /H x V'. The isotropy differentiable subbundle G x H V` closed the M(H) equals if and if H only obviously, g lies in the normalizer gH,g-1 equals group G[(g,v)l N G (H) of H in G. Consequently, MH needs to be the same as the closed differentiable N G (H) /H x VH, as the normalizer N G (H) is closed in G, submanifold N G (H) X H VH and
=
=
=
=
describing
therefore
of the definitions Of
a
Lie
MH7 M(H)
4.2.9 Lemma Let G be
subgroup Ho
closed
c H
Let g E G be
PROOF:
subgroup
a
as
well
as
Lie group and H C G
conjugate an
of G. The relation
and MH
(4.2.1)
of the
a
is
a
direct consequence
lemma.
following
El
compact subgroup. Then every
to H is identical to H.
element such that
Adg(H)
=
gHg-1
=
Ho. Since Adg is
a
subgroup of H of the same dimension, meaning connected components of the unity of H and Ho agree. From this it follows
diffeomorphism of G, Ho that the
needs to be
a
that for every h E Ho the connected component of h in Ho needs to agree with that of H. Due to the compactness of H and Ho both of them possess only finitely many
connected components and therefore the claim will be given if it can be shown that Ho H and Ho have the same number of connected components. But noting Adg(H) =
11
this is the case, indeed.
4.2.10
Proposition Suppose
left
the manifold P.
on
a
on
the
quotient
space
a proper and free way fr,?m the uniquely determined manifold structure canonical projection 7r: P - G\P turns into
the Lie group G acts in
Then there exists
G\P
such that the
a.
differentiable fiber bundle with typical fiber G.
4.2.11
Note and Definition A fiber bundle P
--)
N
occurring
as
in the
proposition
by means of a proper free left action of G will be denoted as opposite G-principal bundle. Usually the structure group of a principal bundle operates from the right on have chosen the additive
"opposite" to of ordinary Analogously express the structure group acting well: bundles fiber to as associate However, they opposite principal bundles one can arise from manifolds F on which G operates from the right, and will be denoted by
the total space, which is the
reason
why
we
from the left.
FGXP
--4
N.
to the
case
158
Orbit
PROOF:
Since the group action is proper,
we
already
know that
G\P
is
a
Spaces locally
compact Hausdorff space with countable topology. With the help of the slice theorem now local charts for G\P. Let x E G\P and z E P be a point with Due to the fact that the group action is free, there exists a G-equivariant diffeomorphism T (T,,'T2) : U -- G x V from a neighborhood U of z onto a
constructs
one
x
Gz.
=
=
product G
V, V C Vz being a zero neighborhood of the slice to z. Then the map s : 7r(U) --4 V, G-y 1-4 T2 (1j) is well-defined in a neighborhood 7r(U) of x, continuous and a homomorphism onto its image. Any two of those charts of G\P are compatible by virtue of the slice theorem, hence the set of all s : 7r(U) ---4 V, defines a differentiable atlas on G\P. Moreover, the projection 7r: P --4 G\P describes a fiber bundle, since by construction this is the case locally in charts: S 7rJU T-1 : G x V ---) V c V, is nothing else but the projection onto the second coordinate. The differentiable structure of G\P is uniquely determined, since by the fiber x
o
bundle property of be the
7r
the sheaf
7r,,e', P
same as
but
on
CG\P
of
infinitely
the other hand C'
G\P
o
times differentiable functions must
determines the manifold structure.
Stratification of the Orbit Space
4.3
The set of conjugacy classes of closed subgroups of a Lie,group G is ordered, defining (K) :5 (H) as to be equivalent to H being conjugate to a subgroup of K. 4.3.1
4.3.2 Theorem Let (D
:
G
Then the orbit types of (D
(1)
There is
a
x
M
--4
satisfy
M be
the
a
proper group action and
G\M
connected.
following relations:
uniquely determined conjugacy class (H') Moreover, G\M(I-i-) is connected.
in G such that
M(Ho)
C
M
is open and dense.
(2) Every compact subgroup H C M emerging as isotropy group of an x E M fulfills (H) < (H'). In other words, (HO) is maximal in the ordered set of orbit types of M.
(3)
For any two compact open and closed in
4.3.3 Definition
principal
(H')
subgroups K, H
C
G with
(H)
0 the G-space Nx1 1[(g,v)l E G XGx VxI JIVII < F-I possesses a the theorem. of the in orbit sense By virtue of the slice theorem and type principal the paracompactness of the orbit space we can now cover M by locally finitely many
then dim Vx
connected,
not
1 and
=
Gx
GL(Vx)
--i
Then the orbit type of each
=
The fact that of such N1. x
N
can
be
G\M
is connected
with each other
joined
gives
chain N F-o,
that any two slice bundles N'0 and P0 0 i.e. NF n N' j` Nk NE, Pj+1 x Pj
by Po principal orbit types (H) a
=
PA;
coincide, their the quotient M, G\M(H) of M(H) from follows we now have Finally, So connected. we (1). is which proved (1). (2) to the We to the of proof according claim last the perform theorem, (3). get to < k. Hence the
for 0
the form
qj
P
-
(4.4.1)
Pj,
j=1
where qj induction.
0 whenever
=
< d is
a
Let
deg pj
us assume now
polynomial
in the pj and that p E
exists
Then, first there
d and qj E C else. This was the initial step of the that for some d > do every H-invariant q of degree
>
T(V)"
homogeneous of degree
is
d + 1.
representation
a
k
P
=
1: Tj
-
(4.4.2)
Pj,
j=1
with rj E T(Y) and deg Tj < deg p. with respect to the Haar measure g
9)(V)H
where qj E
is
P 1)
'
deg qj *
*
)
Toz W
=
De
isomorphism. Consequently, for each Hilbert basis
p
=
(pl)
Pk)
of
J)(V)H
the Zariski derivative
Tz (H \ V)
T,z is
injective
PROOF: a
direct
at any
point
The lemma is
V
proof
see
E
an
-->
Tz W
=
W
V. immediate consequence of the theorem of SCHWARZ. For El [14, Lem. 2.17] as well.
BIERSTONE
We first prove that
XjG\(unM(G..) )
is
mapped G-equivariantly G/G.x X VG , consequently
and
is
an
immersion.
diffeomorphically
Via
onto
an
p the stratum U n
M(G..)
open subset of the bundle
the restriction
_ JG\(unM(G-x)) : G\(U n M(Gx))
-4
G,\VxGx
=
VxG1
has to be
other
the
immersion and
a diffeomorphism onto a zero neighborhood in V .x. On the preceding lemma Lemma 4.4.5 the map fjv x : VGx --- Rk is an
therefore
XIG\(unM(Gx))
=
F'
TjG\(unM(Gx))
as
well.
hand, by
165
4.4 F inctional Structure
Assume now,
RN and
:
y
succeed to show that any two singular charts x : G\U -- R' C --) R' C RN as defined above are compatible that means there
we
G\V
point GZ E G\(U n V) a neighborhood W and RN with 0 C RN open such that
exists around each
H: 0
(5
--)
c
H
=
XJW
o
a
diffeomorphism
(4.4.4)
YJW
Then the restrictions XG\(unm(,,)) are immersive as well. To H. Since the singular chart y around a point GIJ E G\U with G., holds.
