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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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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71
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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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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
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209
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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,dimS
with the
topology
induced
For every element
dpz(x)
=
sup
x
X
E
{k
by
X.
NJ
E
defines its
one
depth by
3SO, S1.... )SkEZ: xESO<Sl<
...
<
Skj-
dpz (U) holds for all elements x, ij of a piece S, hence the depth By definition dpz (x) of a piece S is well-defined by dpz(S) dpz(x). Finally, the depth of X is given by =
=
dpz (X)
=
sup
mapping f : X morphism of decomposed spaces, if for that the following holds: 1.1.6
A continuous
(DS3)
f (S) C
(DS4)
the restriction f Is
By
-)
I dpz (S) I
S E
Y between
every
piece S
decomposed E Z
there is
a
spaces is called
piece Rs
E
a
such
Rs, :
S
--i
Rs is smooth.
the condition of frontier and
continuity
a
morphism
erty:
(DS5)
ZJ.
for all S < S' the relation
Rs
<
Rs, is satisfied.
f also has the
following
prop-
1.1
Decomposed
17
spaces
Obviously, the composition of two morphisms is again a morphism, hence the decomposed spaces together with their morphisms form a category 65po,,. If a paracompact topological space X has two decompositions Z and % we will is finer than Z, if the identity mapping is a or that say that Z is coarser than of from morphism decomposed spaces (X, ) to (X, Z). In most cases we are interested in a rather coarse decomposition of a paracompact topological space. Given two
1. 1. 7
decomposed spaces (X, Z) and (Y, ) one can form their cartesian topological sum X II Y. The subspaces S x R with S E Z then form a decomposition of X x Y, the subspaces T c X II Y with T E Z a decomposition of X H Y. Thus (Espt,,, becomes a category with (finite)
X
product and R E or
T E
Y and their
x
products and sums. A topological space
Y C X is called
decomposed subspace, if for all pieces S E Z S, and the corresponding partition Z n Y condition of frontier. In this case (Y, Z n Y) is again a decomposed
the intersection S n Y is of Y satisfies the
a
a
submanifold of
space.
To
give the reader
long
spaces, 1.1.8
Manifolds
way with
one
a
first
impression about the variety of all possible decomposed examples follows.
list of useful and instructive
a
Every single piece.
smooth manifold M is It is
a
decomposed
space in
a
canonical
easily possible (infinitely) many different All of these decomposia decomposed space. the canonical decomposition and usually will not be considered to construct
partitions of M which turn M into tions
are coarser
than
further. 1.1.9
Intervals The most
simple examples of (nontrivial) decomposed spaces are [a, b [, I a, b] or [a, b] with -oo < a < b < oo, and the decomposition into the set ] a, b [ and one respectively two boundary
intervals of the form
given by obviously points.
coarsest
aM Manifolds-with-boundary Let M be a manifold with boundary, S, M \ aM its interior. Then M comprises a decomposed Mo boundary and S2 space with pieces S, and S2. 1.1.10
=
its
=
1.1.11
Cones If X is
=
a
topological
space, the
cone over
X is defined
as
the quotient
space
CX If
M is
=
[0,
1[XX/fO} X
X.
manifold, the cone CM is a decomposed space with its pieces given In case M S' we obtain the := [101 x MI and the set 10, 1[x M. well-known standard cone Xc_. := CS' (Fig. 1.1), and in case M S' fl} the edge X,,,,,,, := CSO (Fig. 1.2). Instead of M one can take a decomposed space X with finitely many pieces S E Z. The cone CX then is decomposed as well, where its pieces are given by the cusp o and the sets ] 0, 1 [ x S. Hereby by
now
the cusp
a
o
=
=
dim CX
=
dim X + 1,
CX
=
dp
dp
X + 1.
=
Stratified
18
and F inctional Structures
Spaces
10
10 7.5
2.!
_.L U
1.1: Standard Cone
Figure
Figure
Pathological example
1.1.12
LU
S, =f0jx]-1,1[
The space X
and
S1
(x,-y)
S2
U
S2
1.2:
C
Edge
R2 with
EWlx>O,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,
<
oc
>
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)
<
In this
z=sin
-
X
case even
(%01)1
dim R, > 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 <
and the
m
(k + 1)-tupel (jo,
jk)
jo j, jk :5 m, comprise a decomposition of s. The sets will be called open faces of s, the closed hulls S[Vjo, j,, Vj, j, are simplices, again and are named faces of s. By gluing together simplices one obtains polyhedra which are decomposed spaces well. A finite simplicial complex K consists of a nonemPty set of simplices in R', as such that the following axioms hold: <
<
...
<
=
...
,
(SCI)
If the
(SC2)
For two
simplex
s
belongs
simplices of
to
K, then
every face of
s
belongs
K the intersection is either empty
to K.
or a common
face.
One associates to every finite simplicial complex K its geometric realization JKJ, which subspace of Rn consisting of the union of all simplices of K. The partition of
is the
JKJ into the open faces of the simplices of K turns JKJ into a decomposed space. By a polyhedron or a triangulizable space one finally understands a topological space which is homeomorphic to such a space JKJ. Given an explicit triangulation that means a homeomorphism h from JKJ onto X, the canonical decomposition of JKJ can be carried over to X naturally via h. ,
1.3.15
part
of)
Neil's Neil's
and the union
XN il I V 1.1.16
parabola A well-known example for a decomposed space is (the real parabola XN61 j(X,1j) E R2 I X3 =,jj2J. Its pieces are given by So 101 S, of the two legs S1, f (x, 1j) E XNei, I Ij > 0} and S12 I(X) 1j) E =
=
=
0}*
<
Whitney umbrella The Whitney polynomial x2 _'Y2z, i. e.
umbrella
Xw,,,,, (Fig. 1-5)
of the real
XWUmb
='
I (X, 1j, Z)
EV I X2
=
IJ2 Zj.
is the
zero
set
Stratified Spaces and F inctional Structures
20
1.5:
Figure
Whitney
Umbrella
f0j, S, f(O, 0, z) I z < 01, decomposition of Xw,,,,, has the pieces So S2 1(0, 0, z)l z > 01 and S3 1(,x,,y, z) E XWUJ V :A 01. Observe that the partition Of XWUmb into To S3 is not a decomposition, because it 1 (0, 0, Z) I Z E RI and T, does not satisfy axiom (DS2); instead one has To n Ti- =SO U S2 =,4 To. In Figure 1.5 one can see the Whitney umbrella including the "handle", which is given by S1. The standard
=
=
=
=
=
=
example for a decomposed space given by WHITNEY Whitney cusp (Fig. 1.6). It is defined as the zero set Xw,,,,, of Z2 X2 The variety Xwc,, has two natural decomposition, the real polynomial -y 2 +.X3 Xwcsp \ S1 namely first the decomposition into the z-axis S, and the complement S2 and secondly the decomposition by Ro S2. We will see S, \ f0j and R2 f0j, R, 1.1.17
[1921
Whitney
cusp Another
is the so-called
-
.
:--
=
in the section
"right"
on
the
=
Whitney conditions which
one
:--
of these
decompositions
is the
one.
spaces which have been
presented up to now decomposed space is the cone comb X, ,, which arises from appropriately gluing the cones CSn to the real half axis R !'. More precisely, one first forms the topological sum of R-'O and all CSn. From this space one constructs XCmb by identifying every point n E N C R :O with the cusp on of the cone CSn. Intuitively, one thus obtains a comb, the teeth of which are given by cones. The pieces of the natural decomposition Of XCmb are given by R -' \ N, ] 0, 1 [ X Sn and In), where n runs through the natural numbers. 1.1.18
are
Cone comb The
decomposed example
all finite dimensional. An
1.1.19
they
are
of
an
infinite dimensional
Manifolds-with-corners Manifolds-with-corners defined in
appropriate
way,
namely
are
decomposed
like in MELROSE
[126,
Sec.
spaces, if
2.1].
The
by MELROSE entails that one can find a canonical partition of the boundary by embedded hypersurfaces. The usual definition of manifolds-with-corners via charts in model spaces W,' (R :O)' x W` does not allow this in general. This problem is discussed in [127]. In the following we will introduce the definition by MELROSE [126, Sec. 2.1] in a slightly more general form and will give the canonical decomposition of definition
=
manifolds-with-corners.
1.1
Decomposed
21
spaces
Figure
Let M be datum for M
understand
finite family
('qj)jEJ
(CD1) L(M)
=
(CD2)
of smooth functions
fX E
R"'
I ilj(x) :
=
-jij
Then the
E
M
family
I ijj(L(ij))
corner
data L,
N >
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
c
let U C X be
an
open
=
1
X
E
Mi and
1.1.21
Mjo
c
Mi. This
Proposition If X
is
proves
a
glued
2
(DS2).
13
space,
so
is the
cone
CX
over
X.
Let M1, PROOF: Mk be manifolds-with-boundary which after glueing them together give the space X. Using Proposition C.4.2 of Appendix C one can furnish the 10, 11 X Mk with differentiable structures. The gluing funcproducts 10, 11 X M1, tions hi+1 induce gluing functions Chi+, : D ([0, 1 [x Mi+,) ---) CEi by -
-
-
)
-
-
-
,
Chi+, (t, m)
[{0}
x
Ei]
if
t
if
t > 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 <
This proves the claim.
dPZ2 (.9)
+ 1 <
dp Z2 N
'
11
By the lemma the depth dpz(x) of a point x of a stratified space (X, 8) is independent of the specially chosen decomposition Z which defines 8 in a neighborhood of x. Therefore it makes sense, to define the depth of x with respect to the stratification 8 by dps (x) dpz(x). =
1.2.6 Lemma
depth k.
(Cf
Then the
MATHER
[123,
family (Tk)kEN
2.2]) Let Tk be the set of all points of X of E-decomposition of X inducing 8.
Lem.
is
a
separability of X the set Tk has to be the countable manifolds, hence is a E-manifold. The local finiteness of topological the family (Tk)kEN follows immediately from the fact that every point possesses a neighborhood decomposed in finitely many pieces according to (ST1). As the points PROOF:
By (STI)
sum
and the
of smooth
25
1.2 Stratifications
piece of a local decomposition have equal depth, the stratification 8 is induced by that means the set germ [Tkl., coincides for every x E X with 8_'. So it remains show that the condition of frontier is satisfied. Let x E T, be a point of depth
of
a
(Tk)kEN to
Obviously then k > 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
is
S0,d
a
a
partition P' of T, in
decomposition of UI
(1)
The
(2)
For every open subset U c X and every decomposition Z of U every of Zk of depth I < k is the union of pieces R of Z.
:=
with d E N
and T-I. Then T' is
a
piece S
C U
IU
Consider
S.x
is
a
a
point
E
x
and let
Tl,+,
S,, be the connected component of x in Tk+,. Then d,,. Let S be a piece of Zk. We claim that
smooth manifold of dimension
either S., C 9
or
S-,n3=0.
This claim will follow from the fact that the set
S.
of all
Ij E S-x with ij E 9 is open and closed in S-,. That S' is closed follows from the obvious fact that S-, is open in Sx. To prove that S' is open suppose that Ij E 9 and choose an
open
R
=
U n
UjEN Rj
=
hence
UjEN Rj,
S,
n U
Every nonempty Sk+ld,A
=
is
nonempty SA;+l,d,A with d
Sk+l,d,A
condition
one
(1).
checks
R C
a
Thi:s
us now
C
a
one
has
Sk+l,d,A
as
E A but
the union of all d-dimensional
S,
n
-9
=
0 for all S
E
9)k \ A.
d-dimensional manifold, and the family of Tk+1 of all
E N and A C
(2)
a
shows S' to be open in Sx.
define
9 for all S
immediately
Condition
decomposition Z of U inducing shrinking U we can suppose that countable family of pieces Rj, j E N
with
and Z satisfies the condition of frontier
S n U. As -y E
For every subset A C g)k let S-, with x E Tk+1 such that S,,
the
together
in which ij lies. After
of
Z, piece Sx. By induction hypothesis there is
of Z with R c
U of -y in X
neighborhood
S. Let R be the
holds
g)k is
that the
as
a partition of TA;+,. By definition of corresponding partition Zk+1 satisfies
well. To show this let U and Z be like in
(2)
piece of Z of depth k + 1. By induction hypothesis every piece S of Zk satisfies either R C 9 or R n -9 0, hence R lies in some piece Sk+l,dimR,ATherefore Sk+l,d,A n U is the union of all d-dimensional pieces of depth k + 1 of Z which intersect Sk+l,d,A n U. This completes the induction. and let R c U be
a
=
26
Stratified So
finally,
family (S)SETk,kEN comprises
the
Spaces
and Functional Structures
decomposition
a
of X and satisfies the
claim.
Corollary For
1.2.8
R
Cl
every stratum S of X there
only finitely many
are
PROOF:
Assume that
(Rj)jEN
is
sequence of
a
pairwise disjoint
X Vj- for all j, any neighborhood of contradicts the local finiteness of the decomposition Zs.
Choose
1.2.9
E R. As S C
Now,
skeleton Xk
be defined
In this context X' is
even a
we
manifold,
a
has
a
strata
meets each of the
x
Rj Rj.
>
S.
This El
canonical
decomposition,
its k-
the union of all strata of X of dimension < k and
as
stratified space.
will denote the E-manifold X' has to be
X0_ follows. Note that
=
(X, 8)
every stratified space
as can
inherits the structure of
X
strata R with
S.
>
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 <
n.
a singular chart around Suppose that rk x < n.
x
with minimal
n
E N.
Without restriction
we
We can
30
Stratified
achieve after x
induces
mo/m'0
affine transformation that
an
surjective homomorphism
a
nix/m!,
--4
open
neighborhood
form
a
Hi
=
yi,
x
mx/m.'
basis of
x1u..
-
of
.
=
mo
yrk
x
an
hence
=
[Yj I
=
-"
+ M?, i
H1,
Hrk
map
1,
-
-
,
CI(O)
E
an
rk
-
x
with
Ux has been
then be written
can
x
E ci [Hilo
Co +
=
R' and
germ
rk
Iflo
surjective linear
a
E C' (U,,) such that Ux c U is
neighborhood of the origin in fflo of a function f E COO(O)
open
Every
and Functional Structures
0. As mentioned above the chart
--4 ni, .,
and such that the elements YI Then choose smooth functions
where 0 is
shrinked, if necessary. in the form
x(x)
y',
Now choose
.
x*
Spaces
mod M20)
[h]o
+
j=1
where the ci ideal 9 ker 3F
=
By rk Quod
to y.
true.
The
1.3.11
x
=
a
:
0
a
n
=
be chosen to lie in the
can
vanishing
there exist Hrk
Mo/M2. Hence, 0
x(U),
over
x+,,
-
-
-
,
H"
because such
E 9
after further
shrinking diffeomorphism onto its image. By Hi-xIU. =0 for i =rkx+l,... n the map Rn is
--1
singular
a
chart over U., and H a transition map from x n is minimality assumption of n, hence rk x =
x
-
immediately
The function rk: X E X there exists
X
For every
a
after
proof just given.