-
see
by
this, choose virtue of the
would then also imply that YJG\(vnm( ,)) is immersive, Eq. (4.4.4) to prove the compatibility have immersive. So, we Xiwn(G\m(R)) hence XIG\(unm(,)) are the suffices consider to case that y is defined around a of the x and y. To this end, it results proven
point G'1J
E
far
so
and
G\U
given by
y:G\V-- R!,
Gz -4-q-T(Gz),
neighborhood of ij, iP : V -4 G X G V. the embedding belonging to it and q q1) a minimal homogeneous Hilbert basis for T(V., )G Y. The (ql, compatibility of x and y is shown when a smooth embedding H : 0 -4 R' with 0 c R' open can be constructed in such a way that Eq. (4.4.4) is fulfilled for a suitable neighborhood W of Gz. By the theorem Of SCHWARZ there are smooth functions H1, Hk E COO (R') such that for all v from a zero neighborhood in V'J
where V denotes
a
slice
=
-
-
y
-
,
Hi
-4 (G.,v)
R is the
where 7ri Gz from
-
a
neighborhood Hi
-
y
(Gz)
7ti
=
X
o
projection G-y
o
T-1 (G,,v),
i
=
1,
-
-
-
,
k,
onto the i-th coordinate. Then it follows for all
W of
=
Hi
-4
o
-
T (Gz)
=
7ri
-
x
(Gz),
i
=
1,
-
k.
and e,00 are isomorphic, since p is a G-invariant -q*e' G\V Gx\Vx is injective. Since diffeomorphism. Consequently, by Lemma 4.4.5, TG' x TzV- TGz H Tz y x holds, 4.4.5 lemma an isomorphism, too, and Tz is T Lz by Tzy(Gy) H TG1,Jy G-y 'Y y(Gij) Now note that the sheaves
=
o
neighborhood 0 C R' of y(G-Y) the injective. Hence, is an restriction H := (H1, embedding. This was the last constituent in H1)1o the construction of singular atlases U for G\M. Invoking now the chain of equations given by the theorem Of SCHWARZ for
has to be
-
a
suitable open
-
-
,
x* e' (W) the last claim
defined
by
-q *V e' (W) *
=
follows, namely that
U. Moreover,
even
the
by
IV
eG'\m
*
(e' (G "\VX)G.,,)
G\M
orbit types.
of
a
supplied with G\M carries a given by e,'G\m(U)
proper G-action be
Then the orbit space are
of this the orbit space becomes
=
topoG\M open. By means Whitney stratified space. Moreover, the stratification by minimal among all Whitney stratifications of G\M. for U C
logically locally orbit types is
e(G\U)G,
coincides with the sheaf of smooth functions
canonical smooth structure the smooth functions of which
eoo(71-1(U))G
=
following holds
4.4.6 Theorem Let the orbit space
the natural stratification
=
trivial and
a
166
Orbit
4.4.7 Remark The
G-action
stratified has been
proof that the orbit space of a linear given by BIERSTONE [13], see also [14,
Thm.
be
can
2.5]
Spaces
Whitney
from the
same
author. To carry out the
proof of the
(BIERSTONE [14,
4.4.8 Lemma
Hilbert basis. If then y
-y'(0)
C
>
:I
E,
-
Rk is
=
f0j
121 -curve in X
a
need the
and p
=
p (V)
following
(pl,
=
-
-
-
,
with -y (0)
Pk)
be
a
0, then
=
Endow V with
an
H-invariant scalar
product.
Without loss of
generality
that pi describes the square of the distance from the origin. Let constant such that Jpj(v)j < C holds for every i and every unit vector v
assume
can
0 be
a
of V. Let
di
deg p j.
=
Then
j(U1)*** iUk) EW I U1
XC
-
-
=
=
-
=
CIU11dj/2,
!O) IU,1:5
0 holds. From Y' Obviously, (7r, -y)'(0) 0. k. Thus, -y'(0) 2,
for i
Let V`
2.12])
Lem.
e
we
0 holds.
=
PROOF: one
claims not shown up to now,
i
=
101 follows di ! 2,
2,.
..
,
kj.
(7ri -y)'(0)
hence
-
=
,
PROOF
THEOREM:
OF THE
local
Only
holds remain to be shown. Since in both to carry out the
proof for the
Equip V with
G-invariant scalar
a
Then the orbit space
G\V
case
can
0
0
=
-
and that
triviality cases
that M is
product
the
a
condition
Whitney's
(B)
local ones, it suffices linear G-module V and G is compact.
properties
.L and
a
via V be considered
are
Hilbert basis p (P 1) ) P k) stratified subspace of RI. We =
*
*
'
-
as
pair R < S of strata of G\V is (A)-regular at every point G\V. Let W be the orthogonal space of T,(Gv), and W' the orthogonal Gv denotes the isotropy group of v. Let Y : U -4 space of W" in W, where H WI submersive be G-invariant embedding in accordance with the slice x x a W') (G tj
want to show at first that any
GV E R c
=
T(W')H.
Then
G\U resulting
choose
Finally,
theorem.
--->
in
a
one
i
homogeneous embeddings
G\V -14 W
further :=
V
G\U -Y4 H\W'
and
embedding
-
Hilbert basis q
minimal
a
has the two
obtained
IqT-l (-q x id)
-1
:
-
Y
X
=
(qj,
WH " IW
X
qj)
of
WH,
by composition
--
e,
Y:=
q(WI)
X
WH
Rk is injective, virtue of Lemma 4.4.5 the Zariski derivative TOR: TO'Y -4 Tpz,(v)Rk consequently there are zero neighborhoods Wand W in R1 and WH respectively, and a smooth embedding L: W'x W -4 Rk such that Liyn(w,xw) ilyn(wxw). The space Y C RI x WH gets a stratification by the pieces S(K) CI(W(,)) x WH, where K c H NO X WH lies in the stratum H. of of Then the the closed subsets runs through origin
By
-
,
=
S(H)
=
M
X
W"- Consequently, for
converging to
a w (=-
S(jj)
1.1s(H) holds.
The
particular,
pair
(S(H), S(K))
at the
every sequence
(Wk)kEN
of
points of S(K)
>
S(H)
the relation
origin.
=
fo}
X
WH
C
lim k-- oo
therefore fulfills the
Tw, S(K)
Whitney condition (A), hence, Lly : Y ---i G\V describes
Since the restriction i
==
in an
167
4.4 nznctional Structure
isomorphism. of stratified
spaces and at the
same
time
is
L
an
embedding, Whitney
point GV E R. Due to the fact stratification of G\V as a result is (A)-regular.
needs to be satisfied for each stratum S > R at the
(A)
that Gv E
G\V
has been
arbitrary, the G\V is
Since p consists of
polysemi-algebraic nomials, p(V) A semi-analytic set Z C Rk possesses by and consequently a semi-analytic set. LOJASIEWICZ [115] or MATHER [123] a minimal (A)-regular stratification by semianalytic smooth manifolds. By MATHER [123] this stratification is minimal, too, Next it will be shown that c
among all
(B)-regular.
even
R' by the theorem of TARSKI-SEIDENBERG is
(A)-stratifications
of Z
by
a
Due to Lemma 4.4.8 the
smooth manifolds.
G\V by orbit types is minimal among all e'-stratifications. Accordresults the to proven so far the stratification of the orbit space by orbit types
stratification of
ing
has to be minimal among all (A)-regular stratifications of G\V. As a consequence, the corresponding strata of G\V C R' and those of H\W' C R' are semi-analytic.
(They
are even
LOJASIEWICZ
but
semi-algebraic,
[115,
p.
103]
will not need this at this
we
every smooth
semi-analytic
point.) Following (B)-regular over'
manifold is
I result, each one of the strata CI(W(K)) with K < H is stratum that each its own, implies S(K) is (B)-regular (B)-regular over JO}. This, on has be S to over S(H). By Lemma 1.4.4, (B)-regular over R L(S(H)), too. L(S(K)) all strata > R. will S of run through If K now runs through all closed subgroups H, let At end the stratified us directly show the space. Consequently, G\V is a Whitney from follows local triviality of G\V, although this Corollary 3.9.3, too. immediately the origin. Let around local triviality By virtue of the slice theorem it suffices to prove of t, S is G-invariance of the S c V be the unit sphere belonging to R. Then, because of itself a G-manifold again and moreover compact. In the category topological spaces the isomorphy V - CS holds. Furthermore, the G-action commutes with the canonical R>'-action on V and CS, respectively. As a consequence, G\V and C(G\S) are isomorphic as stratified spaces taht means G\V is topologically locally trivial around
a
point of
its closure.