N, x -4 rk x is lower semicontinuous that neighborhood U with rk -LJ < rk x for all -Y E U. --)
shrinking
y
:
U and 0
U
---
Rn and
x :
embedding
an
K
U :
-4
0
0 c
--->
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)
<
11ij
C
x1l,
-
where dGr is the vector space distance defined in Appendix A. 1. Similarly to Whitney's condition (B) one shows that the condition (W) is invariant under diffeornorphisms of
V,
class of
a
so
the notion of
(W)-regularity
at X0 E R of
a
pair (R, S) of C'-submanifolds
manifold M is well-defined.
For every e'-stratified
subspace
of
a
manifold M condition
[175] always topological local triviality of the work of KU0 [105] and VERDIER [175) it follows to
(W) implies according
considered stratified space. By the that for a subanalytic stratification
of
a subanalytic space, which means that all strata are subanalytic, condition (W) implies condition (B). In the category of complex analytic stratifications of a complex analytic variety (W) and (B) are even equivalent (see TROTMAN [174]). In the real algebraic case the situation is different; here the two conditions are not equivalent.
But there is
Using HIRONAKA's desingularization theorem VERDIER [175] subanalytic set (resp. semialgebraic set, resp. complex analytic variety) has a (W)-regular subanalytic (resp. semialgebraic, resp. complex analytic) stratification. Meanwhile there exist proofs of this result by LOJASIEWICZ-STASICAWACHTA [116] and DENKOWSKA-WACHTA [51] which use only elementary methods and do not need HIRONAKA's resolution of singularities. more.
could prove that every
Bekka's condition
1.4.13 open
R
=
(CI)
(C2)
neighborhood
p`(O).
BEKKA
[8, 9]
there exists
an
submersive,
and
C A is
calls the
open
for every sequence sequence of kernels
T_,R
(C) Suppose
T of R C M there is
that
R, S
C
M
are
disjoint
el-mapping p : pair (R, S) (C)-regular at X E given
neighborhood
a
U of
Wk)kEN of points ijk kerT-u,(p1s) converges
x
E
to
T
Rn
such that Pisnu
S a
--
:
and that
on an
R-f'-' such that
-S,
if
S n U
---
R is
converging to x such that the subspace A c T."M, the relation
satisfied.
Now, the chain of implications (B) =:> (C) ==> (A) holds, where the second implica(C) => (A) follows immediately by definition of (C). The reader will find a proof
tion
of
(B)
=>
(C)
in
Corollary
3.4.3. Note that the inverse
implications
are
not true in
general.
According
to the results of BEKKA
[8, 9] an essential feature of (C)-regular strat(C)-regularity already allows to construct control data on the underlying stratified in the sense of MATHER (as they are defined in 3.6.4). A consequence of this is that (C)-regular stratified spaces are topologically locally trivial. ified spaces is that
1.4.14 as
An important
ingredient
for the formulation of the
following condition
for the later defined curvature moderate stratifications is the notion of
a
as
well
projection
valued section.
A C'-mapping P : T -4 End (TM) from a submanifold T c M to the endomorphism bundle of TM is called projection valued section (of class el), if for every x E T the image P.,, is a projection in End(T.,M), that means if P' Px. =
X
1.4 Local
71iviality and
the
Every submanifold S End(RI) by mapping to
C R1
induces
every
x
a
projection valued section PS : S orthogonal projection of R' onto T'S.
canonical
E S the
(5)
The Bekka-Trotman condition
1.4.15
ifolds of R1 satisfies the condition
(5)
Uof xinR' anda 5 >0 suchthat
at
Z) 11
-
We say that
R, if there
E
x
forallij
11PS'-Y0J
41
conditions
Whitney
E
an
SnU and zE RnU
6111J
!
pair (R, S) of submanopen neighborhood
a
exists
Z11
-
(1.4.2)
-
A stratified space (X, CI) with smooth structure is called (5)-stratified, if for every pair (R, S) of strata and every point X E R there exists a singular chart x : U -4 Rn around
x
(x(R n U), x(S n U)) satisfies condition (5) at x(x). (6) has been introduced originally by BEKKA-TROTMAN [11]
such that
The condition
and
[8]. It is very useful for any considerations of metric properties of a stratified In particular, the condition (5) guarantees that the geodesic distance on X. space X is locally finite (see Section 1.6) and, in case X has no strata of dimension < BEKKA
dim X
-
2 that X is volume
One
can
xinRn
and has finite Hausdorff
measure
somewhat weaken the Bekka-Trotman-condition
the local finiteness of the satisfies at
regular
x
(see
the condition
is
geodesic length
(51)
(6)
in such
We say that the
preserved.
R -',
with I E
[8]
if there is
an
a
way that
pair (R, S)
neighborhood
open
U of
anda5>OsuchthatforallyESnUandzERnU
dps'-Ai
-
Z)11
!
51[y
Z11"".
-
(1.4.3)
Proposition If the pair (R, S) of submanifolds of Euclidean Whitney's condition (B) at x E R, then (R, S) is (A)+(5)-regular at x.
1.4.16
PROOF:
That
(R, S)
introduction of the
satisfies
(A)
at x, has been shown
Let
Whitney conditions.
In other words this
x.
B EKKA
[55, 581).
and FERRAROTTI
means
V k E S and Zk E R with
that there
linlk-- oo IJ k
=:
are
us now
liMk--4oo Zk
x
=
already shortly after the
(5)
suppose that
sequences
space satisfies
('IJk)kEN
and
does not hold at
(Zk)kEN
of
points
and
1
k
After selection of and on v
(PS;Y,)kEN
subsequence
to the
the sequences
cannot be
a
an
111J-k
-
ZkJJ
(11'Jk-Zkll
and
element of
r.
a
recent work
Whitney stratified
JJPS,yk(1Jk
(Zk)kEN
sets.
Zk)JJ-
then converges to onto
a
the relation
But this contradicts
by
-
a
unit vector
V E
R'
kEN
orthogonal projection
(1J-k)kEN
1.4.17 Remark Stratified sets which
ered in detail in
'?
subspace
JJPvJJ
:5
r C
-1 holds k
R'.
By assumption N, hence
for all k E
Whitney (B).
El
satisfy conditions (A)+(5) have
BEKKA-TROTMAN
These spaces form
an
[12]
been consid-
and have been named
intermediate class between
stratified spaces and (C)-regular spaces. We have to postpone the implies (C) till 3.4.4.
weakly Whitney
proof that (A)+(6)
42
Stratified
The
1.5
sheaf.of Whitney functions
A smooth structure functions via
Spaces and F inctional Structures
on a
stratified space X generates besides the sheaf IS' of smooth
fixed
a
covering by -chart domains the sheaf of so-called Whitney funcplay an important role for the extension theory of smooth funcand 3.8) as well as for cohomological considerations of X (Section
tions. This sheaf will
(Sections
tions
1.7
5.4). For the definition of
1.5.1
chart
singular
x:
and there exists
X
Whitney
R. Then A:=
-i
functions let
x(X)
is
a
suppose first that X has
us
locally
closed stratified
a
subspace
global of Rn
open subset 0 of Euclidean space such that A n o is closed in 0.
an
m E MO U fool. For every in X locally closed set U the subset x(LQ locally closed, hence by Appendix C there exist the spaces Jln(x(U)) and 811(x(U)) of m-jets resp. Whitney functions of class CT, on X. Via the chart x we now pull back these spaces to X, that means in other words we set
Furthermore let
of Rn then is
J'
X'X
Now, if
U
(U)
through
:=
J'(x(U))
F_'
and
X'X
(U)
F,'(x(U)).
:=
all open subsets of
X, then we obtain two sheaves Jx' Rn and corresponding argumentation are left to the reader. The first sheaf is called the sheaf of m-jets on X with respect to the chart x, the second one the sheaf of Whitney functions of class C' on X with respect to the chart
81 X'R
x.
runs
where the details of the
n,
is a subsheaf of J" According to the construction in Section C.3 P_' X'X X,X' For the following a new representation for P-m will prove to be useful. X'X
known that
sheaf is determined
a
uniquely by
its espace 6tal6
(see
It is well-
GODEMENT
[60]
for the necessary sheaf theoretic notions). This suggests to determine the espace 6tal6 n) for Whitney functions. To shorten notion we will often write F'M instead of
lt(F-ln X'R X'X
As
.
a
set
look like? To
It (F-')
answer
is the union of all stalks
8,
x
X
this
question, let
z
=
x(x)
E Rn.
E X.
How do the stalks 8m X
Then let
us
recall that 01
Rn'Z
denotes the stalk of all germs of smooth functions on Rn at z and 9'T, (A, Rn)z C C T' Rn'Z the ideal of of function germs which are flat on A of order m (see Section C.3). That means
0' (A, Rn), consists of all germs [f],,
of f up to order
vanish
m
on
A.
identify the stalk 8' is given as follows. t(F-M X,R n) one can
X
open denote
by f
A n V
According
with the
E
CR%,z
quotient
:
V
--)
The
topology
-4
It (EI)
8'(U)
with U C X open consists of all sections F
there exists
an
open set V C 0 with
x
E
:
U
--i
V and
of
R with V c 0
the mapping z -4 [f]z + 0'(X, Rn)z. equipped with the finest topology such that all T are continuous. :
is
E U
partial derivatives
CR'n,z/3'(X, Rn)z.
For every smooth function f
t(P_') x
such that all
to WHITNEY's extension theorem C.3.2
Then
Thus
t(F,1) a
such that for every function f E C .. (V) such
[flx(,j) + 0'(X, Rn)x(,j) for all ij E x-1 (V). Now, setting FM (x) := f (x) FM (x) does obviously not depend from the special choice of f that means we can assign to F a function FM : X -4 R which lies in 12"(U) by definition of E'. Altogether 'one thus obtains a canonical epimorphism 8' --) C' of sheaves of
that
[F],j
=
the value
,
commutative
algebras.
The definition of the sheaf of Whitney functions does of the
global
chart x, but in most
applications
this
depend on the special choice dependence does not play an
1.5 The sheaf of
essential role. Therefore Let
1.5.2
will often not denote it.
general
only singular charts xj
Uj
:
-4
Then
instead of
Fuj,Xi
the space
JX''U (U)
*
Oj
R'i, j
C
E J.
Moreover,
pair
The
by
F,1
X'U
=
=
0)
subspace of all families F
analogously for F-X'.U(U)
JX,U(U)jEi
let
Finally,
we
TU(F)
as
smooth
or
consisting of
abbreviate and write
us
Jj'
result
iunuinui
(Fj)jEJ X'U (U)
J'
with -4
immediately from
Fj
J'
X'U
E
Fj'
=
well,
(Fiifinui
as one
.
Then
JX''U
derives from the
will be denoted
with U C U C X
(U)
those of X': for F
=
3
1
sheaf axioms
a
(or even locally closed) subset U C X (Fj)jEJ Of M-jets Fj E J;I(U n Uj) such
iunuinui
The restriction morphisms TV U
(U).
open and
let
atlas of X
of indices j, i the relation
F(0) i is satisfied.
stratified space X with
define for every open
we
.
that for every
a
(countable) singular
a
the set of all families F
as
of
case
C'-structure. Then choose
a
even
we
consider the
us now
43
functions
Whitney
(Fj)
E
but fulfill the
and
F_XmU are presheaves, corresponding properties of Jj1
thus have obtained the desired sheaves Jm
X'U
and 8'
X'U
.
and
E;'.
We will call them
respectively the sheaf of Whitney functions on X of class Cm. Let us mention explicitly that both sheaves depend on the special choice of the atlas U. Like in the Euclidean embeddable case there exists a natural sheaf morphism F(0) the function 50) E Vn(U) with 00) F_' -4 Cm by assigning to F E FTn i X'U iunui X'U
the sheaf of m-jets
=
Example Let X be an n-dimensional manifold. Then, on the one hand there embedding x: X ---> RN into some Euclidean space of large enough dimension and on the other hand an open covering (Uj)jEJ together with differentiable charts and F-m associated to these xj : Uj -4 Rn. The sheaves of Whitney functions 8m X'U X'X two initial situations are different in general, and comprise in a certain sense the two is equal to the space of extreme examples of such sheaves. If N > n, then F,1 X'X (X) the algebra of m-times to Em is in and of canonically isomorphic m-jets x(X) RN, X'U (X) 1.5.3
exists
an
continuously differentiable 1.5.4
Proposition
stratified space
The sheaf El
(X, Cm) X'U
The
proof
on
of
X'U
holds for the sheaf Jm PROOF:
functions
X.
Whitney functions of class ism associated
with Cm-structure and
an
atlas U is afine sheaf
The
to
a
same
of m-jets.
can
be
performed analogously
to the
one
for Theorem 1.3.13. n
1.5.5
The well-known
norms
defined in
Appendix C
on
spaces of jets and
Whitney
and F,1 (U). To see this let X (Kj)jcj X'U X'U (U) be a family of compact sets Kj c Uj n U and I C J a finite family of indices. Then F'm (U) by define seminorms I Jjc,,,,I and 11 llx,m,j on Jm X'U X'U (U) resp.
functions
can
be carried
over
to JM
=
-
-
IF1 _qc,m,j
Jm X'U (U),
JFj1Kj,m)
F E
IlGjllKj,m,
G E 8m (U). X'U
iEI
JJGJJj,c,m,I iEI
and
44
Stratified Spaces and F inctional Structures
If now Kj runs through a compact exhaustion of Uj and I through all finite index and E' (U) become Fr6chet algebras with seminorms sets, then J' X'U (U) X'U resp.
Jjx,,,j.
1.6
Rectifiable
and
curves
regularity
points of a connected differentiable manifold can be connected by a curve length. In a stratified space with smooth structure this need not be the case anymore. Moreover, for such spaces it is not immediately clear what to understand by a curve of finite length or in other words under a rectifiable curve. The goal of
Any
two
of finite
this section is to introduce the
appropriate notions. Besides that we will introduce regularity locally in singular charts relate the length of a rectifiable curve to the Euclidean distance. These regularity notions will serve to better understand the metric properties of a singular space, but also to formulate and different
notions which
prove extension theorems for smooth functions
First let
1.6.1
us
consider the
metric space. Then for every
5(x, y) is
well-defined,
[t-, t+], J'yj
f
(Y, d)
x, y E Y the
points
space.
be
a path connected geodesic distance
f 1-yj I -y E C([O, 11; Y), 'y(0) x and -y(l) yj E where e, ([t-, t+], Y) is the set of all curves in Y defined
and the sup
=
pair
(A)-stratified
situation. Let
following of
on an
=
inf
=
length
of the
curve
-y
:
[t-, t+1
E d(-/(tj),'y(tj_j)) I k E N,
Y is
--
t-
=
to
=
[0, oo] on
the interval
given by
<
tj
<
<
tk-1
<
tk
=
t+
1<j
Explanation To avoid any confusion with respect by a curve or a path in a topological space Y with real numbers a continuous mapping -y : [t-, t+1 -understand the image of a curve. 1.6.2
us
recall that
The
distance need not
geodesic
always
be
finite,
as
the
to the used notation let
Y
we
t- <
always understand t+. By an are we
following example
from
[591
shows. 1.6.3
Example Let
X=
f0j
U
X C R' be the stratified
EW
I (t,Tt COS O,Tt sin 0)
Then X inherits from R'
subspace
It> 0,
0 E
[0,27t],
Tt=2t+tsin(l/t)1.
metric
by restriction of the Euclidean distance. Any two points x, -y E X \ {01 can obviously be connected by a rectifiable curve. But 5 (0, x.) oo holds for every point X E X \ 101, as the following calculation shows with a
=
x
=
(t, Tt COS 0, Tt sin 0),
'
s,,
27rn
and s'n
t
5 (0,
X)
1 :=
27rn+
:
t
fo, V/11+1
srds>fo if.1
+
ds
t
-3 +
fo
Sn
S
I cos(1/s) I ds
fS
> -3 + nEN>0
-3 +
00.