As
a
=
=
GO,
where the link is
given by G\S.
This
11
to show.
was
A further consequence of the lemma on page 166 is the derivations and vector -fields on an orbit space.
following
statement about
Proposition (BIERSTONE [14, Prop. 3.91) Let the Lie group G act properly M, and let 6 E Der(1S'(G\M), (G\M)) be a derivation on the space of smooth functions of G\M. Then 6 E XOO(G\M) holds, if and only if 5 is tangential to every 0 for any smooth function f vanishing stratum S of codimension 1, hence iff 5(f)js on such a stratum S. In particular, the relation Der(e`(G\M), (G\M)) X'(G\M) is satisfied if and only if G\M does not possess a stratum of codimension 1.
4.4.9 on
=
=
PROOF:
We follow the argument
fices to prove the claim for the
case
given
[14]. By virtue
in
of compact G's and for
a
of the slice theorem it suflinear G-action
on a
finite
dimensional vector space V. Again, due to the slice theorem, it suffices to show that if fOl is a stratum of codimension > 2, every derivation 6 E Der(E!'(G\V), eoo(G\V))
being tangential an
induction
to the strata of codimension 1 vanishes at the
argument
we
mension less than that of
can
JO}.
assume
Let p
=
that 6 is
(pi,
-
-
-
,
tangential
pk)
be
a
origin. By
means
of
to all strata with codi-
Hilbert basis for
9)(V) G
and
168
:ff
Orbit Spaces
G\V
:
-4
tor field V
X:=
vector field V
small
p(V)
Rk the induced diffeomorphism. Then there is
generates
is defined
4)t
c
Rk such that the restriction of V
on
local group
a
a
smooth
vec-
X
equals the derivation V" (6). The of diffeomorphisms ( t, where for t sufficiently on
of 0. If the curve -y(t) 4)t(O) for sufficiently 0 by Lemma 4.4.8. Hence, V(O) -y'(0) 0, implying F.(6) E X'(X) and thus 6 E 100(G\V). Assume now on the other hand that there are arbitrary small t with 4)t(O) X, where we can evidently achieve after a possible small t lies in
change
to
of 0 and the
a
on a
neighborhood
it follows
X,
-4)t that these t > 0 such that
=
=
=
are positive. Since X is closed, there is a neighborhood U 0. Let x E U n X. Due to the induction hypothesis 4)t (U)
t
=
-y(s) 4)s(x) lies in X for sufficiently small s < t. Let so be the largest s < t such that -y ([0, so]) C X. Since V is tangential to the strata of X \ f0j, 0 must (Pso (x) hold. Consequently, U n X is contained in the curve ( _s (0), 0 < s < t, contradicting curve
=
=
the assumption.
Finally,
Again, by
need to show that
we
codimension I there exists
a
derivation 6 not
virtue of the slice theorem
the stratum with codimension I is
we can
Z2. Then, in the first case, the second case to R>-o. In both cases - - is ax
4.4.10
on
G\V
being given by
given by the origin
=
vector field
orbit space with
a
of
linear
G-representation G/Go fel or orbit space G\V is diffeomorphic to D in the a derivation that is not induced by a smooth as a
a
consequence either
=
for this needs to vanish in GO.
At the end of this section
we
stratum of
smooth vector field.
a
restrict ourselves to the situation where
space V. But then V has the dimension 1, and
G/Go
on an
13
would like to present another
important class of
stratified spaces with smooth structure, the so-called orbifolds. These have been intro-
duced into the mathematical literature Orbifolds represent in
ticularly
by SATAKE [146]
under the
name
V-manifolds.
natural way stratified spaces with very mild singularities. Parfor that reason, orbifolds often allow results and constructions usually known
for manifolds
a
For
only.
example, KAWASAKI [99, 100] succeeded to prove a signature literature, the exact definition of orbifolds
theorem for orbifolds. In the mathematical is rather technical
however,
this
section.
A
U an
=
can
(see [146] be very
or
also
easily
[47, B.2.3.]),
done
by
Whitney (A) space (X, C"0) (Ui)iEJ of X by open sets in such a
orbit space of the form
Gj\Mj,
where
means
in
our
language
is called
an
orbifold, if patch Uj
way that each
Gj
is
a
of stratified spaces, provided in this
of the orbit spaces
finite group and
there is is
Mj
a covering diffeomorphic to
is
a
differentiable
Gj-manifold. The projections 7rj : Mj -- Uj belonging to it are named orbifold charts. Obviously, it is possible by the slice theorem to choose the manifolds Mj as open zero neighborhoods
in
a
linear
Gj-representation
space.
Chapter
5
DeRham-Cohomology complex
The deRham
5.1
Considerations
the deRham
on
cohomology
on
of
singular
singular
spaces
spaces have
a
long
tradition.
early years of complex analysis one was interested in the question, what Already the relation between the (smooth) deRham cohomology of a singular analytic variety in the
example NORGUET [1351, 1959). In the year example that differently to the regular case the by [82] of deRham cohomology a singular analytic variety need not coincide with the singular the of underlying topological space. In another work BLoom-HERRERA cohomology have shown [181 that for every complex analytic space (X, 0) there exists a canonical and its classical
(see
is
cohomology
could show
1967 HERRERA
for
an
splitting H* (X; where H* (X; n
)
denotes the
fl )
H* (X; C) ED
A*,
with values in the sheaf
complex Q complex cohomology of X. cohomology vanishes for regular X, but if X has The reason for that lies mainly in the fact that
hypercohomology
of Kdhler differentials of 0 and H* (X; C) the "classical"
complement A* to the classical singularities then in general A* =,4 0. in the singular case the sequence The
0
need not be
singular means
Let
a
case
us
CX
__
(9
__4
f1lX
__
f12X
locally constant sheaf CX and this is because in the holomorphically contractible (see REIFFEN [145]) that need not hold for holomorphic differential forms on X.
resolution of the
X need not be
the Poincar6 lemma
question
now
consider
a
stratified space X with smooth structure E!'. Then the (yet to be defined) deRham cohomology
arises what the relation between the
of X and the be
__
proved.
singular cohomology of X is, and
In this section
sections it will be
we
computed
the definition of the deRham
further
approaches
5.1.1
Let X be
a
whether
define the deRham
a
kind of deRham theorem
cohomology
of
X;
in the
for several different classes of stratified spaces. will
complex give cohomology theories
to construct
we
a
historical overview about on
can
following After some
stratified spaces.
stratified space with smooth structure Coo. The sheaf Coo of smooth
M.J. Pflaum: LNM 1768, pp. 169 - 181, 2001 © Springer-Verlag Berlin Heidelberg 2001
170
DeRham-Cohomology
functions
on
X induces
(Q*, d) of sheaves f2l obtain
a
Coo
:
flxk,p
:=
further
called the deRham
___
f1l (X)
___
of
complex cohomology
already a
complex
f1k
j12 (X)
(X, C').
___4
The
.
.
.
__4
f1k (X)
___4
cohomology H,*,,(X)
.
.
.
,
of this
complex
is
of X.
by HERRERA et al. there exist other cohomology or other meaningful cohomology theories on following we explain some of the most important ones, but
also refer the reader to the article on
[20]
of BRASSELET for
deRham theorems for
a
detailed exposition about
singular
varieties.
Controlled differential forms VERONA has introduced in
every controlled space X
a
complex of so-called
families of smooth differential forms
similar to the
we
above mentioned studies
the present state of research
are
f12
a
deRham
stratified spaces. In the
5.1.2
___
B
After application of the global section functor
k E N.
,,
the so-called deRham
Besides the
Ell
__,
Appendix
complex
Coo (X)
approaches for
to Section B.3 in
according
[176, 179]
for
controlled
(as)SE&
which
differential forms. These satisfy a control condition
for controlled vector fields.
Hereby it is not necessary that the family (ocs)SES can be put together to a global continuous differential form, as in particular it is not immediately clear what a globally continuous differential form on a controlled space should be. The important fact now is that the corresponding sheaves of controlled forms comprise a fine resolution of the sheaf of locally constant real functions on X, hence a deRham theorem holds for controlled differential forms. one
[57] has generalized infinitesimally controlled forms
FERRAROTTI
the method of VERONA and introduced
of
which also
Intersection the
theory
rise to
a
a
complex
deRham theorem.
homology Already POINCAR9 knew that singularities could particular duality on the homology of manifolds which nowadays is known
5.1.3
destroy
under his
gives
name.
for for
Therefore mathematicians have tried to set up
a
(co)homology
singular spaces which satisfies a kind of Poincar6 duality. This has been achieved by the intersection homology theory of GORESKY-MCPHERSON [63, 64] which appeared in the mid 80's. In intersection homology one considers the homology of complexes consisting of singular chains which intersect a stratum only in an allowed dimension given by a so-called perversity. Hereby a perversity is nothing else than a special integer valued function on the set of strata. A particularly elegant approach is the one via perverse sheaves, which comprise complexes of sheaves or more precisely objects in the derived category of sheaves. The Poincar6 duality in intersection homology then is an immediate consequence of Verdier's duality in the theory of derived categories (see for example KASHIWARA-SCHAPIRA [981 for derived categories and Verdier duality). A deRham theorem for intersection homology has been proved by BRASSELETHECTOR-SARALEGI [22]. More precisely the authors of this article introduce a special class of forms named intersection forms and show that integration of intersection forms over chains leads to an isomorphism between the cohomology of intersection forms and intersection homology.
5.2 DeRham
cohomology
171
e,'-cone spaces
on
5.1.4
L2 -cohomology At the end of the 80's the importance of L 2 -cohomology
for the
study
of
singular
apparent by the work Of CHEEGER [39] and
spaces became
ZUCKER'[194, 193]. Hereby
supplies
one
a
stratified space
better the top stra-
or
of the complex By the result [39, Thm. 6.1] of CHEEGER one knows that for a Riemannian pseudomanifold with conic singularities the L'-cohomology coincides with the intersection homology of middle perversity. CHEEGER-GORESKY-MACPHERSON [42] have extended this result to analytic locally conic varieties and have posed in their work the famous conjecture which says that for any projective algebraic variety with restriction of the Fubini-Study metric as Riemannian metric the L 2-cohomology coincides with the intersection homology of middle perversity. The Cheeger-Goresky-MacPherson conjecture has been shown for the case of isolated singularities by OHSAWA [136]. According to SJAMAAR [161] orbit
tum of it with
a
Riemannian metric and studies the
the L2
consisting of
differential forms.
-integrable
spaces of Riemannian G-manifolds have
coincides with the intersection
cohomology
as
homology
well the property that their L'-cohomology perversity. An essential tool for
of middle
many of these considerations is the sheaf theoretic
to L
approach
Some further and intuitive
2-cohomology for
better
as
explained by [131, 1321. derstanding of intersection homology and its connection to L2_ or more generally to Lq -cohomology is given by the concept of shadow forms by BRASSELET-GORESKYMACPHERSON [21]. We cannot go into this concept at this point but refer the interested reader again to [20], where it is explained in greater detail. NAGASE
DeRharn
5.2 Before
we
space let on
cohornology
first recall
a
a
un-
ff-cone spaces
on
computation of the deRham cohomology for a Coo-cone essentially entails the Poincar6 lemma
start with the
us
means
classical result which
manifolds.
5.2.1 Lemma Let M be t E
[0,
[ the embedding by
oo
Q'(M)
a
differentiable manifold and Lt
x F-4
Define the operator
(%, t).
t
K m,t (w) (y)
(vi,
W
Then
Km,t
homotopy
satisfies the H
:
M
x
[0, t]
fI1+1 (M
E
where
PROOF:
div + ivd and
H,
=
[0, t]), +
'y E
M
Km,t
a ,
as
M, L*t
Km,td
-
X
f11+1 (M
[0, 00 [ with x [0, t])
vi) ds,
vi, V1,
M
-) :
-
-
-
,
vI G
TuM.
hence for L*, 0
every smooth
M the relation
-
H(-, s)
for
The claim follows =
X
equality dKm,t
dKm,tH* follows,
fo, w(y, s) (
vt)
:
CV, where ZV
s
E
by
+
Km,tH*d
[0, t] an
=
H*t
-
H*0
(5.2.1)
-
easy calculation
using
Cartan's
magic formula
is the Lie derivative with respect to the vector field V
iv the insertion of V. Hereby let V be the
vector field
on
M
x
[0, t] given by
172
DeRham-Cohomology
V(x, s)
For
as
further
a
of the lemma
proof
see
HOLMANN-RumMLER
[89, 13].
0
5.2.2 Theorem Let
RX
comprises
a
(X, COO) e00
---
be
- 4 f2l _ 4 f22 _ 4
PROOF:
We
sequence 5.2.2. In other words
we
the
(S
basis of
U)
W+1 around
x
d
a
local
(f2*, d)
x
that
there exists
x
In this
=
U
=
JIJ
=
smooth
(US) lJrad 'JL)
neighborhood
Hs (ij s, 1)
=
-Lj s
on
x
of x,
Wv,,r By assumption
chart k
cone
Hs (-Lj s, 0)
=
x
--4
(S
U)
n
CL
x
C
1W+1
I
[0, 1 [
'Yrad E
(x, 0).
and that
embeddings
Q.
and IJL E
Now let V C S be
smoothly
a
El 0, 1[, and
{Ij
=
U
:
is
and such that S has lowest
x
identical
are
has coordinates T
V there exists
and
E S X
*
E X there
x
0 for k >
=
S' be the link chart for L. As the claim
--
suppose that k and I
presentation the point
contractible
X.
on
0, where jfk the quotient sheaf ker dk/im dk-j.
means a
Jfk(W)
such that
such that S is the stratum of
one we can now
X
.
=
dimension among the strata of U. Let I: L is
(5.2.2)
..
have to show that for all k E N and all
W of
neighborhoods
cohomology sheaf of By assumption on X
n
.
by Proposition 2.3.2 that the sheaves f2' are fine. 0, hence it remains to prove the exactness of the
know
already
-
a
1 4 r1k - 4
...
fine resolution of the sheaf of locally constant real functions
Moreover it well-known that d
exists
121-cone space. Then the sequence of sheaves
a
X11JS
E a
E
smooth
V and Vrad
-rj.
2 a Whitney form Pm of class 12m with dpm oc. By the same reason there exist Whitney forms -v' of class e" d-.v' with pm-1. Next choose a compact exhaustion X Pm (Kj)jEN of W and M
=
-
M
transfer in in Section in every
a
11
canonical way the seminorms
1.5,
to the spaces
Em(W) (see
for
C2k.(W) (see e
example [1181),
Hence there exists for every
m a
11 Pm
-
-V.
also
-k
fl
-
on
Em(W),
Appendix C.1).
(W)
Ebk(W)
Pm-1
Ilyj,m
-
which As
were
F,'(W)
has to be dense in every
defined is dense
8k (W). m
with
3
As the is
8'(W) together with the seminorms 11 IIK-,m comprise n6chet spaces, *
well-defined,
hence lies in
8k-1 (W)
and satisfies
dp
oc.
=
this
P
This proves the claim.
M
5.4.5
Corollary
The
Whitney deRham cohomology of an (A)-stratified space X hav-
curvature moderate control data of every order M E
ing cally tempered resolutions cohomology of X. PROOF:
The
proof
5.4.6 Remark As in
of class (!' coincides in
is similar to the
one
a
N>0 and which possesses lo-
canonical way with the
given for Corollary
5.2.3.
singular
1:1
particular subanalytic sets fulfill the prerequisites of the results section, the Poincar6 lemma holds for Whitney forms on subanalytic sets, hence the Whitney-deRham cohomology on subanalytic sets coincides with the singular in this
cohomology.