12n + 1 nGN>0
s n
I cos(1/s) I
ds
1.6 Rectifiable
curves
and
45
regularity
(Y, d) can be connected by a rectifiable length then 5 (x, ij) < oo holds for all X, y E Y. In this case we say that Y is a finitely path connected space. -If every point of (Y, d) has a basis of finitely path connected neighborhoods, then (Y, d) is called locally finitely path connected. A connected locally finitely path connected metric space is finitely path connected. A curve y : [t-, t+1 -4 X in a stratified space (X, 1?') with smooth structure is called rectifiable, if there exists a neighborhood U C X of -Y([t-, t+]) and a singular In
1.6.4
case
that
curve
points of
two
by
means
a curve
a
metric space
-y of finite
,
Rn, such that the curve x -y is rectifiable with respect to the Euclidean analogy to the metric case one calls a stratified space (X, 'C"o) (locally) finitely path connected, if any two points of X can be connected by a rectifiable curve, respectively if every point has a basis of finitely path connected neighborhoods. One now checks easily that the rectifiability of a curve in X is independent of chart
x :
the x :
U
--
special
U
Rn
-)
choice of the
chart that
a
y
singular singular chart, then
with
compatible 1.6.5
-
In
metric.
Let
us
x
-
means
-y is rectifiable
if as
x
-
y is
a
rectifiable
suppose that A is
lower bound from below
a
versa
V
with
-)
R1
subset of the Euclidean space Rn and that d is the
Obviously the geodesic distance 6(x,-y) given by the Euclidean distance, that means
d(x,ii) Vice
:
and which satisfies imy C V C U.
Euclidean distance restricted to A. a
curve
well for every chart y
=
jjx-ijjj
:
we
make the
so
A has
x,,Lj E A.
6(x,ij),
this need not be the case anymore,
equivalent. Therefore
on
the two metrics
are
in
general
not
following definition.
1.6.6 Definition
set K C R1 is called I-
regular with I E
exists
(TOUGERON [170, Def 3.10]) A compact R ', if K is finitely path connected and there
a
constant C >
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
<
C1111i
-
follows from
X11
:5 C1 11-Y
I I-Lj
-
xJ I
< T <
the strata R > S. For every such stratum define
VR (IJ)
X111/')
-
a
(1.6.1)
I and I > 1. Now
vector field
VR : R
we
TR
--
consider
by
11X--Y11 _P'Y(X--Y)11p,Y(X y)112
=
-
For.
every -y E R let
there is
means
a
QU
0 be the
>
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
<
'YIU(t))112
-
Y-Y M
(X
TV (t)),
X
7, W)
=
1
t.+u lht'i M
-
-XII
=
-t +
11-Y
-
-XII
(1.6.2)
We calculate further t
fo,
t
11- ,Js)ll
ds
fo,
lht-y(s) I I Ry" (.) (X
-
-
X11
-Y' W) I
Ids <6
1 1Y'J (S)
1
51
(11-Y
-
X111/'
-
ON
-
X11
-
t)'/'),
-
X1 I'/
ds
(1.6.3)
1.6 Rectifiable
curves
hence -y,, has finite
and
length. As
-y.u ([0, t,+J [) \,yu ([0, t,+,[)
47
regularity U is
cannot be
relatively compact
a
empty. Moreover, by
subset of
(1.6.2)
R1, the
set Z
:=:
the set Z is contained
Suppose that Z has two different elements z and z'. Then there exist two disjoint balls B and B' around z resp. z' andy has to oscillate infinitely often between B and B'. In particular -/ cannot have finite length.. Consequently Z consists of exactly one point -U, contained in in the stratum R, :5 R. By the maximality of t+ the stratum R, cannot be equal to R, hence R, < R. If R, =,k S we continue and obtain as end point of the path -y,,, a point 'Y2 of a stratum R2 < R1. Recursively we thus in U.
> Rk with ij A; and of strata R > R, > points ij ij 0, ij 1, < < As U contains I k and I where E only R1, t', limt-"+,-I -Y'J' -, (t) t,+J, -1. 1that after terminates means the recursion steps, finitely many finitely many strata, S for an appropriate k. Now, we assign to the point ijl, E S the path -y'J, C S Rk defined above; it connects IJk with x within S. The total length of the path 7 composed by -yj, -yj,, -y,,, then sums up by (1. 6.3), (1.6.2) and (1. 6. 1) to
obtain
a
sequence of
=
-
-
...
-
,
=
=
1'Y1 :5
C
((11-9
=
C
11Y
by
a
-
111J1
-
X111")
+
(11-Ul
f -L, C11. Consequently, rectifiable path in X, and
where C x
-
X111" X111"
=
max
5(x, -Y)
-
X111"
every
:5 C
-
point
IN
-
11'Y2
-
X111/1)
y of X n U
+
can
+
111Jk
-
X111/1)
be connected with
X11
holds. This proves the claim. In
possible to connect any two points of a connected regular (A)piecewise differentiable path, but at least it is possible to connect so-called weakly piecewise differentiable path. it is not
general
stratified space them
by
a
by
a
1.6.9 Definition Let X be
a
stratified space with smooth structure. A curve T in a relatively open subset I C [t-, t'l with
X is called weakly piecewise C' , there exists
complement and a singular chart x continuously differentiable, such that
countable x
-
yl, is
J, I I Tx(- (t)) I I
U =
dt <
--)
X around
[t-, t+1
-y([t-, t+])
such that
holds and
(1.6.4)
oo.
differentiability set of -y. If I has the shape I < tk t+, and if x -y1[t,,t,_,1] is continuto < It-) t_1 \ fto) tkJ with tk 1, then one calls -y piecewise C'. 0, ously differentiable for j The space of all piecewise-differentiable el-curves from [t-, t+1 to X will be denoted by C" ([t-, t+1; X). The set I then is called the *
'
=
...
*
=
-
)
=
-
,
A weakly piecewise Cl-curve is obviously rectifiable, where the length of x -Y is given by the integral in (1.6.4). By Proposition 1.3.16 every curve composed of finitely many rectifiable resp. weakly piecewise Cl-curves is rectifiable resp. weakly piecewise again. -
48
Stratified
1. 6. 10 Lemma Let A
chart around X
the
fulfilling
[t-, t+1
:
/\([t-, t+1). following
--
X be
Spaces
rectifiable
a
Then there exists
curve
and F mctional Structures
and
x :
U
--4
R'
weakly-piecewise el-curve -Y
a
:
singular [t-, t+] a
estimation:
Ix -,Y1 :! Ix Al.
(1.6.5)
-
Without loss of
PROOF:
generality we can assume X c Rn. As every stratum It E [t-, t+1 I A(t) E S} has to be a locally closed locally closed, Is subset of [t-, t+1 With the help of the fact that every open subset of R is the union of countably many open intervals one checks easily that Is is the disjoint union of countably many open, closed or semiclosed intervals.'In particular, Is has countably many connected components. Explicitely S C X is
:=
-
U
is
IS,k)
kEN
where
0 for k =7 1 and IS,k is either empty or one of the intervals It- t+ S,k) S,k or [tS,k t+S,k I with t < t+S,k* We now define an open set S, S, k
IS,k n Isj
[t-S,k t+S,k]
5
=
ItS,k) t+ k]
s
I
U
I'S,k)
(S,k)EJ
where J consists of all
[t-, t+1 we
E
8
x
N with
0.
IS,k :A
The
complement
countably many points, [t-, t+1 strictly monotone decreasing sequence (tik,-I)JEN (S, k) "Mill", ts,k,-l ts,k and a strictly monotone increasing sequence (ts+,k,I)IEN E J
choose for
with
pairs (S, k)
and I
contains at most
holds.
=
a
=
tS+,k. Then define for tsi,,k,O < t+S,k,o and 1'M1l00tS-,k,1 and := tS+,k,1-1. Finally set for I E Z := tS,k,l tS,k,-,+, S,k, -1
with
=
I E N" further
Next C
C
Is,k IS,k
points
t+
IS,k,l According holds. and
can
[tS-,k,l) t+S,k,l]
A(IS,k,l)
then lies in the stratum S, and UIEZ IS k I0 S,k chosen in such that the a obviously points way ts,kJ A(t-S,k,l) be connected by differentiable curves Of'YS,k,i : IS,k,l --4 S of minimal is equipped with the Riemannian metric induced by the Euclidean
to construction
The times
A(tS+,,k,l)
:=
can
length, where S scalar product. Then
1'YS,k,11 holds. We
now
put together
the
curves
<
1AJIS,A:,J1
'YS,k,i to
a
path -y
:
[t-, t+1
---
X in the
following
way: t
[A(t) to construction -y then becomes
According tiability set
Next for
we
Whitney
I
=
1
\
F4-
U(S,k,I)EJ XZ I "S,k,,}
will introduce functions
on
if t E
IS,k,17
if t
1.
-4,y(t)
some
first
a
weakly piecewise C'-curve inequality (1 -6.5).
and satisfies
with differenEl
implications from Whitney-Tougeron regularity
the considered space.
1.6 Rectifiable
curves
and
regularity
(TOUGERON [170,
1.6. 11 Lemma
49 Rem.
2.51)
Let x, ij E R' be two
with ii.
F irthermore, let function flat on x of order (m 1) that means D1g (-k) I ocl < m 1. Then g (-y) satisfies the following estimate: rectifiable
connecting
curve
x
g E
M
0 holds for all
=
-
points and
CI(RI),
E
a
-Y
a
N10 be
a
E N' with
-
n,/2 JyjTa
g (ij)
ID'g(&)I.
sup
(1-6.7)
&EIM-Y
PROOF:
According
to the
value theorem
mean
one
has for every function g E
C-(R-)
Ig( y)-g(x)l
<
,Fnjx-igj
ID'g(&)I,
sup &E[-,-Y] I.1=1
[x, U] C Rn is the segment connecting piecewise linear path -y connecting x with 1j:
where
g (-tj)
Vn- jyj
g (x)
-
x
with -U.
Then
we
have for every
I D'g
sup
(1.6.8)
&EiMT
to the limit
Passing If
now
g is flat
estimate
one
over x
shows that
of order
(m
-
(1.6.8) is true for arbitrary rectifiable paths -y. 1), then an easy induction argument proves the
(1.6.7).
El
an I-regular compact set, and F a Whitney function of class C', According to WHITNEY's extension theorem C.3.2 there exists a function f E e,'(Rn) such that F JI(f). For 0C E Nn with locl < m and X E K the function g D'(f T, 'F) is an element of Cm-1011(Rn). Moreover, g is flat on x of order m locl, and equal to the rest term (RTF)() over K. Hence by Lemma 1.6.11 we have for all X, Ij E K the following estimate
Now let K C Rn be m <
oo
K.
over
=
=
-
-
n-
(RmF) () (ij)
1''I 2
X
6 (X, 1j) 'I'l sup
I F(O)
F(O) (x)
&EK
(1.6.9) -
< 2n
This proves the first part of the 1.6.12
2"1 5(x, -y)'I'l IFIK,,Tt.
following proposition.
Proposition (WHITNEY [187],
means
in other words if K is
a
1-regular set,
on
I IK,m *
m E
Let K C Rn be
equivalent metrics, N the seminorms
that K,m
are
JIFIly,,, PROOF:
K define
then for all
equivalent on 8"(A). (TOUGERON [170, Prop. 3.11]) If K is I-regular constant C,, > 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
<
with I E
R -',
then there exists
C, IFIK,mt.
< m. Then there exists
a
constant C > 0 such that for all
x, ij C K and all F E el (K)
I (RTF)()(ij) 1
:5
1 (R'tF)()(ij) I+ Cjx-ijj'-1'1 IFIK,mlX
a
50
Stratified
On the other hand there exists by assumption and for all F E
X
:5 2 n- 21'1 5(X, 1j) '"I :5 Djx
1.6.13
(1.6.9)
and FVnctional Structures
a
constant D > 0 such that
800(K)
I (R"F) () (ij) 1 The claim
Spaces
-
IFIKjal
yj'1'-1jF1K,-nit-
follows from these two estimates.
now
Corollary
For every
The notion of regularity
space X with smooth structure and
regular stratified
is every atlas U of X the space F,' X,U (X)
according to
closed subset A C R' does not make
closed
a
El
subspace of J1 X,U (X).
TOUGERON a
given
in definition 1.6.6 for
statement about the behavior of the
a locally geodesic
distance of two
points x, ij E A with respect to their Euclidean distance, when the approach the boundary aA of A. FERRAROM and WILSON have introduced in their work [59] a new notion of regularity which considers the behavior of the geodesic distance near the boundary. The important application of this new kind of regularity is the extension theory of smooth functions which will be the topic of the next section. To be compatible in our notation with I-regularity as defined above we have appropriately adapted the regularity notion of [59]. The (T, t)-regularity defined in the following corresponds to (T, I-')-regularity in [59]. two
points
x
and ij
1.6.14 Definition
(cf.
FERRAROTTI-WILSON
[59,
Sec.
1])
Let A C R' be
a con-
nected, locally closed and locally finitely path connected set, and Z C A a locally connected finitely path connected boundary set, that means Z C a (A \ Z). The one calls A (r, I)-regular relative Z, where I, T E R !', if for every point z E Z there exists a neighborhood V C RI of z such that v n A is finitely path connected and such that the following axioms (RAI) and (RA2) or (RBI) and (RB2) hold for some constants CI) C2, D > 0: 5 (x,
(RAI)
Z)
(RA2) d(x, -y)
:5 C, :5 D
d(x, Z)0 max
for every
f d(x, Z), d(-tj, Z) I'
5 (X) 1J) :! C2
(RI31) S(x, -y)
: C,
d(x, 1j)1/1
(RI32) d(x, ij)
:5 D
max
says that A is
Z.
with x, ij
max
e
vnA
\
Z
implies
Z
implies
Z.
with x, ij E V n A
max
\
f d(x, Z), d(-y, Z)
for all x, ij E v n A
C2 d(x, 1j)
\
\
I d(x, Z), d(-Lj, Z) I-r(1/1-1)
(T, I)-regular
of type
RA,
in the second of
type
there exists for every z E Z a neighborhood in A satisfying axiom (RAI) (RI31)) we will say briefly that A satisfies (RAI) (resp. (11131)) relative Z with
RB. In
(resp.
case one
d(x, -U)
I d(x, Z), d(-U, Z) I'
5 (X) 1J) :!
In the first
E Vn A
X
case
exponent 1.