Chapter
6
Homology
of
Algebras
of Smooth
Functions
Hochschild homology theories and the closely connected cyclic cohomology introduced by ALAIN CONNES [45] have proved to be very useful for the structure theory of algebras (see LODAY [1121). In particular in the framework of noncommutative geometry invented by ALAIN CONNES, where one wants to introduce geometric notions like forms, connections, deRham cohomology and so on for (noncommutative) algebras, these homology theories play an important role. But even for "commutative geome"
Hochschild
(co)homology
becomes
and
important, because one can help deep geometric analysis example the index theorem of ATIYAH-SINGER (cf. NEST-TSYGAN [1331). Moreover, there is hope that it will be possible to formulate and prove appropriate index theorems for singular manifolds with the help of methods of Hochschild homology (cf. MELROSE-NISTOR [128]). Therefore, in this chapter we will study the Hochschild (co)homology of the algebra of smooth functions on a stratified space. try
prove with its
In many
more
results of
and in
particular
more
like for
in the
of function
algebras it turned out that homology is better suited, if one wants to obtain geometric information on the algebras under consideration in the spirit of ALAIN CONNES. For example one can compute the topological Hochschild (co)homology of the algebra of smooth functions on a manifold (see [45, 143, 102, 165]), but one knows only very little about the general Hochschild (co)homology of these spaces. In the following we will first introduce topological Hochschild (co)Homology as a relative homology theory and will then derive some useful properties of this topological homology theory. As a reference for further results on the (co)homology theory of topological algebras see TAYLOR [164]. the
cases
topological or
6.1 6.1.1
case
in other words local version of Hochschild
Topological algebras
and their modules
Let k be the field of real
or complex numbers. A k-algebra A together with the topological k-vector space is called a topological k-algebra, if the product : A x A -4 A is a a continuous mapping. In case that the underlying topological vector space structure is locally convex and and if for every continuous seminorm 11 11
structure of
a
-
-
M.J. Pflaum: LNM 1768, pp. 183 - 199, 2001 © Springer-Verlag Berlin Heidelberg 2001
184 on
Homology
A there exists
a
11abIl then A is called
a
11'
continuous seminorm
locally
space, then A is called
:5
llall'llbll'
convex
of Algebras of Smooth Fbnctions
such that
for all a, b E
(topological) k-algebra.
Fr6chet
If
A,
additionally A is a Fr6chet misunderstandings let us
To avoid any
algebra. always assume an algebra to possess a unit element. An A-module M of a topological k-algebra A is called a topological A-module, if M has the structure of a topological k-vector space and if the structure map (a, M) F-4 am is continuous. In case that A is a locally convex algebra, M a locally convex topological
explicitly that
mention
a
we
11
k-vector space, and if for every continuous seminorm. seminorms
on
M and
11amli then M is called
a
locally
:5
11 11'
on
-
11all' Iml
convex
Fr6chet spaces, then M is called
for all
topological
a
on
M there exist continuous
A such that
a
E
A,
M E
A-module. If
M,
additionally
A and M
are
Fr6chet A-module.
For a topological algebra A and two topological A-modules M and N we denote by HomA(M,N) the set of all continuous A-linear mappings from M to N. Then the topological A-modules together with the continuous A-linear mappings form a category A-VoDt,,p. Note hereby that as objects of A-9noDt,,V even non-Hausdorff topological A-modules are admitted. By tHomA(M, N) C HomA(M, N) we denote the subset of all topologically A-linear homomorphisms that means the set of all continuous A-linear mappings f : M -4 X such that the induced mapping M/kerf -4 imf C N is a topological isomorphism. The composition of two topologically A-linear homomorphisms f : M -) N and g : N -4 T gives again a topological homomorphism g f E tHomA(M, T), as the following consideration shows. Let f : M/ ker f - im f and -g : N1 ker g -- im. g be the induced topological isomorphisms induced by f and g and let h : im f/ ker g -4 o
f-'(ker g))
+
Then h,,
-91im (g
be the continuous
map with
T-1
: im f -4 M/ ker f the A-linear inverse of g f : NC/ ker (g f) ---) im (g f). comprises f) Hence the topological A-modules together with the topologically A-linear homomorphisms form a category.
M/ (ker f
quotient
o
.
-
o
.
Proposition Let A be a topological k-algebra. Then the category A-Tzo-otov topological A-modules and continuous A-linear maps is additive. In case that A is a locally convex algebra (resp. a FMchet algebra), then the locally convex (resp. complete locally convex resp. R&het-) A-modules form a full additive subcategory A-MoDE, (resp. A-MD,1, resp. A-MD, ) of A-VoDtov. 6.1.2
of
PROOF: space is
additivity of A-VoDt,,p is clear, because the zero dimensional vector object in A-MoDt,,p, and because continuous A-linear morphisms can multiplied by scalars. The further statements then follow immediately.
The a zero
be added and
o
6.1.3 Remark One could
modules and the
conjecture that the category consisting of all Fr6chet Atopologically A-linear homomorphisms is an Abelian category. But
185
and their modules
Topological algebras
6.1
C and consider an following example shows. Let A choose a compact Afterwards X. infinite dimensional separable complex Hilbert space closed image. Then idX and operator k: X -4 'K of norm < 1 which does not have a f idX + k are bijective and continuous, and both are topological homomorphisms. f But the difference operator k idX is not a topological homomorphism, as by
this is not the
case
the
as
=
=
=
assumption imf
-
is not closed.
homological algebra in the category A-9XoV(,,p. To this end it morphisms f : M -- N in A-VoDt,v the kernel and cokernel one regards f as a morphism in the category A-VoD then one already ker f k M and a cokernel N - 4 cokerf. We supply kerf with the
We want to do
6.1.4
is necessary that for all
of f exist. If has
a
initial c.
kernel
topology with respect to both topological A-modules, and comprise the kernel
with respect to k and coker f with the final
topology
Then kerf and cokerf
are
N of f as the im f resp. cokernel of f in A-Voiotq. Moreover, one obtains the image kernel of c. Thus the topological spaces ker f , coker f and im f satisfy the universal
category A-VoDt. In the following topological kernel, topological cokernel and topological image of f By the universal properties of im f the morphism f can be i m with a uniquely determined morphism m : M -- im f factorized in the form f i m will be called the canonical factorization of f. Altogether The factorization f
properties of kernel, cokernel and image we
in the
will therefore call ker f , coker f and im f the .
=
=
we
thus obtain the
0
-
.
-
following canonical
-4
ker f
k)
M
-"-')
sequence of f
im f
-i-4
X
-c-)
coker f
--
0
.
locally convex algebra and f a continuous A-linear mapping between locally A-modules, then kerf, cokerf and imf are again locally convex A-modules in a canonical way and comprise kernel, cokernel and image of f in the category AVoDt,,p. If on the other hand A is a complete locally convex or even a Fr6chet algebra and f a continuous A-linear mapping between complete locally convex (resp. Fr6chet) modules, then ker f is again complete (resp. Fr6chet), but not necessarily coker f and im f Now denote by V the completion of a locally convex A module M and define f := im f the complete cokernel by coker f := N/im T and the complete image by If A is
a
convex
.
.
are complete locally (resp. Fr6chet) Moreover, ker f coker f and im f satisfy the universal properties of kernel, cokernel and image of f in the additive category A-VoD,[, (resp. A-VoD'a). Finally let us 1M f holds in the Fr6chet case, if mention that by the open mapping theorem im f Fr6chet A-modules. between and only if f is a topological homomorphism and possess kernels and cokadditive all and As A-Voot,,,, A-Vool, A-9)W, are
Then both coker f and im f
A-modules.
convex
,
=
ernels, the notion of exactness of More precisely a sequence
a
sequence of
f
Nlk-1 'k4 JVk
topological
modules is well-defined.
f
N(k+l
mappings is called exact at Mk in A-9XoD(,,v (resp. A-VoDf,) or in other words is called topologically exact, if ker fk and iM fk-1 coincide as topological A-modules. If the modules -Vk are
of
topological (resp. locally convex)
A-modules and continuous A-linear
186
Homology
of Algebras of Smooth 1 mctions
all
complete locally convex or Fr6chet, then we say that the above sequence is exact Mk in A-MD,1, (resp. A-MDX) or briefly the sequence is weakly exact at Mk, if ker fk iM fk-1. The notion of topological exactness obviously makes sense as well in A-M'Or, and A-MD, , where by the completeness of the kernel ker f" every topological sequence exact at -VA; has to be weakly exact at JVA, as well. Finally we call a sequence of topological vector spaces topologically resp. weakly exact, if it is topologically resp. weakly exact at every one of its points. at
"
6.2
Homological algebra Let
6.2.1
that A is a
for
topological modules
consider the category A-M'O,f, more closely, where it is assumed complete locally convex k-algebra. In the following we will introduce homology theory in A-9Ao?),r,. According to HILTON-STAMMBACH [85, us now
a
relative
Chap. IX]
needs for the definition of
a relative homology theory in an additive epimorphisms in that category with respect to which the projective objects, the so-called E-projective objects have to be defined. Afterwards we will define exactly like in ordinary homology theory 8--projective resolutions and E-derived functors, which then will give the desired (co)homology objects. As epimorphism. class, which will fix the relative homology theory in A-9ROD"r, we take the class T of all surJective continuous A-linear and k-splitting mappings -4 X. Hereby means k -splitting, that there exists a continuous k-linear e mapping -- M which satisfies e s s idx and which sometimes is called a continuous kone
category 9A
a
class E of
section of
Let
=
-
e.
Proj(T) COb(A-9RoD,t,)
projective objects that e :
M
-4
be the class of all
the class of all
means
objects
T-projective or topologically morphisms
T such that for all
N from T the sequence
Hom(T, M)
Hom(T, N)
--
0
(6.2.1)
By CT we then denote the completion of T that means the class of all epimorphisms e : M -- N in A-MD,[, such that for all topologically projective objects is exact.
P the sequence
(6.2.1) is exact. (see Appendix A.3) category by the functor (t,
Next recall tensor
that the category of the
completed
A-9Ro'O,,[, gets the product.
6.2.2 Proposition For every complete locally convex k-algebra A plete locally convex A-modules are topologically projective:
(1)
every
topologically direct
(2)
every
complete locally convex A-module T of the form T a complete locally convex topological vector space,
summand T of a
structure of
a
7r-tensor
the
following com-
topologically projective A-module, =
Aa7N,
where V
denotes
(3)
in
case
that A is
a
Fr6chet
algebra,
every
finitely generated projective A-module
T which has the structure of a R6chet A-module. If conversely T is
topologically projective A-module, then complete locally convex A-module of the complete locally convex.
summand in
a
a
T is
a
form
topologically direct
Aa"V,
where V is
6.2
for
Homological algebra
topological
modules
187
(1) is obvious. Let us show (2). Let e : M -4 N be a morphism of T, hence M be a continuous section of e, and particular A-linear and k-splitting. Let s : X L: V -4 T 10 v. Then one associates to every ACD7,V be the canonical injection v f L and sets g:= s. V. By continuous A-linear mapping f : T --) N the mapping V the universal properties of the tensor product A7, there exists a uniquely determined continuous A-linear mapping g : T --i M such that g' g L. As e g is continuously PROOF:
in
=
=
-
=
-
A-linear and e
the relation
Now
e
-
g
g
-
-
L
(3). Let (vi, (vi, Vk). Then
k-vector space
to *
-
-
*
:
the
*
A0 V
=
AtD,V
-
)
)
f
g'
-
=
e
-
s
-
V
f
=
f follows. Hence T is
=
we come
e
=
--)
topologically projective. be a generating system of T, mapping
Vk)
T,
a
(D vj
-4
avj,
j
=
1,
and V the free
k
ker f is a closed continuous, surjective and A-linear. Moreover, the kernel Q subspace of the Fr6chet A-module AOV and by the projectivity of T has a complement T' which is linearly isomorphic to T. As Q c A (9 V is closed, T' inherits the structure of a Fr6chet A-module of A(9V. By the open mapping theorem T' -- T is a topological isomorphism, hence T is a closed subspace of A 0 V. By (1) and (2) the topological projectivity of T thus follows. It remains to show that every topologically projective A-module is the topologically direct summand of an A-module of the form A6,V, where V is complete locally convex. Now set V := T and define e : V -4 T by A 0 v t-4 av. Then e is continuous, A-linear, surjective and possesses a k-linear continuous splitting s : T -- V, v -4 1 ov. By the topological projectivity of T there exists a morphism f : N( --> V of A-Voio,(, such that e f idu,. Now set T' := im e. Then T' is a complement of ker f hence a closed subspace of A&,V. Moreover, fg,, provides a topological isomorphism from T' is
-
=
,
to T with continuous inverse
e :
T
-
V. This proves the claim.
El
Corollary The class T is projective that means for every object M ofA-VOD& an epimorphism e : T --) M of T with T topologically projective.
6.2.3
there exists
Define T
PROOF: a
(2)
v -
av an
:=
ACD7,N(.
Then 9) is
topologically projective
and q
:
T
M,
epimorphism.
In the next step
we
will fix
a
class of sequences in
so-called T-exact sequences and then construct
A-9)ToD&, namely the class of appropriate resolutions of objects.
Q85, IX.1.Defl) A morphism f in A-VoD& is called T-admissible, if topological isomorphism with closed image and if in the canonical decomposition
6.2.4 Definition
it is
a
f
i
=
-
m
T-exact,
of f the
if all its
mapping m in T lies in T. An exact sequence in A-VoD& morphisms are T-admissible. A complex in A-9AoD& C
:
-
-
-
-4
C"
--+
C"_1
---
-
-
-
---
is called
CO
T-projective or topologically projective, if every A-module C,' is topologically projective. Moreover, the complex C is called T-acyclic, if the augmented will be called
188
of Algebras of Smooth Functions
Homology
complex Ck
Ck i
---
-N Ho(C)
Co
:=
coker a,
-4
0
is T-exact. A
T-projective or topologically projective resolution of an object M topologically projective and T-acyclic complex C with Ho(C) - M.
a
6.2.5
then is
theorem Let
Comparison
-24 Ck-1
k4 Ck
C
*
*
*
- 4 CO
and D
be two
5k
4 Dk
...
in
Dk-1 n4
-::->
-
Do
complexes A-9AoZ),I,, topologically projective and T-acyclic. Then there exists for every continuous A-linear mapping f : Ho(C) -4 Ho (D) a morphism of topological complexes f : C - D which induces f that means all components fk : Ck ) Dk of f are continuous and Ho(p f holds true. The class of f is determined homotopy uniquely by f. where C is assumed to be
D
---
PROOF:
We
recursively
Ho(C), D-1
:=
suppose that
f-1,
the
HO(D)
=
construct the chain map f To this end
and
fk_1
f_1
are
:=
f
well
as
C-2
as
:=
D-2
:=
we
0 and
defined for k E N such that for I
already
first set C-1
f-2 =
:=
Now
0.
-1,
-
,
k-1
diagram C,
Cl-I
1fk
Ifl-1
DI commutes.
there exists
Then iM
fA;
a
the induction
inducing
DI-1
(fk-lak)
Ck
im 5k, hence by the T-projectivity Of Ck C ker 5k-1 Dk such that the diagram (6-2.2) commutes for I k. As
-4
=
hypothesis
holds for k
0,
=
thus obtain
we
inductively
a
chain map f
f.
Let f and g be two f
inducing
inductively
a
first s-1
:=
S-2
S-1)
Sk-1 such that for I
homotopy :=
chain maps from C to D.
EDkEN Sk,
Sk
0 and suppose that
we
s
=
=
-1,
fi is true.