1.6 Rectifiable
A stratum S of
9
closure
open subset Z C
regularity
51
stratified space X with smooth structure, or more RA resp. type RB) relative a
a
(T, I)-regular (of type
is called
if there exists
aS,
x(S n U)
such that
and
curves
is
an
a
(T, I)-regular
covering of
set
(of type
X
charts
by singular
RA resp. type
RB)
precisely the relatively DS x :
U
-4
Rn
with respect to
x(Z n U). Finally to
respect
we
(T, I)-regular
(resp. S)
call A
Z, if for
every
Ferrarotti- Wilson
point
regular or briefly FW regular with a neighborhood in A which is for appropriate constants 1, T > I depending on z. of Z there exists
z
with respect to Z n U
Example Every I-regular space A C pn is (1, I)-regular (of type RA and type RB) with respect to any closed, finitely path connected boundary set Z c A. 1.6.15
1.6.16
Example For
respect
to the
Next
S
a
boundary
assign for
we
subanalytic
every
N>'
M E
(T, I)-regular t): c-(T, S
to every
resp.
M(tT
-
CA/S (-r, I)
mT(t Hereby V
z E
There is
an
holds
:=
If
inf
< 5
(X, Z)
is of type RB.
\
=
Z and
an
Lem.
1.1])
Let V C Rn be
a
neighbor-
connected and t E R>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 <
(resp. S)
15 (x, -y), 5 (x, Z) + 6 (-y, Z) I
(RAI) holds,
<
curves
if A
V n A.
over
(X, 1j)
If
1) Ll'
RA,
estimate
Moreover, (RAI) implies the If finally the axiom (RB 1)
j
is of type
to
for all X, -y E V n A
PROOF:
(resp. S)
Z such that V n A is finitely
equivalent
(x, -y)
(RAI)
if A
(FERRAROTTI-WILSON [59,
1.6.17 Lemma
(RAI)'
1)
type RB case an exponent such that (RAI) is satisfied for A exponent exists according to the following lemma.
an
hood of point is
-
space A resp. to every stratum
< I is in the
resp. X. Such
(RAI)
X is FW regular with
(see [59]).
aX
c'(T, I) A
critical constant
set X C Rn the closure
of
:,
2 C,
then choose for
max
x
-FD17 d(x, Z) for all j
E
I d(x, Z) 1/1, d(ij, Z)
V n A \ Z points Yj E V n A \ Z, ! 1. Afterwards choose rectifiable
yj from yj to -jLj+l such that the length Lj of the part passing through V n A \ Z Composing the -yj to a path -y in V n A, then y is continuous by
28(ljj, -yj+,).
Stratified
52 construction and the part
there exists
Spaces
and Functional Structures
(im -y) \ Z lying in v n A \ Z has finite length. By assumption
C > 0 with
a
5(x, Z)
<
i <
2 (x, iji) + E 2 (ijj, ijj+,) j>1
< 2C
(d(x, Z)
111
E d(-yj, Z) 111 )
+
j>1 <
2C
1
(d(x, Z)111
E YJ d(x, Z)111
+
<
4Cd(x, Z)111,
j>1
that
means
(RA1)
is satisfied.
1, then (RA1) would entail for inequalities: If I <
d(x, Z) which is
<
5 (x,
Z)
x
close
C, d(x, Z) 111
<
to Z the
enough
following
chain of
d(x, Z),
<
impossible.
Now let
to the last claim. First shrink V to
us come
a
neighborhood V
VnA\Z
of
z
such
there exists Ij E V n Z with d(x, Z) d(x, 1j). Then one can find for & > 0 a sequence of points -yj E V n A \ Z converging to ij such that 1j_O x, for all j ! 1. Additionally there d(-tj o, ij )'I' < d(-x, -y) 111 + F_ and d(iij, ljj+,) 1/' < -L 2 The path are rectifiable curves -yj from -yj to ijj+l such that I'Yj 1 :5 5(iij,-yj+,) +
that for every
x
E
=
=
22
composed by the 'yj connects has the following upper bound:
,y
1,Y1
<
x
with ij and has finite
E I-Yj 1
(-Yj, -yj+l)
+
length which by assumption
Yj 2i
jEN
jEN
C,
<
( E d(-tjj, ijj+,)
+ 2&
jEN
C, d(x, b) 111 + 2 (Cl +
<
As
arbitrary,
F_ was
Finally
5 (x,
Z)
in this section
:5 C, d(x,
we
the
common
hence the claim.
Z) follows,
will introduce
distance of two subsets of Euclidean space
a
1) F_ 13
notion which rules the behavior of the
or an
(A)-stratified space while approaching
intersection.
1.6.18 Definition
(cf. [170,
Def.
4.4], [118,
Sec.
of 0 c Rn open. Then one says that A, Z are point z C A n Z there exists a neighborhood V
1.5])
Let
as
well
A, Z be
as
two closed subsets
(in 0),
situated
regularly
constants
c E
if for every
N and C > 0
such that
d(x, A)
(RS) If
A, Z
situated every
R'j.
are
d(x, Z)
closed subsets of
(relative Z),
singular
+
an
>
(A)-stratified
if there exists
a
for all
Cd(x, A n Z)c
singular
chart xj the sets xj (A n
Uj)
space X then
A,
E
Z
V.
are
called
=
regularly
of X such that for
(Uj, xj)jj xj (Z n Uj) are regularly
atlas U
and
X
situated in
theory for Whitney
1. 7 Extension
An open of
9n o
c
0 resp. X
\
result follows
following rather useful
only
if for every
relatively closed
C'
E N and
point
>
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
<
locl :5
ml, ml.
given in Appendix C, if A is an open gm(Z; A) kerJm, where Jm: em(A) -4 F-m(Z) one
=
map.
have all necessary
ingredients to formulate
and
proof the announced
extension
(FERRAROTTI-WILSON [591)
Let A c
result.
Generalized Lemma of Hest6nbs
1.7.2 Rn be
locally closed,
Z C A
relatively closed, locally finitely path (T, I)-regular, and F a jet of order m on A. If the restriction Flk\z is a Cm-Whitney function on A \ Z, and F flat over Z of order c ! cAm(T, 1), then F is a Whitney function of class Cm on A. ary set with
The
to which A is
respect
generalized
connected bound-
a
lemma of HESTAAS
immediately
entails the
following main
result of
this section.
Extension Theorem Let X be
1.7.3
(T, I) -regular
stratum, and U
which is flat F E Em
(X)
PROOF
OF
X'U
Fis
=
G and
>
c
Flx\s
1.7.2: We follow the
theorem C.3.2 and that for every
Z
by the fact
E Z
and all
0C
a
stratified space with smooth structure, S an by charts. Then every M-jet G E J- (S)
of X
covering
aS of order
over
with
a
CM(T, 1) S =
X'U
be extended to
can
a
Whitney function
0.
presentation given in [59]. By WHITNEY's extension FIA\Z is a CT' -Whitney function it suffices to show N, I ocl < m the relation
that E
(R 'F) () (y) Ix 1JI-lad
-->
(1.7.1)
0,
-
if
holds,
%
connected
and -y converge to
neighborhood
chosen such that V n Z is be
a
rectifiable
curve
of
z.
with
-y(O)
unique partition the Ai are rectifiable. As V n Z
sup
23(x,,U) (with
d(&, Z)
:5
max
open finitely Obviously V can be Let furthermore -Y : [0, 1] -4 V n A x and -y(l) ij. The path A := imy possesses a with U A, A2 A, c X \ Z and A2 C Z- Obviously finitely path connected, we can choose -Y such that an
according finitely path connected. =
of the fqrm A
JA11 :5
Now let V n A with V C R' be to Definition 1.6.14.
z
given
is
in Lemma
I d(x, Z), d(-y, Z) I
1.6.17)
+
and such that
2 (x, -U)
:5 3
max
15 (x, Z), 6 (-LJ, Z) 1. (1 .7.2)
&EA
Whitney function, hence there exists f E C'(0) with FA\Z \ Z c 0 closed. The function g DO-F(f with oc E Nn, locl < m then is flat on x of order m lccl, and
By assumption FA\Z
is
a
where 0 C R' is open and A
J'(f), T,'nF) E C'-I"1(0)
=
-
-
1. 7 Extension
theory
(RIff)(")(ij)
g(-y)
=
X
Whitney functions
for
regular spaces
on
55
holds for all ij E A \ Z. By FIZ 0 and the compactness of A an to the one in the proof of Lemma 1.6.11 shows the following =
argumentation analogous estimate:
21'1
-
(RF) () (ij)
<
X
n
8(X, Jj)M-10C1
I F(O
sup
F(O) (x)
(1.7.3) C
IFIA,m.
On the other hand the flatness assumption
FIA,,,, where
we
(&EA d(&, Z)')
o
=
sup
=
Ix
-
0
1j1-1'X1
holds, hence the
( (Ix
max -
Ix -'Ulm-lod
Ix --V1
< I is
will from
-
-y 1 :5 D
max
now on
the other hand
Ix
-
-y I
(X' -jL ) Ix--Ul By (1.7.6)
in either
(Rx'F) -
(1.7.5)
case
I
to Lemma 1.6.17. To derive
14IM-10cl =
0
=
0
cases
separately.
(RA2)
and
9
< 5
(RA2).
imply
I d(x, Z), d(ij, Z)
I d(x, Z), d(ij, Z)
C' < -
(1.7.6)
1'. Let D be the constant in axiom
then
-
max
=
C/V
,
(RAI) according
< C , max
! D
I d(x, Z), d(ij Z) I
max
D
then
we
have with
(RAI)
I d(x, Z), d(ij, Z)
case
( (max I d(x, Z), d(y, Z) I Yn-lal 1/1-T
o
Ix
max
type RA and the type RB
I d(x, Z), d(ij, Z)IT,
1X-'j1 on
)
treat the
9(X, -Y) If
15 (x, Z), 5 (ij) Z)1C
M-1al
the exponent in
TYPE RA: First note that in this
Ix
V
0
=
If
Appendix). Consequently
C.2.1 in the
-10d
((S(x,-tj)
X
we
Eq. (1.7.2) entail
relation
J(Rx'F)(')(ij)j
follows, where V
F and
(max 16 (x, Z), 6 (-Li, Z) I c) ,(1.7.4)
o
(see
have used LANDAU's notation
I (RTF) (00 (-y)
(1.7.1)
=
on
(max f d(x, Z), d(-y, Z) (max f d(x, Z), d(ij, Z)
holds, where we have used the fact that by 1, T > 1 the relation (1. 7. 1) follows.
max
d(x, Z), d(-U, Z) I
c"
(1/t-T)(Tn-jVCj)+C/1 JOCI(T-1/1)
c >
)
CAM(T, 1)
I
=
m(IT
-
1).
As
(T
-
1/1)
0
Stratified
56 TYPE RB: First let
Ix
-
-y 1 :5 D
8(X, 1J) Ix If
on
the other hand
Ix
-
-
Using
c >
cl A (-r,
-
I)
1J 1
=
:5
-y I
-5D7
Tm(I
(R'F) vi-io"
IxX
-
-
! D
I
X
(1.7.1)
-
1)
o
0
This proves
C2
<
I d(x, Z), d(ij, Z)JT
max
-YI
C,
Ix
max
max
f d(x, Z), d(ij, Z) I", < -
one
.
and Functional Structures
by (RB2)
Then
8
and
< 5
f d(x, Z), d(iU ,Z) IT(111-1).
-1)
I
-LU
Spaces
then
by (RB 1)
I d(x, Z), d( y, Z) IT(1/1-1)
concludes
(max {d(x, Z), d(-y, Z) (max {d(x, Z), d(-Lj, Z)
also for the type RB
El
case.
1.7.4 Remark The classical lemma of HESTLAS
(see [84]
or
[170,
Lem.
4.3])
follows
Then A
generalized lemma of HESTI NhS, if one takes for A an open subset of is 1-regular, in particular (1, l)-regular with respect to Z C A. Therefore
m-jet F
over
must be
a
from the
A which vanishes
Cl-function
over
over
Z and which is
a
Whitney function
over
A
W. any
\
Z
A.
might ask the question, whether the seemingly rather complicated (T, I)-regularity is really necessary to prove a generalized HESTgAs lemma. In their article [591 FERRAROTTI-WILSON have shown by a counter example that Whitney-Tougeron regularity alone does not suffice to identify flat jets like in 1.7.2 as Whitney functions, and that a further notion of regularity which rules the behavior near the boundary is necessary for an extension result h la HESTLAS. 1.7.5 Remark One
1.7.6 Remark The first extension result for smooth functions
on a
special class of
from the work Of SEELEY
stratified space, namely manifolds-with-corners, originates [1571. SEELEY's result says that for every manifold-with-corners X which is embedded in
some
R'
as a
closed subset there exists
e:
For
an
C'(X)
arbitrary stratified subspace
a
-4
Y c W
continuous extension operator
ff(R7). an
analogous
statement does in
general
not hold.
Multiplying a jet F over A which is flat over Z of order c with a Whitney function on product FG is again flat over Z of order c. But if G is only a Whitney function over A \ Z, the product need not be flat over Z of order c. In the following we will give criteria on G which imply that the product FG is flat of order d < c. A the
relative Z.
1. 7 Extension
theory for Whitney
functions
on
regular
57
spaces
locally closed, m E N, and Z c A a locally closed in A. A Whitney function G E FM(A \ Z) is called
1.7.7 Definition Let A C R1 be
\ Z is dense 0 or if for tempered relative Z of class IS' and order c E R -O, if either Z and oc E Nn, locl < m there exists a neighborhood V c Rn such that
subset such that A
=
A
Whitney
function G E
P,m(xj(S
n
Uj))
is
C
tempered
z
E
Z
G()(ti)d(y,Z)c
sup .uEvnA\Z
as of class C' and order
every
F,'(S) U R -',
over a
stratum S of X is called
tempered relative
if in every chart xj the Whitney function relative xj(aS n Uj) of class C' and order c. E
Gj
obviously linear space of all Whitney functions over A \ Z (resp. S) which tempered relative Z (resp. as) of class Cm and order c. will be denoted by Mm,' (Z; A) (resp. M',c (D S; 9)). Moreover we set Mm(Z; A):= Uc ,,o M',c(Z;A) and U The
are
MM U (as;
S)
:=
UC'0 mum'c (as; S).
tempered relative Z of class IS', if an open neighborhood 0 such that for every M G N one x R :' G is tempered relative 0 n Z of class em and order such that choose can cm E M1 will be denoted This (Z; A). Analogously we define Mu (Z; 9), and call c,. space it the space of relative Z tempered Whitney functions of class C!' over S. A
Whitney function G
E F,-
(A \ Z)
is called
E A there exists
for every
Proposition Under the prerequisites of the preceding definition and the additional assumption that Z is closed in A let c, d be two nonnegative real numbers. If then F E Jm(A) is a jet flat of order c + d and G E EI(A \ Z) tempered of order d, then the product F G E Em(A \ Z) can be continued to a jet E Jm(A) which is flat over Z of order c. In signs: 1.7.8
-
Nrm,d (Z; A) JM,C+d (Z; A) .