(6.2.2)
-
-
-
-
,
91
k
:
Ck
already -
1 the
si-lal
-
--i
=
We then construct
from f to g. To this end set have defined for some k E N maps
Dk+1
equation
61+01
(6.2.3)
By
5k(fk the relation iM
-
9k
-
(6k (fk
Sk-13k)
=
(fk-I
-
9k-1
Sk-1 3k)) C ker
-
6k
6kSk-1)ak
=
Sic-23k-lak
=
iM
0
6k+1 then holds as well. As Ck is topologically projective 5k+1 morphism, there exists a continuous A-linear mapping s,, : C,, -- Dk+l, such that (6.2.3) is satisfied for I k. As for k 0 the induction hypothesis is true, we thus obtain the desired homotopy s. El -
9k
-
and
a
=
T-admissible
=
=
6.3 Continuous Hochschild 6.2.6
Now
have
we
this end let 9A be
an
189
homology
enough
tools to introduce
Abelian category and T
topologically derived
A-TW,1,
:
-i
%
a
functors.
(covariant)
To
additive
object M a topologically projective resolution CM. homology groups Hk(TC c) of the complex TCm the special choice of the resolution CM, we thus obtain for functor LkT: A-TW,j, - A, M -4 Hk(TCm) called the k-th
Then choose for every
functor.
As by the comparison in A do not every k E N
depend an
theorem the
on
additive
left topologically derived Junctor of T. If S : A-Mo,(, -) A is a contravariant additive functor, then we analogously define the k-th right topologically derived Junctor RnS H k(SCM). by RkS(M) MaA- is the functor of the completed tensor product For the special case that T and S HomA (-, M) the functor of continuous A-linear mappings, the corresponding =
=
=
derived functors have their
Note that the
own names:
TbrAk (X, M)
:=
LkT (N),
EXtkA (N' jq)
:=
RkS(N),
(co)homology groups LAJ and RkS
of these groups is Hausdorff if and
6.3
(6.2.4) N E
if it is
only
Ob(A-9AoD,r,).
need not be Hausdorff and that each
complete.
Continuous Hochschild
homology
Aa,,A'. complete locally convex algebra and A' the algebra Ae the has It A. same to convex opposite algebra complete locally Hereby, " the A but like vector b) multiplication (a, opposite space underlying topological the in is A A. an in the object denotes category Obviously, b a, where multiplication A'-9)ToD,,,, where the Ae-module structure of A is given by A' x A D (a (9 d, b) --) a b d E A. Now let M be an object in A'-9XoD,r, or in other words a complete locally of A with convex A-bimodule. Then one defines the continuous Hochschild homology 6.3.1
Let A be
=
a
A' is the
-
-
values in M
by H n (A,
and the continuous Hochschild
A)
A'
=
Torn (A, M),
cohomology of A with
H n (A,
Instead of Hn (A,
M)
n resp. H (A,
M)
A*)
values in N(
by
n
=
we
EXtAe (A, M).
often write
these vector spaces the continuous Hochschild
HHn (A)
homology
resp.
and call
HH'(A)
resp. continuous Hochschild
cohomology of A. 6.3.2
To compute the continuous Hochschild
lution turns out to be very useful. For
by
a
(co)homology the topological
complete locally
convex
algebra
- 4
---
Bar
A it is
the sequence
CBar A
-k4 A'6,A6,---6,A I
"I
k-times
A ea,,A
A'
A
---
0
reso-
given
190
Homology 5k ((CL 0 b)
where
(aa,
0 a, (2)
(9
b)
0
CLO
0
(a2
of Algebras of Smooth Fbnctions
0
ak)
(2)
+
k-1
E (-l)j (a 0 b) & (a, o
+
(9 aj aj+l 0
...
...
(2)
ak)
b,
a,,
+
j=1
(_l)k (a (2) cLkb) (9 (a,
+
One checks
(9
...
0
ak-1))
easily (see for example [112, Sec. 1.1])
a,
that
5k-1
aA, E A.
6k
'
0-
Moreover,
the
mappings Sk
:
CBar
CB
A
A',
A,k
and s_1
:
-
for
homotopy
(a 0 b)
A,k+l)
CA
a
--)
that
0 a, 0
(10 a)
are
0 ak
-4
continuously
(10 b)
0
a
0 a, 0
k-linear and induce
a
...
0 ak
contracting
has for all k E N
means one
6k+1
...
'
Sk + Sk-1
'
5k
=
id.
This
is acyclic. Hence CBa implies that all 5k are T-admissible, but also that CBA A a topologically projective resolution of A in Ae_9Xo0,r,, if we can yet show that all components CABa AeaACD, aA are topologically projective Ae_modules. ,!
is
=
But this follows
directly from the statement (2) in Proposition 6.2.2. If now X complete locally convex Ae_module and if one applies the functors MaA- and HOMAe (-, NQ to the Bar complex, then one obtains the topological Hochschild complex
is
a
C., (A,
M)
and the
C *(A,
:=
JACDAe CB.r A
topological
M)
:=
b :
Hochschild
HOMAe (CA
I
X)
a,,A 24
Ma,,A&,
...
M
-N
0
cocomplex :
0
---)
M
4 131-1
The
Jqa,,A 24
Hom. (A,
NQ
&,A, N[)
Hom.(Aa.,
13k -
-
-
-
.
of (C. (A, M), b.) resp. (C* (A, M), 0 ') gives as desired the continuous (co)homology ofA with values in M. In analogy to ordinary Hochschild hoker bA; (resp. Zk ker pk) continuous Hochschild mology we call the elements Of 71k ker 0") continuous (co)cycles, and the elements of Bk imbk+l (resp. B k Hochschild (co) boundaries.
homology
Hochschild
=
=
=
6.3.3 Proposition Let A be a complete locally convex algebra and M a complete locally convex Ae-module. Then the low dimensional continuous Hochschild homology groups compute
as
Ho (A, NQ
follows:
=
MA
M/1 E ajmj
-
mj aj
a, 0 M, E
iEN
Ho (A, M)
=
H1 (A, M)
=
NO
fm E MI
am
=
ma
Der, (A, M) / Deri (A, M).
for all
a
E
Al,
ACD,,MJ,
6.3 Continuous Hochschild
191
homology
Hereby, Der,(A, M) denotes the space of continuous derivations on A with values in M, and Deri (A, X) the set consisting of all inner derivations that means of all derivations of the form a i-- ad(m) (a) [m, a] with m E M. Moreover, if A is ma holds for all a (-= A commutative and M a symmetric Aa-module, i.e. if am and m E M, then =
=
HHj(A)=?iA/k where For
?!Alk
is the space of
explanation:
Hj(A,M)=JVCDA?!A1k
and
K5hler differentials.
topological
topological Kdhler differentials fIA/k as defined complete locally convex topology as shown in B.2.
The space of
Section B.2 carries
canonical
a
in
If
one supplies the space of continuous derivations Derc(A, M) with the strong operator topology that means with the topology of uniform convergence on bounded sets, then
Derc(A, M)
becomes
complete locally
a
By definition PO(m) (a)=
PROOF:
convex
vector space
am-ma
as
P'(f) (a0b) =f(ab)-af(b)-
and
f (a)b hold for every continuous k-linear mapping f : Aa,,A cohomology groups Ho (A, X) and 111 (A, M), As bi (a 0 m)
homology Now let
group
us
to the :
:
that A is commutative and M
case
MaA
(MOD,A) /im b2. Therefore, E)
the
M is
-4
further
a
gives the obtains
one
Hi (A, M)
WAIIAlk)
--)
Together with continuous mapping
m
0
ai 0 Ci E
E aidjm 0 ciEj + im b2
symmetric Aquotient
is the
--) Ta
0 da
1 of Section
B.2
jEN
1 and E ELj
0
Ej
E
1 the relation -
aiciEjm 0 ELj)
+ im b2
=
holds true, T factorizes to
H, (A, M),
where
to each
?!Alk
other,
=
as one
a
continuous A-linear
morphism mappings
j/-92.