Moreover,
an
a multiplier algebra for algebra, and
M' (Z; case
Z is
a
A) g' (Z; A)
C
-
.
is Likewise M' U (Z;'g)
a
multiplier algebra
essentially
in detail first choose
an
a
on
0 such that F
=
C
G E
M'(Z; A)
can
one
has
analogously
easily
from Definitions 1.7.1 and 1.7.7.
consequence of the theorem of TAYLOR. To
f mod 01 (A;
P-00 (A)
gn,c(Z;-g). U
open subset 0 C R' such that A is closed
partial derivatives of f vanish
c
g' (Z; A).
to WHITNEY's extension theorem C.3.2 there exists for F (-=
f
A)
for goo C 800 U (3). U (Z; -9)
The first part of the claim follows
The second part is
the ideal 01 (Z;
boundary as of a stratum S of X
closed subset of the
Mm,d(Z;'g ) jm,c+d(Z;-g) U U
PROOF:
J",(Z; A).
the space M10 (Z; A) is
that'means M' (Z; A) is
In
c
0).
If F is
on
Z.
Consequently
be extended
by
0 to
a
jet
even
the
over
A.
an
P,'(A)
on
a
0.
see
this
According
smooth function
element of 0' (Z; A), all
product
of F with
an
element
It remains to show that the
Stratilled
58 thus defined
is
jet
again
Whitney function. locl <m
a
Spaces
More
and Functional Structures
precisely
one
has to show that for
allZE Z, MENand OCE Nn with
(IZT (F G)) (-) (1j) Ix -'Ulm-ILXI -
if x, -Lj E A converges to
tempered
But
z.
relative Z of class
all
as
C',
I
(1-7.11)
-)
partial derivatives of f vanish
Z,
over
and G is
relation 1.7.11 has to be true. This proves the claim. El
following considerations of this section we will construct nontrivial tempered To this end we first provide in Lemma 1.7.9 a special partition of unity for the complement of a compact set K C R1. The corresponding result plays an essential role for the proof of WHITNEY's extension theorem and will be introduced here in a somewhat more general form than usual (Cf. MALGRANGE [118, Lem. 3.1] or TOUGERON [170, Lem. 2.1]). Afterwards we will show that two regularly situated sets can be separated by a tempered function, and finally that the solution of an ordinary differential equation given by tempered functions of class C" with m E N10 is again tempered of class C'. In the
functions.
1.7.9 Lemma Let d E
N>0 and 5
functions
(1)
(j
For all
(2)
The
j
> 0
be
real, where
we assume
6
=
1 in
case
d
=
1.
compact set K C R'
Then there exists for every E
a countable family of smooth given e,' (R' \ K), j E J, with the following properties:
E
J the relation
of supports supp
family
sets supp
(3) E dpj(x)
with
qbj =
X
E supp
1 for all
X
4)j
is
locally finite, where the by 4' that means
is bounded
4)j
E Rn
holds.
> 0
4)j
number N (x) of the N (x) :! 41.
\ K.
jEJ
(4)
For all
(5)
j the relation 26 d(supp ( j,
For every
m
E N there exists
such that for all
X
E Rn
I a'q)j (x) I PROOF: via the
<
\
hyperplanes
xk
>
constant
diam.
(supp ( j)
holds.
Dm depending only
on
d, 5,
n
and
m
K
I+
D,
For every p E N let
a
K)d
where k
=
(52)P Let u, be the set of all such cubes.
)
10d
,
=
M,
OC
E
Nn.
Rn into closed cubes of side
decompose
us
I
d(x, K) dl al
1,
n
and jk
run
length
through all integers.
We now define Jp Let J0 be the set of all cubes W of E0 such that d(K, W) ! "Qn. 5arecursively. The set Jp consists of all those cubes W E Ep which are not contained in a
J
cube of =
and which
Jp-1
J0,
satisfy
the estimate
d(K, W)
' n5112P
UpEN JP*
If
a
cube W E
Ep
meets
d(K, W)
:
a
cube W' of
d(K, W')
J0,
Vn -
6d2pd
> -
J,-1,
then
Vn
V n
6d2P-1
Td2pd
Vn -5d 2P'
Finally
we
set
theory for Whitney functions
1. 7 Extension
hence W is either contained in
family and
J forms
a
covering
of R'
a
cube
\ K,
J0,
-
-
R'
n, and
k now
,
is
or
and the cubes of
Jp
element of
an
Jp.
Thus the
meet at most cubes of
set
1) [0, 1] be a Cl-function such that *(x) ip (x) 0, if there exists an index k with 1Xk I --4
`w
ipw(x)
1W
)
where xW denotes the mid
I
4)w(x) One checks to
easily
Jp-,, Jp
that the
=
*W(X) EjEJ *j N
R7
x c
+
-1. 4
.1 2
for every
For W E J
point of W and 1W the side
locally finite
then is
2
1Xk 1 :5
\
and
covers
Rn
\
K
K.
family consisting of the 4)w (W
J)
E
satisfies conditions
(3).
If W E
then the
jp,
d(K, supp 4)w) hence
if
=
length. Obviously the family (SUPP*W)WEJ by compact sets. Therefore we set
(4)
follows
Obviously
following
d(K, W)
!
as
for all
chain of
equalities holds:
6d2(p+2)d
6d2p+2
well. cc
with
I ocl
< m
I < I
loci
have for all
X
depending only \K
D
x-xw))
D"*
<
1W
1W
where D is constant,
\,fnVdiam(supp 4)w), 2,rn5d-1
-
D'*W(x)l
we
Jp-1
-
59
spaces
Jp+,.
Next let
(1)
regular
on
on m
(and
I
loci
1W
the choice of
iP).
On the other hand
E Rn
1 <
E
*w(x)
<
4,
WEJ
hence the LEIBNIz rule and the
D,
>
0, which depending only
preceding inequalities entail the on m
existence of a constant
and n, such that
JD'(Pw(x)j
D, < I
loci'
1W
For W E J0 the we
inequality JDN w(x) I Ep-1 containing
have for every cube W' of
d(K, W') hence for all
X
x
Vfn<
6d2P-1'
E W
d(x, K) and for all
holds. Now let W E W
E supp
+
bd2P-1
diam(W')
V n
<
5d2p-21
( w
d(x, K)
+
5d2P-2
<
6d2d(p+2)
D2
V1W)
Jp,
p > 1. Then
Stratified Spaces and Fbnctional Structures
60
where D2
0
>
depends only
c, 6 and
on
So
n.
finally
for all
djocl
D'(Pw(x)j
<
D,
+
E Rn
x
\
K
W)
D2 _.
d(x, K )d
which proves the last claim of the lemma.
(cf
1.7.10 Lemma
[170,
TOUGERON
El
4)
=
0
on a
PROOF:
neighborhood We
of A \
like in
proceed
(A n Z)
[170].
4.5])
Lem.
subsets of 0 C Rn open. Then there exists
and
For the
Let
A, Z be
two
regularly
function qb E M' (A n Z;
a
(P
=
I
on a
neighborhood
proof of the claim
0)
situated
such that
\ (A n Z)
of B
-
it suffices to
assign to a point open neighborhood (A n Z)) such that 4)z 0 on a neighborhood of (V, n A) \ (A n Z), 4),; I on a neighborhood of (V, n Z) \ (A n Z) and such that for every 0C E N' there exist constants c' E N>0 and C' > 0 such that the following estimate holds: z
every
0
E
Vz C 0 and
an
function 4)., E F, (V-, \
=
=
1
I D'4)z(x) I if
C'
<
1 +
d(x,
An
z)c')
for allx E
V,\(AnZ).
has constructed such 4), then one can set (P Ej EN *j 4)y-j where finite of smooth functions 0 such that EjEN *j on 1 locally family (*j)jEN and supp *j C V,,, for every j E N, where the points zj E 0 have been chosen appropriately. We can suppose z E A n Z, the case z A n Z is trivial. Then there exists an of open neighborhood Vz z, relatively compact in 0, such that over Vz an inequality of the form (RS) (see p. 52) holds. Under these prerequisites let K C A be compact such that d(x, A) d(x, K) for all X E V.,.
Namely,
one
is
=
.,
,
a
=
=
Now
we
consider for K and d
and associate to every W E J
=
an
6
=
1 the construction done in the
integer Aw
in the
following
preceding lemma
way: if there exists
an
(Pw with d(x, K) < 12 d(x, Z), then let AW 0. 0, otherwise let AW Consequently, the function 4)z can be extended to a (!'-function on Rn \ (K n Z) (denoted by ( z as well) which vanishes on a neighborhood of K \ (K n Z). On the other hand 4)z I on a neighborhood of Z \ (K n Z), because if W runs through all x
E
supp
=
=
=
indices such that supp
(ow
n Z is
nonempty, then for every
E supp
x
4)w
I
d(x, K) hence AW
'd(x, Z),
=
Let
1.
then
d(supp q5w, K)
>
E
X
V, \ (A
( ,;(x)
has
n
Z),
diam
! 2
in
particular d(x, K)
if otherwise
0, Lemma 1.7.9 and (RS') the inequality one
=
I D '( ,; (x) I holds, where C'
is
an
appropriate
<
C'
d(x, A)
!
>
-
2
d(x, Z),
d(x, A). If d(x, A) then according 2 'd(x, Z), =
<
to
d(x, A n Z) -cl 0-fl,
constant. This prove the lemma.
To formulate the last result in this section
neighborhood. Though
(supp (PW)
this notion will be
we
11
will need the notion of
explained
in detail
only
a
tubular
later in Section
theory for Whitney functions
1. 7 Extension
3.1,
already introduce
we
more
it here in its most
61
spaces
form and refer to
simple
Chapter
3 for
details.
By
neighborhood of
tubular
a
a
submanifold S C Rn
borhood T C Rn of S such that for every
with
regular
on
d(x, 7r(x))
mapping
7r
d(x, S),
=
1.7.11 Theorem Let
7r(x)
and such that
will be called the
projection
N7",
M E
understand
we
E T there exists
x
+
t(X
-
7r(x))
of the tubular
S C Rn be
a
open neighpoint 7r(x) E S T for all t E [0, 11. The
exactly E
an
one
neighborhood.
submanifold of class of fOl
situated tubular
S C R
Rn.
12', and T c Moreover, let
neighborhood regularly a relatively 101 x aS tempered function of class 121. Then there exists a regularly situated tubular neighborhood T' C T of 101 x S together with a unique solution -y: TI -4 Rn, (t,x) F-4,y(t,x) =,y,,(t) of the initial value problem
R
f
x
:
Rn
T
a
x
x
Rn be
--
_K (t)
Moreover,
TI and -y
-Y."(0)
f (t I T. W) I
=
X.
tempered relative fOl
be chosen such that -y is
can
=
x
aS of class
em. PROOF:
equation without
By the existence and uniqueness of solutions of an ordinary differential glue together local solutions to global ones, hence we can suppose loss of generality that S is a bounded subset of R. By assumption on T we can
there exist continuous functions Cl) C2 E N and
C1, C2
> 0
e(x) and
Bl,(.,,)(0)
secondly
4
1j- with radius
and define
priate
T.
Next
S
-4
RII and 6
6(X)
and
Bn
Hereby
x we
4e (_X)
(.X)
set for
C T. E
X
:
S
-4
R"
as
well
as
constants
E S first
X
C, d(x, aS)cl
>
! 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
y (s, ij)) 11
dsI< T(7r(ij)) M(7r(*Lj))
:!
F,(7r(ij)),
(t,'J.)
E
To,
62
Stratified K p (t, -y)
hence
-
7r(-y)
PICARD-LINDEL6F
< 2e
(7r(ij)).
and Fbnctional Structures
Like in the standard
shows that K is
one now
Spaces
a
proof of
contraction. For
the theorem of E
o,
F,
we
namely
have t
K o (t, -y)
fo,
K* (t,
-
f (s, o (s, -y))
-
f (s,
iP (s, -y)) ds
t
hence
denote
JJK p by
t
I fo,
L(ij)
<
-
ds1:5
114) (s, -y) -(s, -y) I I
K*11p,
111( 2
:5
y and which is
a
-
L(ij) II(P
So K has
*11p--
-
ds<1 e2L(-y)ltl, fo, IPIIEIe2L(-y)lsl -2
exactly
one
fixed point in
solution of the above initial value
'E, which we problem. Moreover, by
definition of E the solution -y is tempered relative f0j x OS of class 0". Next shrink To to a regularly situated tubular neighborhood T' of that for every
v (E
R' with
jjvjj
=
1 the
fO}
x
S such
integral operator t
Kv has
:
E,
exactly
-)
C (T1,
one
fixed
E,
:=
Iy
p
point
-yv in the Banach space
-4
Kv y (t, ij)
+fo
R),
C(T1, RV) I
E
y (t,
By definition of Kv the fixed point
1j)
=
E
v
D f (s, -y (s,
B2n,(,,(,,)) (v)
for
-U)). (0,
E
y (s,
ij))
ds
T'
-yv coincides with the derivative
D'y(t,1j).(O,v),
and the restriction TITI : T' ---) Rn becomes a relatively {0} x aS tempered function of class C'. Recursively one thus obtains a sequence of regularly situated tubular C T' C To c T of {01 x S such that TIT- : T"' -- Rn is neighborhoods T' C ...
tempered
relative
{0}
x
aS of class C'. This proves the claim.
11
2
Chapter
Differential Geometric
Objects
on
Singular Spaces Stratified tangent bundles and
2.1
Whitney's
con-
(A)
dition
will construct for every stratified space X with stratification 8 and smooth structure C' the stratified tangent bundle TX The Whitney condition
In this section
we
(A) crystallizes hereby
in
stratified tangent bundle is 2.1.1
As
set
a
we
natural way
a
again
define the
a
as
the property which guarantees that the
stratified space.
stratified tangent bundle of TX
=
a
stratified space X
by
U Ts. SES
Like in the
case
of manifolds 7r
where 7rTS
:
TS
-4
=
7r-rx
one
:
has
TX
--
a
X,
canonical TS -3
S denotes the canonical
projection,
V
-4
7rTS (V) E
projection
S,
of the tangent bundle of the
If X possesses a smooth structure, then we can 0 C Rn be U way. Let U C X be open, and x
stratum S E S.
the
following
topologize a singular
TX in chart.
'
Then let
us
(Tx)iTsn-ru
=
set TU
:=
T(Xisnu)
7r-1 (U) and define Tx for all strata S.
Now
TU we
can
TO
C
supply
Wn by requiring that TX with the coarsest
topology open and all Tx are continuous. The projection continuous becomes then a 7r mapping which is differentiable on all the spaces TS. concludes easily from the corresponding properties for X that the TS Moreover, one such that all TU c TX
are
locally closed subspaces of TX and that the partition of TX in the spaces TS is locally finite. Hence the question arises, whether TX is a stratified space with strata TS and whether 7r is a projection in the topological sense that means whether X carries the quotient topology of TX In other words we have to show under which conditions TR C TS_ for all pieces R < S and simultaneously 7rl (Y) 7r-'(Y) for all subspaces when X satisfies Whitney's the this is exactly Y C X. It will turn out that case, are
=
condition
(A).
M.J. Pflaum: LNM 1768, pp. 63 - 90, 2001 © Springer-Verlag Berlin Heidelberg 2001
64
Differential Geometric
2.1.2 Theorem The stratified
(X, CI) X
smooth structure
tangent bundle _FX of
Singular Spaces
on
stratified space space with
a
stratified space again and 7r : TX -- X a topological if and only if X satisfies the Whitney condition (A). In this case TX inherits
projection, from
is
Objects
(X, Cx')
(weak)
canonical
a
charts Tx: TU
---
TX
R2',
where
7r:
PROOF:
Without loss of
-4
and that X is the identical
the condition that TX is
S, liMk,,,. T,.,S
-4
RI
smooth, and
a
which is determined
C7rX
through morphism
runs
generality we can assume mapping over U. We will
all
singular charts
by the
of X. The
of stratified space.
that X is
subset of
a
show first that
stratified space and and 7r a (xk)kEN a sequence of points of S
E R and
x
smooth structure U
x :
X then is
projection
let R <
a
(A)
RI,
U
=
X
follows from
topological projection. So converging to x such that
a
For the
exists.
proof of the relation T.,R C r choose V E TxR and By assumption V E 7C' 7r-'(Y), hence there is a of vectors tangent sequence vi, E T,,,S converging to v. But then v E -r must hold, hence Whitney's condition (A) holds as well. Now let us assume conversely that the condition (A) is satisfied. Then TR c TS_ has to hold for all x E R, hence TR c TS-. Therefore TX becomes a decomposed space. T
-:::::
let Y be the set of all Xk.
subspace Y of X, and V E n-1 (Y). Then there exists a sequence of X:= 71(V)- We can suppose that X E R and xJ, E S for an points liMXk incident pair of pieces R < S; the other case is trivial. By (A) the existence of vectors v follows. Hence 7r-I (Y) c 7r-'(Y). As the inverse inclusion Vk E Tx,S with liMVk Next choose
a
Xk E Y with
=
=
is
trivial,
obtain 7cl
we
(Y)
7r-I(Y)
=
It remains to show that the Tx
Let Tx E
TU n
we can
TU
:
Tft
ft)
is
fl).
TfI
:
that TU
-4
is
means 7r
--i
topological projection.
a
R11 induce
R2fi be
two
After transition to smaller
a
smooth structure
TX
on
singular charts and v a vector neighborhoods U and ft Of 7r(V)
neighborhood 0 C R' with m > max(n, A) diffeomorphism H : 0 -- 6 C RI satisfying xlu,,a
an
a
R21 and TR
T (U n
==
find
there exists
T(U
-4
:
open
of v, and
of =
x(Unft) H
-
RIU,,a.
such that
But
now
neighborhood TXIT(UnQ) TH TRIT(Unfl)' Of course, are compatible. By definition of the topology of TX the Tx comprise homeomorphisms which act as diffeomorphisms on every piece TS. As the union of all the TU covers the stratified tangent bundle TX, the Tx form a singular atlas and define a smooth structure on TX By the just explained construction 7r becomes a smooth morphism. of stratified spaces. This finishes the proof. El this
n
means
an
open
=
-
that Tx and TR
2.1.3 Remark
in the case, where X is
Only
ified tangent bundle is
geometric properties
locally
in
trivial. But to
common
given by a smooth manifold, the emphasize the fact that TX has
with the tangent bundle of
have chosen the notion "stratified tangent bundle" for manifold.
Example Let M be sphere S'. The cone CM
2.1.4
a
a
can
Ity
E
and inherits
as
becomes
(A)-stratified,
an
such
a
T_X,
a
smooth
even
Rn+1
then be
I
t E
canonically
[0, 1[,
smooth structure. even
Whitney
Obviously
SnJ
C
a
we
smooth
to be embedded in
identified with the
IJ E M C
many
manifold,
if X is not
compact manifold which is supposed
strat-
subspace
Rn+1
CM with this smooth structure
stratified space.
Consequently
TCM is
a
65
2.2 Derivations and vector fields
(A)-stratified. space
well-defined TCM
I (t1j, Sli + V)
=
E
and has the
W(n+l)
I
t
following shape:
E10, 1[,
M,
*Lj E
R,
s E
V
E
T-gMI
U
101.
homeomorphic, as one might expect, shows that singular charts of the example Moreover, stratified tangent bundle of a stratified space do in general not map the stratified tangent bundle to locally closed subsets of Rn. In the special case of the edge and the standard cone let us first identify S' resp. S' Note that TCM
as
to the
TM
cone over
with the we
topological
a
subspaces
space is not
this
R.
x
2 L' 2
( 1, 1) 1
S' and
C
1
L2 2
obtain the stratified tangent bundles of the
edge
2
_rXEdp
=
(cos y, sin y, 1)
I (x, Jxj, sgn (x) -y, sgn (X) Ij) I X, Ij ERJ [0, 1[,
t E
where sgn
By
2.1.5
R
:
a
u
2
2
--
W E
J-4- 1, 01
stratified
R,
s E
is the
vector
R,
E
U
RI
C
over
X
we
C
S2
.
Then
follows:
cone as
(- sin
o,
cos
y,
Offl I
R ,
signum function (which vanishes
field
RI
o E
CV
(tl,'2 (cos y, sin y, 1), sgn(t) ("12 (cos y, sin y, 1) +
_XCone
1
and the standard
understand
in the
section V
a
origin).
:
X
TX such
-4
that for every stratum S E 8 the restriction Vis : S --) TS is a smooth vector field over S. Note that the notion of a stratified vector field does not require that X has case X carries a smooth structure we will speak of a X, if V : X -- TX is continuous, and, in case X is an (A)stratified space, of a vector field of, class C' with in E N U {ool, if V : X - TX is a mapping between the (A)-stratified spaces X and TX of class Ic'; the space of all such vector fields of class C1 will be denoted by X'(X). If m oo, then we speak of a a
smooth structure.
continuous vector
But in
field
on
=
smooth vector one
field. Assigning
to every open U C X the space
obtains two further sheaves X' and XOO
over
X-(U)
resp.
1-(U),
X.
Derivations and vector fields
2.2 Given
a
of smooth manifold M the space of derivations Der((!', R) of the stalk (!00 X X at x. the to is M x E tangent canonically isomorphic space footpoint
functions at the In
stratified, an analogous result examples 2.2.1 and 2.2.2 will show.
case
two
X is
Besides the derivations of
does in
fields
over
X
gives general 2.2.1
a
derivation,
hold,
as
the
following
well.
For the
case
of
a
to the space
e'(X)
manifold M the space
X'(M)
of smooth vector
the other hand every smooth vector field on will show in this section, but the inverse does in
case on
as we
not hold.
R -O. Then the derivative D : CI(X) Example Consider the half line X -4 f comprises a derivation, but it cannot be represented as a smooth vector =
e0o (X), f
field
as
M. In the stratified
rise to
not
C' with values in R the derivations from X
on
importance COO(X) are Der(e,-(M), C!-(M)) is canonically isomorphic
to
general
on
X.
'
66
Differential Geometric
2.2.2
Let
Example
(x, 1j, z)
X,,c
induces
the stalk C' 0
on
e,'
:=
E
R31 X2 +
by R3. The
the smooth structure induced
XdC,O
S,y : Co
-4
curve
Objects
z2J
y2
Singular Spaces
be the double
1, 1
-y
on
X,,
t --)
cone
with
(t, 0, t)
then
the derivation
atl
[flo
R,
t=O
Of course, the tangent space of X,c at the origin is equal to 0, but S. does not means the derivations Der (E!o cannot be generated by T0XdC , R)
vanish,
that
-
Proposition Let X be an (A)-stratified space, x V E T-,,X induces a derivation
2.2.3
a
point of X and S the
stratum
of x. Then every vector
lflx -4v(fisnu)Conversely,
if 6
vector
T.,X with 6
Co"IfIs =01 X
PROOF:
R denotes
--
a
X
E
V
C'
:
C 1200 X
a,
=
if and
derivation, then there only if 5 vanishes on
exists
a
unique tangent
the ideal as
E
"
every vector V E TX generates a derivation Dv : Cxl -) R by 6, vanishes by definition on the ideal as. Vice versa, if Moreover, v(fls,u). 6 : C' --) R is a derivation vanishing on as, then 6 induces a derivation 5 : IS'las -4 R. As COxO/gs is canonically isomorphic to the stalk of smooth functions S over the v. Hence 5 footpoint x, there exists a V E TxS with 5v follows. This finishes the proof. 1:1
[f].x
Obviously,
-4
X
X
X
=
2.2.4 Remark One
interpret the results of this
can
tangent bundle TX of
section in such
a
way that the
(A)-stratified general TzX, but that always TX C TzX holds. Note that only for manifolds (without boundary) TM T'M. Nevertheless there are singular stratified spaces with I' (X) C' (X, T'X) := Der(C' (X), C,' (X)) like for example the space does in
an
not coincide with the
Zariski tangent space
=
=
standard
In the last section
2.2.5
of
a
X,_.
cone
smooth vector field
following
we
give
we
as a
introduced for every
(A)-stratified
smooth section for the
projection
space X the notion
7r :
TX
X In the
--
further and for many purposes very useful characterization of
a
smooth vector fields.
First note that vector field V
Vf
:
X
-
that Vf is
:
X
one can
--
define for every
(not necessarily
TX and every smooth function f E
R
by requiring (Vf)ls
an
element of
E!00(X)
=
in
Vlsfls
case
suppose without loss of
continuous
Cm(X)
for every stratum S E S.
a
or
smooth)
new
function
We
now
that V is smooth. As smoothness is
generality that X
a
show local
closed subset of an open set 0 C R" and that X inherits the smooth structure from R". By the smoothness of
property V: X
--
we can
TX C V'
one can
find functions Vk
V(X)=
E
E!00(0)1
X)EVk(X)3k k
k
is
=
a
1,
n, such that
(2.2.1)
67
2.2 Derivations and vector fields
for all
Hence
E X.
x
we can
satisfied for every f E
C'(X),
0 such that VkJX vA; onto the k-th coordinate. over
Altogether 2.2.6
we
smooth vector field V
a
0.
on
Likewise,
CI(O) fIx f. As Vf is differentiable of any order and Vf is true, the claim follows. Vice versa, if Vf E C-(X) is
(Vf)lx
the relation
as
extend V to with
there exist functions f
=
then
=
we can
V7rk for k
=
conclude that there exist smooth functions
1,
-
-
Hereby
n.
-
,
Eq. (2.2.1)
But then
7r,, denotes the
true, hence V has
is
projection
to be smooth.
thus obtain
Proposition For any are equivalent:
V
:
X
TX
-4
(A)-stratified
on an
space X the
following
conditions X
TX is smooth.
(1)
V
(2)
For every f E Coo (X) the function Vf is
:
--
Given
.(3)
a
chart domain U
-x-> 0
c Rn
element of C' (X).
an
with 0 E Rn open and
the vector field V is the restriction of a smooth vector field V
x(U) :
0
n 0
-4
=
TR ,
x(U) more
precisely Tx-
2.2.7
For any two vector fields
X' (X)
by
Vlu
V, W E X' (X)
IV, VVI Is
V
==
-
define their Lie bracket
we can
IV, W1
c
S E S.
IVIS, WIS],
=
(2.2.2)
x.
tangential to S for every. piece S, hence IV, VVI is a vector field 0 C R1 be a chart on X. It remains to show that IV, VVI is smooth again. Let x : U V x like in Prop. 2.2.6 (3) and let V, W be vector fields on 0 such that Tx Vlu W x Then we have for every piece S and Tx Wlu
By
definition. IV,
W
is
-
=
-
=
-
-
.
Tx
hence Tx
-
IV, VVI ju
-
IV, VVI isnu
=
IV, W1
-
=
x.
Tx
As
[Visnu, WisnLd
-
IV, W1
is
=
IV, W1
-
Xisnu,
smooth vector field
a
0,
on
the claim
follows. Now
2.2.8
we come
back to
Proposition
our
original topic, namely the derivations
Let M be
a
manifold, X C by M and 0
with the smooth structure induced in M.
Denoting by
satisfy V(3)
C
0
we
XO (M) have the
X'(X)
C
M C
a
closed
C'(M)
C ' (X).
on
(A)-stratified subspace
the
vanishing
the space of those smooth vector fields V
following
on
ideal of X M which
canonical relations:
X'(M)/OX'(M) a
-
Der(COO(X), COO(X)).
PROOF: Obviously X'(X) C Der(C'(X), C'(X)), hence it remains to prove the iso- Der(C'(X), C'(X)). By C'(X) C'(M)/O we obtain morphy XOO(M)/'OX'(M) a a canonical mapping =
TT
:
XO (M)
-
Der(C' (X), C' (X)),
V
1--4
(f
=
f +0
1-4
Vf +
0),
f E C' (M).
68
Differential Geometric
objects
Singular Spaces
on
First let
us show that IT is surjective. So let 6 be a derivation on CI(X). Then choose locally finite covering (Uj)j,,-j of M by chart domains Uj A RI and a smooth partition of unity ((pj)jrj subordinate to (Uj)jEj. Moreover, let Cpj E CI(M) be a
functions with supp Cpj c Uj and Cpj 1 over a neighborhood of supp oj. Then there exists for every index j and every i C' n a E 1, mapping vji , (M) with where is the i-th component of xj. Then V vji+g oj Ei vji ax is a smooth vector field on M. We will show that V E XOO(M) and IT(V) =: S. Now, =
=
.
6(Cpjx, +O),
=
5 induces for every the other hand
E X a derivation 6'
x
i with
V(O)
by
ker TT
the relations
x
C
6'([fl,, )
e'(M)
-R, [fl.,, by definition of
holds
E supp
Consequently,
pj
TT(V)
0 and
=
6(f
F-
+
0)(x).
On
the functions vji for Vf (x) 5'([f] x). By =
5 follow. Hence IT is
V(Cpjx, )
Ej
surjective. Now,
have to be elements of a for every
this proves the claim.
Altogether,
Differential
2.3
=
OXOO (M) the functions vji
=
V E kerTT.
:
X
Ei vji(a Xil1)(x)
bx(g)
fOl
.
=
all f E eOO (M) and all =
.
x,
forms
11
and
stratified
cotangent
bundle Given
a
Whitney (A)
space X with structure sheaf e'
e'-module sheaf fIX of Kdhler differentials
X
=
e,"o X
one
can
form the
Appendix B and the sheaf of exterior algebras fl EDkEN ilkX where f2kX Alf2x. Together with the Kdhler derivative d : fl -i Q we thus obtain a complex of sheaves over X. Like in differential geometry we call the elements of the space Q k (X) differential X forms. They provide geometric information about the underlying stratified space, even though not every result about differential forms on manifolds can be carried over over
according
=
to
=
,
to the stratified
In this section
case.
we
properties of X; cohomological aspects X will be considered in Chapter 5. 2.3. 1 Lemma If x
has
a
:
U
Rn is
--
a
or
will derive
some
of the
more
in other words the deRharn
singular chart,
fundamental
cohomology
every differential f(orm
W E
of
k
f2x (U)
representation of the form
where the functions Before
we
x',
PROOF:
i
start with the
write f2k instead of Qk X
=
1,
n
proof let
as
long
We prove the claim
First choose
dx" A
f
W
as
...
A dxk
eoo(u)
denote the components of the chart
us
(2-3.1) x.
agree that for the rest of this section
we
only
this does not lead to any confusion.
only for neighborhood 0
k
=
1; the other
cases are
proved analogously.