The two thus defined
checks
by
some
_VaA?!A1k and 0
are
inEl
short calculation.
Proposition Let A be a commutative, complete locally convex algebra and M CBar complete locally convex A-module. Then the antisymmetrization ek : CBar -A,k) A,k
6.3.4
(a 0 b)
0 a, 0
...
0 a,,
-4
sgn
(a) ((a 0 b)
(D a,(,) 0
...
0
a,(k))
CFESk
is
0
ij
ij
a
one
m0Eaj0cj,- Eajm(&cj+imb2-
E (aim (D cidjEj
=
a
the closed ideal
jEN
As for two elements
a
mapping
T:MaAj-4Hj(A,M),
verse
ma,
trivial, hence H,(A,M)
is well-defined and continuous.
thus obtains
am
-
Ho (A, NC).
come
Then b,
bimodule.
That
X.
--
=
the
well.
continuously Ae-linear F-k
:
Ck (A) JV[)
and induces -4
morphisms
Ck (A) M)
and
Fk
:
Ck (A, JA)
-4
Ck (A, JA)
192
Homology
the continuous Hochschild
on
bk PROOF:
The
(co)chains
*
continuity of
Ek
=:
0
of Algebras of Smooth Functions
such that
Ek
and
pk-1
.
0.
=
ek is clear
by definition. We now show that bk ek 0; analogous argument. We denote by Irl E Sk) 1 < I < k the transposition of 1 and 1 + I and by Sk,1 the set of all permutations a with (T(l) < cy(I + 1). Then for all I < k the set Sk is the disjoint union of the sets F-k
*pk-1
Sk,l'rl.
Hence
the
equality
Sk,1
and
bek(moal
0
0
...
=
0 follows
by
=
'
an
ak)
E (a(,(,)
m
0 a(,(2) (9
...
(9
acr(k)
a(T(k) M (9 a,(2) 0
-
(9 aa(k-1) 0 a,(,)
...
aESk k-1
+
1: E 1=1
(- 1)' sgn ((Y)
(7n
(2)
a,(,) (9
o
...
a,(,)a,(,+,) o
o
...
acr(k)
(3-ESk,i m
-
0 a,(,) (9
...
0
a(,(,,(,)) aCr(TI(1+1)) 0
...
0,
0 aa(k)
which proves the claim.
M
Proposition Under the assumptions of the preceding proposition following continuous mappings:
6.3.5
the
antisym-
metrization operators induce the =k
IF-] k
VCMI /k
IF]
17flk
--
Hk (A, M),
-k
k
A Der, (A, X) :
Hk (A) M)
_4
--
H
k
a
(9
a(,) 0
A
...
A
o
...
dak
"
IF-k (M 0
[Ek(f,
fk
a(k)]
topological algebra
then there exists
'
*
'
a, 0
0
...
0
ak)],
A
da(k)
fk)])
a
E m(o) 0 da(l) A
-4
and M
a
continuous
...
.
finitely generated topologmapping
k
Hk (A, X)
-4
AA Der, (A, N())
f(i)
-k
For the in
da, A
MaA-k rl /k)
If A is finitely generated as ically projective A-module, k
0
(A, M), f, A
[ E m(o)
[7.r]
m
(9
0
f(k)
f(l)
A
A
f(k)
-k
explanation of the symbols fl /k and A we refer the reader to Section B.2 the Appendix. Moreover, denotes the (co)homology class of a continuous
Hochschild PROOF:
(co)cycle. By
the relation bk' F-k
operator induces
a
continuous
0 from
mapping
:
Ek
Proposition 6.3.4,
M6k Xk A
--
the antisymmetrization Hk (A) _V). To show that this
-k
Ek factorizes
by
NCakQA/k
to
a
map
[Elk,
we
first have to prove that the continuous
mappings A 3) a, are
"
IF-k (M o
a, 0
...
Oato
all derivations. For that it suffices to prove
F-k (Tnb 0
c
0 a2 0
...
0
ak) +F-k(MCObo
CL2(9*
...
0
ak)]
E
Hk (A) NC).
(cf. [112, Prop. 1.3.12])
that
-Oak) -E-k(mObco
a20
*
...
0
ak)
6.3 Continuous Hochschild is
a
b, a, setting ao following relations =
(
E
But this is the
Hochschild boundary.
(continuous)
bk+1
193
homology
=
and
c
sgn(cr)m (D
TI,+1
case
Sk+1 I a ' (0)
12
(Y
after
indeed, because
a-'(1) I
11pVW11
=
dGr (V) W)
and
1,
< I
dGr(V) W)
11PVPWV11 11PWV11
=
dGr(V)W)-
of symmetry dGr (V)
reasons
W) ! dGr (W) V) holds as well, hence we obtain (3). proof of the triangle inequality (4) choose a normal v C= V, with
For the
dGr(Vl) V3)
11v
=
dGr(Vl) V3)
-
Pv.,vll.
IN
=
This finishes the
Then
PV3V11
-
:5 dGr (V1)
A.2
spanw with
=
on
Pwv is well-defined follows, hence w 11PWV11 reason Pvw :A 0 follows. Hence altogether
0
dGr(W) V)
By
=
V2)
:5
IN
+
dGr (V2) V3)
-
PV3PV2V11
:5
11V
-
PV2V11
+
11PV2V
-
PV3PV2V1
-
proof.
Polar
decomposition
The
polar decomposition of a linear isomorphy is well-known in linear algebra and analysis. We will need this result several times in this work and will need in particular that the polar decomposition is differentiable. This special property is often not shown in the literature, hence we will prove it here.
functional
A.2.1 Theorem Let V be
(., -)
(Euclidean mappings n : GL(V) the
unitary g E
values and
as
maps
finite dimensional
(real or complex) vector space and Hermitian) scalar product on V. Then there exist smooth -4 GL(V) and s : GL(V) -) GL(V), where u assumes only a
or
s
only positive definite linear g
Moreover,
u
PROOF:
NJ,
maps, such that for all
GL(V) s are
Assign
where
g. The
and
11
-
11
=
U9 S9,
uniquely determined by these properties.
to every N > 0 the open set
denotes the operator
UN then provide
an
open
norm
covering of
UN
of
(-, GL(V).
Ig
E
and
GL(Y)l 11g*g -N idvjj -
g*
the
Note that
and
XEK
Kj
C
the space
Kj,
all seminorms
by
the value of F at E N' with
by
D
1PI :5
,
and
I IK,m *
-
J'
=
we
-
sequence of
a
with
Fr6chet space.
by F(x) := F(')(x). mapping Jm(A) -i defines the so-called jet mapping is defined
the linear
Together
then have
J'1'1-
becomes
x
ax"
C'(0)
a
Jm(A) together
m
(0"g) 1A) 10C,
By the monotony increasing with respect to t as long as s EJ2. t be bijective. If and that has concludes to one Cp Cpl CP2 extends Cp by finally
which
implies bj
to be
of the component functions one
(0, 1
CP (s, 0)
(S to
a
(1)
continuous function
(5)
to
and is
on
tempered
[0, 11
can
[0, 1] X ] 0, 1]
:
O
be extended to
morphism
(SI t)
=
(211 0).
t)
M
product x
N)
structure
x =
on
Altogether
DM
x
21
51
21
then the thus extended
0 satisfies
conditions
J(I, 0)}. 2 (S' t)
[0, 1],
-4
(((S
X
[0, 1]
on
-
1)2 + t2)-100) CP (s, t)
2
[0, 1]
x
and that y is
a
homeo-
but obvious computation shows that the
0)
and that D Y (s,
t)
vanishes for
Let M and N be two manifolds with nonempty boundaries aM a
N U M
(not unique) x
manifold structure
aN
the
topological manifold-with-boundary, such that holds, and finally such that the differentiable the canonical product of the manifolds M' and
N becomes
x
N' coincides with
x
s