C Rn of
x(U), such that x(U) is closed in 0. Then realize that e,'(0) --4 fl'(U), f -4 dflu comprises a derivation on el(U), hence induces an obviously surjective mapping fIR'. (0) -1 Q(U). Now, every form cv, E fIRL(O) has a (unique) representation of the form oc IR Ein--1 f i d7ri, where f i C e' (0) for every i, and where the ni are the coordinate functions on Rn. By an
open
=
7EiIU
=
X' the claim then follows immediately.
El
2.3 Differential forms and stratified 2.3.2
The sheaf f1l of k-forms
Proposition
69
cotangent bundle on a
Whitney (A)
space X with structure
sheaf C' is fine.
immediately from the fact that C'
This follows
PROOF:
is
a
fine sheaf and f2k
Besides the stratified tangent bundle
2.3.3
a
1:1
Coo-module sheaf.
we
also want to construct
cotangent bundle T*X for X. In the following we will describe how this As a set T*X is defined analogously to the stratified tangent bundle by
a
stratified
can
be done.
U T*S,
T*X
SES
and the
projection
7r
=
7tT.x
:
T*X
--)
chart
x :
U
--
R'
now
let
given as the mapping which projection 7rT.s : T*S -4 S. For
X is
every stratum S E 8 with the canonical
Lx*(u) (T*R)
=
I (IJ, a)
E T*R'
I -Li
c
x(U) I
coincides every
over
singular
C T*Rn be
the
embedding Lx(u) : x(U) - Rn. Then the mapping x induces a T*U by (1j-, oc) t--4 (x-'(-Lj), a Tx-i(,j)x). We supply T*X with -4 T*x : (T*Rn) map Lx*(U) the finest topology such that all T*x are continuous. Obviously all T*U are then open. Moreover, we have for every additional singular chart R : ft -) R' with transition map H : 0 --- C) from x to R over U, c U n a a commutative diagram of the following pullback of T*Rn
via the
-
form:
LX* (U. ) (T *Rn )
T *x
(2.3-2)
T*U-,
T*HJ R*(U,,) (T*Rn) LX TH-1(,j)H). Thus T*U Hereby T*H : T*() -- T*O is defined by (U, oc) quotient topology of L.*(U)(T*Rn) induced by T*xIn the following we will show that T*X is a decomposed space with pieces T*S, -
carries the
S E S. Without loss of Rn via
the
a
chart
x :
continuity of
X 7t
-4
generality
we can assume
that X
can
be embedded into
:A 0 be two pieces R, S E 9 :A 0 holds, hence R C -9 and R is
Rn. Now let T*R n T*S
the relation R n
some
8. Because of incident to S.
P. (LJ, OC) E T*Rn with -y x(7r(p)) and T*x(ij, a) X :=:-- 7t(P) Then there exists a sequence (Xk) kEN C S with liInk--ioo Xk Consequently, the sequence (X(Xk)) a)kEN converges in T*Rn to (V, oc), and (T*x(X(Xk)) a-))kEN c T*S Next choose
some
P
E T*R and
=
=
=
-
to
P. So finally T*R
C T*S.
theory of differentiable manifolds the question arises whether the stratified cotangent bundle T*X is isomorphic to the stratified tangent bundle TX But this in general not the case as the following examples shows. By comparison
with the
edge X,,,,,, I (x, JxJ) 2 x E RI has the stratified tangent bundle TXEd., ={ (x, JxJ, sgn (x) 1J, sgn (X) -Y) E x, ij E RJ. On the other hand its stratified cotangent bundle is given by T*XEdge XEdge x R/101 x R. More generally, if M is a submanifold of the sphere Sn' then the stratified cotangent bundle T*CM of the cone over M can be identified as a topological 2.3.4
Example
It has been shown in
Example
2.1.4 that the
=
=
space with the
cone
C(T*M
x
R).
70
Differential Geometric Objects From the stratified cotangent bundle
2.3.5
construct the
one can
Singular Spaces
on
alternating powers
A'T*X by setting A kT*X
U AA;T*S. SES
supplies A kT*X
One
AkT*x:
with the finest
Lx*(U)(AkT*R) Lxl A
are
continuous.
...
A
ak)
topology
such that all
--
A kT*X,
1-4
(x-1(ii), oc, Tx-i(,j)x A
mappings
-
...
A "k
-
Tx-i(.U)x)
Moreover, one has a canonical projection 7r 7rAL-T*X : Ak T*X by a section of A kT*X a mapping w: X -4 A kT*X such that 7t =
and understands
X,
-4 -
W
idx.
By the
2.3.6
universal property of Kdhler differentials
in differential forms. V
T.,X,
E
form
E
a
X
E
x
(or
More
more
precisely, generally of
a
insert tangent vectors
one can
defines the insertion of
one
derivation
tangent
a
Der(C', R))
E
v
in
vector
differential
a
X
f2'(X) by 0CX(V)
(0C'V)
:=
:=
i'(0(X)'
where i, is the
morphism associated to the derivation v in the universal property Appendix B. One now extends the insertion canonically to mappings
from i
f1k (X)
:
T_,X ok
x
Every
k-form
2.3.7
Proposition
one
oc
then
differential form
of X there exists
a
__
(OC) VI
R"
provides in
a
0
case
'
*
Vk)
0
natural way
X with
(a)
-4
iviO
a
section -a: X
For every continuous section
...
ovk
X
w :
-4
=
--
OCX
(V1
w
there exists
a
A kT*x
uniquely defined
-
e
-
'
'
*
x
0
Vk)
-4
ocx.
A kT*X there exists
=
=:
0
A kT*X,
W, if for every singular chart differential form e E f1k (0) such that a on
w1u In this
'
U
x :
(KA)
---)
exactly 0 c X
(2-3.3)
x.
continuous section dw
:
X
-4
A k+'T*X
such that
(dw)lu PROOF:
With the
=
(E)lU
=
A k+'T*x
-
doc
-
(2.3.4)
x.
help of Lemma 2.3.1 the claim follows almost immediately by a Proposition 2.2.6; therefore the details are left to the reader.
similar argument like in
M
2.3.8
For later purposes let
us provide some universal constructions on the stratified fi tangent and cotangent bundle. Let F: QJecfiR' -- ZecR" be a (covariant) functor on the of finite dimensional real vector spaces. category As an example for such a functor we name the k-th tensor product (9 A; the exterior product Ak' or the k-times symmetric tensor product SYMk Then we define for every ,
.
(A)-stratified
space X stratified spaces FTX and FT*X
FTX
U SES
FTS
and
FT*X
by
U SES
FT*S.
2.4 Metrics and
Now
length
71
space structures
by the methods from Section 2.1 and the again (A)-stratified and that FT*X is stratified, where
checks easily
one
that FTX is
FTS resp. FT*S with S E S. Thus it is clear from the spaces TX ok , AkTX, Sym kT*X and so on.
by
If f
2.3.9
X
:
--)
with f induces
a
Y is
a
smooth map between
morphism
f2*f
=
f*
fl
:
-4
the stratified tangent bundle: the smooth
which
through
runs
between the stratified tangent bundles
spaces then the
given by
pullback
:] 1, 1 [ -4 X,,, from Example
does not induce
cone
]1, 1[
are
But the situation is different for
fl . -y
in this section
the strata
on, what to understand
now
(A)-stratified
curve
the cusp of the double
ones
xR and
a
TX,,. Insofar
2.2.2
smooth
it is in
mapping general not
arbitrary smooth maps f : X -i Y a meaningful tangent map. But if f is additionally a morphism of stratified spaces, then one can define a tangent T (f Is) for all strata S of X. TY by setting (Tf )is map Tf : TX to define for
possible
=
---
Metrics and
2.4 The
goal
length
of this section is to carry
space structures
over
the notion of
Riemannian metric
a
a
on
differentiable manifold to stratified spaces. Moreover, we want to examine, under which conditions such a Riemannian metric provides a geodesic distance which reflects the
topology
of the
underlying
space.
2.4.1 Deflnition A continuous section t: X -4 T*X (9 T*X over an are
=
I ocx (9 &1
Mx
Px
Tx*Xj
E
(A)-stratified space X is called a Riemannian metric,
x
E
if the
Xj
following conditions
satisfied:
(ME1)
For every stratum S the restriction
:
Ljs
S
T*S 0 T*S is
-4
a
Riemannian
metric.
(ME2)
For every
pair V, W of smooth
vector fields
(V'V%%:= L(V'W):X--4R' is
a
smooth
We call
t
a
(Xj Isnu) If
Oj
on
X --4(V-X'W_') '-:= LX(V"'W-X)
X.
smooth Riemannian metric, if there exists
Uj A Oj
C Rn and
a
covering (uj)jEj of
X
by
a
*
t is
Lj coincides with the restriction Lisnuj a
Riemannian
smooth Riemannian metric
Whitney (A)
2.4.2 Remark For the
boundary
MELROSE
case
[1261
manifold-with-boundary.
on
-
X, then
we
often call the
pair (X, L)
a
space.
that the
underlying stratified
has introduced the notion of
notion coincides with the notion of a
X the restriction
family of smooth Riemannian metrics Lj on the for Rn that such C every stratum S and all j GJ the pullback metric
chart domains open sets
mapping
on
a
a
space is
a
manifold-with-
b-metric. By definition this
Riemannian metric defined above for the
case
of
72
Differential Geometric
2.4.3
part)
Objects
on
Singular Spaces
Example The Fubini-Study metric on projective space induces (via its real every projective algebraic variety a smooth Riemannian metric in the above
on
sense.
2.4.4 Example The so-called PL-spaces, where PL stands for piecewise linear, are of significance in the integral geometry of singular spaces. PL-spaces provide beautiful examples for (A)-stratified spaces with a Riemannian metric. The importance of PLspaces lies in the fact that one can define and study intrinsic curvature measures on
PL-spaces, as CHEEGER-MfJLLER-SCHRADER [43] (see also KUPPE [106]). As the underlying stratified space of a PL-space one has given a polyhedron X with an explicit triangulation h: X -4 JKJ C R' as singular chart. Hereby K is a simplicial complex (see 1.1.14). Suppose that X is equipped with a continuous section L: X -4 T*XOT*X which is smooth
2.4.5
simplex
on
is called
a
K there exists
a E h
composition h:-'(a) 4 a _'4 R' is an PL-space. Obviously L then is a Riemannian
R' such that the
a embedding L : a -4 isometry, then the pair (X, L)
metric
If for every closed
every stratum.
over
linear
-
X.
Proposition For
(A)-stratilied
every
space X there exists
a
smooth Rieman-
nian metric.
PROOF:
Oj
C
First choose
Rli, and
Euclidean scaler
product p,,, (v,
A a
covering (Uj)jEJ of X by coordinate neighborhoods Uj the partition of unity ( oj)jEi. Denote by
a
subordinate smooth
a
simple argument
w)
now
R'j =
.
Then define for
E X
x
E (pj (x) (Txj.v, Txpw),
v,
shows that the section R: X on X.
4
w E
T-,X.
T*X 0 T *X,
x F-i
g.,
comprises
smooth Riemannian metric
Let X be
Whitney (A) space, and t every weakly piecewise 1?'-curve -y : [t3--, tJ+1 -lyl , by 2.4.6
a
fjj
lyl, iEJ
Hereby
UjEj Ij
I is the
=
connected components and that K'.
1-yl,,
metric 11
over
0 such that K C U
matrix-valued function such that
a
1-yl,
E iEJ
CK
for all
z
c:
we can
compact
singular and x*il a
=
then
its
chart
tlu.
0 and V, W
z
E
Rn. Then
X. To
on
geodesic length
dt.
x:
U
-4
I of -Y into its E
Ij
.
0 C Rn and
Let b E e00 (0; is
we
equal
We show
image in a
GL(Rn))
smooth be the
to the Euclidean scalar
have
JIj
(Tx(- (t)), Tx(- (t))),1(x(,Y(t)) dt 1: J,
E
f1j
iEJ
assign
set K C X such that 'Y has
(b-1.v, b-1.w),1(, ) z
product (v, w)
smooth Riemannian metric
X
decomposition of a differentiability set - (t) E TSj the tangent vector of 'Y at t
g there exists
on
a
C-y(tF),T(t))
is finite. To this end choose
By assumption
M
=
JJbx(,y(t)).Tx(- (t))JJ
dt
iEJ
Tx(- (t))
dt
=
CK Ix
,
'y I < oo,
(2.4.2)
2.4 Metrics and
where
CK
:--:
length
sup
73
space structures
Illb.(,,)Il I
E
x
KI.
Of course, this estimate holds for every
piecewise C'-curve in K. Moreover, the
constant
CK depends only
which later will turn out to be essential.
on
weakly
K and L;
a
fact
note that the
geodesic length I-YI,' Finally depend on the choice of the singular charts, in contrast to the Euclidean length Ix -y 1. Setting D K sup bx-(' I x E KI we also have an estimate inverse to does not
-
(2.4.2):
(2.4.3)
Ix -yj :5 DK ITIg-
2.4.7 Theorem Let
(A)
space and
L
a
5,,:XxX--)W 0, comprises
a
The metric
metric
6.
(X, IC')
be
a
connected locally finitely path connected Whitney
smooth Riemannian metric
(x,-y) on
--4
X. In
is called the
inf
case
11-yl,, I -y
X is
geodesic
X. Then the map
on
ewl([O, 11;X), -y(O)
E
regular, 5,,
induces the
x
=
and
-y(l) ='yj
original topology
of X.
distance of t.
By the estimate (2.4.2) the geodesic distance 5,,(x, -y) of two points X, -y E X ! 0 holds, and 5,, is symmetric in Obviously, the inequality x and 1j. For the proof that 6. comprises a metric on X it remains to show only the 0 is satisfied, if and only if x Y. Let x, 1j, z triangle inequality and that 5 L (%, -U) be three points of X and'e > 0. Then there exist weakly piecewise C!'-curves -Y and A from x to -y respectively from ij to z such that PROOF:
is well-defined.
=
=
J-yj,:56 L(x,-y)+e The
then has
composed path A-y
6,,(x, z)
:!
and
geodesic length Jyjg +'J;kj L,
1-ylg + JAIR
:5
6,,(x, -U)
hence
5,(y, z)
+
+ 2c
arbitrary, the triangle inequality follows. To prove positivity assume x O'g. Then choose a singular chart x: U -4 0 C RI with finitely path connected U such that x, 'y E U. Also choose a compact and finitely path connected neighborhood K C U of fx, ijj. Finally choose a finitely path connected open neighborhood V c K of x, and e > 0 such that B, (x(x)) n x(U) C x(V), where follows. As
F_ >
0 is
,
B,(z)
is the ball around
z E
R' of radius
& >
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
<
<
=
k.
e
P(X, J) O
+ 2
2.4 Metrics and
length
Then there exists for i
a
weakly piecewise C'-curve
-y
:
[0, 11
X such that
--
-y(ti)
=
A(ti)
k and
0,
=
75
space structures
IT11ti'ti+11
+
Tk
Consequently
5'(X' J) As
F_ >
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
--
<
=
On the other
equicontinuous. By the theorem of AsCOLIa subsequence (;kl)lErq converges uniformly
< lim
JXIJ
=
liminf h1kI,
1-+00
and
L,
further transition to
J-yI Namely,
< p +
JAI (t) I I E N} c X is relatively compact. following relation holds:
the sequence
BOURBAKi and after to
d(Al (0), Al (t))
+
for every t the set
hand for t1, t2 E d
d(x, A, (0))
<
t,
<
...
<
d(y(t), Al(t))
<
IA, I : ,
achieve that
t,, '
4n
=
(2.4-6)
k-4oo
1 be
a
for all t E
sequence of intermediate
[0, 11
and 1 >
points, E > 0 lo. By possibly enlarging
liminf 111k I + E. Then 2 k-4oo
n-1
n-1
E d(y(ti),'y(ti+i))
8
<
+ 2
i=O
hence
(2.4.6)
2.4.14
< liM
inf IYk I + F-,
k-- oo
i=O
and thus the claim follow.
Corollary (cf. BUSEMANN [37, 1.1.(4)])
if the closed ball a
E d (A,,, (ti), Al, (ti+,))
! , (%)
segment connecting
around x
X
with 1j.
E X is
El
If (X,
d)
is
an
inner metric space and
compact, then there exists for every -Lj
E
-9, (x)
2.4 Metrics and
PROOF: curves
length
First
Yk from
d(x, -y) < e. -4 d(x, ij) compactness of B, (x)
assume
x
77
space structures
0 <
to ij with
lyk 1
Then there exists and
IYk I
< C
.
a
sequence of
Consequently
geometric
every
curve
yk
and the proof of the theorem of B USEMANN By the 2.4.13 for compact,length spaces the sequence (Yk)kEN converges after a possible transition to a subsequence to a curve -y c 9,(x) connecting x with ij and having length d(x, -y) Consequently, -Y is a segment connecting x d(x, -y) :5 1-YI :5 liM infk,. IYk I
lies in B, (x).
=
-
and ij.
d(x,ij)
Now let 'Yk E
B,(x) converging
compactness of that after x
=
9,(x)
e.
and
transition to
a
Then there exists
Lemma 2.4.11
by
to ij. Choose for everyYk
a
segment from
sequence of
points
Using the of the theorem of BUSEMANN checks one proof recalling a subsequence (07k)kEN converges to a segment a connecting a
X
to 1Jk-
the
with -y.
Now
we can
following
fundamental result.
Hopf-Rinow (cf BUSEMANN [37, 1.1.(8)] and GROMOV [69, locally compact inner metric space (X, d) the following conditions are
Theorem of
2.4.15
Sec.
prove the
1.13])
For
a
equivalent:
(1) (X, d)
is
(2) Every
closed ball
(3)
[t-, t+ [-- X is a semiopen segment, that path y1[t-,t] is a segment, then -y can
-If 'y
:
complete,
hence
9,(x)
a
length
c X is
space.
compact.
restricted
R-, t+1 If a
one
---
of these conditions is
-y))
=
be extended to
a
segment 7
:
X.
connecting segment. that
-y (d(x,
if for every t < t+ the
means
satisfied, then there
means a
segment y
:
exists for every
[0, d(x,-y)]
pair of points
X with
--
-y(O)
=
x
x, Y
and
-y.
2.4.16 Remark HOPF-RINOW
were
the first who showed in their work
[90]
dated
from 1932 that for any two points of a complete Riemannian manifold there exists a minimizing geodesic. Some time later COHN-VOSSEN [44] succeeded to prove that the main result of HOPF-RINOW is valid without the
assumption of differentiability.
Thus the "modern" version of the theorem of HOPF-RINOw as one can
find it in BUSEMANN
[37]
or
GROMOV
[69]
as
goes back
formulated here and
essentially
to COHN-
VOSSEN.
(2) ==> (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.
<
1
and because
k
Choose for every i < k
to Xk-
an
accumulation
2.4.14 there exists for every
By Corollary
a
point xik
the
on
curve
Tk
Hence pi. Let -yik be the restriction Of Yk which connects x with x. k subsequence (k1j)iEN of (k)kEN the sequence converges by the theorem
(-ykllj)jEN
segment yi from x to a point -y, E Bp(x) with d(x,-yl) pi. Furthermore, there exists a subsequence (ki&N Of (klj)jEN such that the sequence of BUSEMANN to
a
=
(y2k2j)jEN
of segments
converges to
a
segment -y'2 from
x
to
point
a
Bp(x)
V2 E
with
Continuing like this one obtains a sequence of segments -Y1'1 c -Y2' c such that the initial point of 'yk is x and the final point Vk has distance Pk to X Now U -yk' is a semiopen segment, hence can be completed by a point -tJ to
d(x, IJ2)
=
P2
...
-
segment -y'. Obviously 1Jk
well, d
for
as
k.-
(ii, xk!.) kjj
Choose for
j
ij. The
--4
sequence
(Xkij)jEN
converges to -y
a as
> 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.
<
point Of (XA;)kEN, and the claim is Now
diagonal
kjj
all
+ p
i !
(y,
< d
-
pi <
+ d
(y i, x,'kii )
+ p
e/2. Afterwards
N. Therefore ij is
an
-
p i.
choose N
accumulation E3
to the construction of Riemannian metrics
on
stratified
spaces.
2.4.17 Theorem Given
a
smooth Riemannian metric
connected t
on
regular Whitney (A)
X such that
(X, 5,,)
space X there exists
becomes
a
length
space.
a
In
that case there exists for every two points x,ij of X a weakly piecewise Cl-curve -y connecting x and -y and which satisfies 1-yl, 5,(x, ij). Moreover, every connected compact regular Riemannian Whitney (A) space is a length space. =
PROOF:
If X is
a
(X, 6,J
has to be
Now let X be in the
a
compact connected regular Whitney
(A)
space with
L, then every closed set is compact, hence,
Riemannian metric
length space. arbitrary connected regular Whitney (A)
an
the existence of
smooth Riemannian metric
a
smooth
by Corollary
space.
2.4.9
We will prove (X, 6,) is a
L such that
a following length space. To this end choose a compact exhaustion (Kj)j(=N of X and then according to Lemma 1.3.17 an inductively embedding atlas with respect to (Kj)jEN consisting
2.4 Metrics and
of
point
Ko and
E
x
one can
xo(x)
that
79
space structures
charts xj : Kj+2 -- Oi C R"j. As X is connected and locally finitely path choose the compact sets Kj to be finitely path connected. Fix a
singular
connected
length
0. After
K-1
set
an
appropriate transformation
construct Riemannian metrics Lj
Inductively k be the Euclidean scalar product multiplied with a factor C > for all Ij E xo(Kl \ Ko). Assume that one has given k, ij =
0.
we now
we can
on
suppose
R'j. First let
0 such that C I hi I I > 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
C'(0)
an
=
od
D (x
'
-
ii) (-y),
-Y E X.
da', the operator D lies in D k(0). expansion of the following kind: f
fo, (7r locl:! k
-
1J)
"
+
E IPI=k+l
fp (71C -
Next let
-
-Y)
us
write down for
2.6 Poisson structures
83
where
f,, E R and fp Eq. (2.5.4):
Df(y)
E
C' (0).
But this
D and calculate with the
odf,,,d,, -L)
IPI=k+l
means
If D
D c
doa'
for all ij E
apply
1: D(fp-(7t--Li) ) (y)
f,,D(7r--y)'+ I ocl!5 k
Then
G
hence
X,
of
help
D(f+0)(Y).
=
locl
Dgl(O) and TT k(D) D, hence TTI is surjective. is in the kernel of TTk, then do,(y) Dk(O) -LD(n ij)01(y) 9 od d,, E a and therefore ker TT k OD'(0). Consequently D(U) =
=
=
-
=
0 is
isomorphic to DO(O)/OD(O). Obviously, the family (DS)SE8 of differential operators DS E Dk (S) defines a uniquely determined differential operator D E Dk (X) with (Df)ls Dsfls for f E CO')(X), if the function X E) x 1-4 Dsxf1sx(x) is smooth. So it remains to show that every stratified differential operator is given by such a family. Let D E D k(X) be stratified. For a point X E S choose a singular chart R : U -- R' around x such that =
SnU is
a
=.R-'(Rm
x
f0j),
where
differentiable chart of S.
have for all
m
=
By
dimS <
n.
In other words
x:=
the above considerations and
as
Risnu: D is
S nU
--
stratified,
Rm we
f CC' (X)
(Df)isnu
=
1:
with
dx axxA1snu
do,(ij)
od
D(x
-
)'(y)
for ij G S n U.
Jal
Now, Elocl
.
2.6
Poisson structures
Besides the differential fer the most
geometric notion of a Riemannian metric one can also transimportant objects of symplectic geometry to stratified spaces. With
the Marsden-Weinstein reduced spaces, which play an essential role in mathematical physics, we then have a rich and important class of examples for symplectic stratified spaces in
our
hands.
2.6.1 Definition A smooth section A: X
called Poisson
(PS)
TX 0 TX
over a
Whitney (A)
space X is
is satisfied:
For every stratum S the restriction
Als: provides A
--
bivector, if the following condition
a
S
--
TS&TS
Poisson structure for S.
Whitney (A) space X together with a Poisson bivector will be called a Poisson stratified space; if every stratum S with the Poisson structure induced by As is symplectic, then X is called a symplectic stratified space.
Differential Geometric
84
A Poisson bivector A induces
bracket, by setting
bidifferential operator
a
=
,
singular
If, g}
chart to
then is smooth
a
a
smooth function
2.6.2 Remark The notion of
[162]
general
X
on
only
use
indeed,
on
a
on
X, the so-called Poisson
as
A
bivector
can
be extended
locally
in every
hence in every chart
field,
If, gJ
[161],
BATEs-LERMAN
[6]
and
somewhat different form than in 2.6.1. The difone
of SJAMAAR-LERMAN lies in the fact that
rather weak notion of
a
(2.6.1)
stratified space has been introduced
a
smooth structure which in
a
stratified tangent bundle and therefore
smooth vector fields.
or
dg Is).
Euclidean space.
does not allow the construction of
also not of continuous
0
well SJAMAAR
as
definition and the
our
the authors
(df Is
-i
symplectic
a
by SJAMAAR-LERMAN [1621 (see ORTEGA-RATiu [137]), though in in
Als
(antisymmetric)
smooth
is the restriction of
ference between
I., .}
Singular Spaces
on
for every stratum S and all functions f , g E C' (X):
If g}Is The function
Objects
More
SJAMAAR-LERMAN
precisely,
stratified space a subalgebra A c eO(X) such that for every function f E A the restrictions f Is on the strata S are smooth. If now A comes with a bracket I -, .1 : A x A --i A, then S JAMAAR-LERMAN call the triple understand
by
(X, A, 1-, .1)
a
smooth structure
a
symplectic stratified
1. every stratum is
(A, I., .})
2.
3. the
is
a
embeddings
S
space, if the
--)
algebra, X
are
and
Poisson
mappings.
immediately that symplectic stratified spaces sense of SJAMAAR-LERMAN as well, but
One checks
stratified in the 2.6.3
is
The
given by
holds:
following
symplectic manifold,
Poisson
a
on a
probably
in
our sense are
not vice
symplectic
versa.
important class of examples of symplectic stratified spaces symplectic manifolds with symmetry. In the regular case
most
reduction of
independently MEYER [1291 have introduced this physics; SJAMAAR-LERMAN [162] have extended the In it to the singular case. following we will explain symplectic reduction and will this construction symplectic stratified spaces. Hereby we obtains by sketch, how one MARSDEN-WEINSTEIN
[119]
and
reduction scheme in mathematical
will need notions and results from the
theory of G-actions
and their orbit spaces
as
use of later proven theorems will not lead to they will be provided in Chapter of this paragraph will not be needed anywhere any circular arguments, as the results
4. The
else in this we
monograph.
For
possibly
refer to ABRAHAM-MARSDEN Let
(M, w)
g E G the map
phism
of
M,
supplements
from
symplectic geometry
[1].
together with a proper and symplectic Lie : G x M -4 M. That G acts symplectically hereby means that for all (Dg is a canonical transformation or in other words a symplectomor-
be
group action (D
necessary
a
symplectic
manifold
i.e.
(D*w
=
w.
9
Suppose further for all
that there exists
the relation dJ'-
=
that
means
iE.,,w holds, where &M is the fundamental
vector
a
G-invariant moment map J
:
M
-4
9*
85
2.6 Poisson structures
field of & and V- the smooth function M E)
(M, w, G, J)
the form
(J(x),
moment map
Hamiltonian
as
for every
E R.
A
of
quadruple
Hamiltonian G -space. The
symplectic geometry comprise integrals of the motion for every G-invariant G-invariant function h E Coo(M)G and every & E g:
components of the
flow,
F-4
x
is called in
a
th, J J
=
XJ&h
=
L&,,h
=
0,
Xj& denotes the Hamiltonian vector field of JF-. One can then decrease the degrees of freedom of the Hamiltonian system. Thus one comes to a more simple description of the system and possibly to a complete integration of the system. In where
mathematical detail this so-called Marsden- Weinstein reduction works
as
follows.
zero level set Z J-'(0) of the moment map. Under the assumption regular value of J, Z comprises a submanifold of M. But in many cases this assumption does not hold, and then Z is "only" stratified. The essential observation now is that Z is a G-invariant subspace, hence one can form the orbit space Mo G\Z, and thus obtains the desired reduced space. This space has the following properties.
Consider the
that 0 is
=
a
=
(SJAMAAR-LERMAN [1621, SJAMAAR [161, 3.1.1. Thm], BATES(M, w) be a Hamiltonian G-space with proper G-action and J :
2.6.4 Theorem
LERMAN
[6])
M
a
--
9*
Z with
a
Let
G-equivariant
stratum
M(H)
manifold.
moment map.
of orbit type
Then the intersection of the
(H),
where H is
the orbit space
a
closed
subgroup
Moreover, (Mo)(H) G\(M(H) Z) symplectic structure (wo) (H) the pullback of which to Z(H) := M(H) n Z the restriction of the symplectic form w to Z(H). The decomposition into the manifolds (Mo)(ji) induces a stratification of Mo. PROOF:
In
[162]
n
=
zero
carries
level set
of G is
a
canonical
a
coincides with
of Mo
=
G\Z
the theorem has been proven for the case of a compact Lie group. case of proper G-actions in [6]. We do not give the prove
It has been extended to the of the claim at this
point but refer the reader
to the two cited articles.
El
Besides the natural stratification
by orbit types the reduced -space Mo G\Z a given by C,(Mo) Coo(M)l/gG, where 0 is the ideal of smooth functions vanishing on Z. The function algebra C-(Mo) stems from a smooth structure on Mo. Apparently this follows from the fact that Mo can locally be embedded in some Euclidean space and that the corresponding local smooth structures coincide with the one given by 'C'(Mo) (see PFLAUM [144, Sec. 6] for details). Moreover, as a consequence of [162, Sec. 5] we obtain that Mo is even Whitney stratified and locally trivial. carries
=
canonical functional structure
2.6.5 Theorem
Every
=
reduced space Mo like in Theorem 2.6.4 comprises a The Poisson bracket on sense of definition 2.6.1.
tic stratified space in the
symplec-
e,01(Mo)
satisfies
ff gJ
-
,
Hereby,
7r :
such that f
Z -
-4 7t
=
7r
=
ff "011Z,
f,
g E COO (MO).
Mo denotes the canonical projection, and
T1z
and g
o
7r
=
-glz.
7, -g
(2.6.2) E C!oo
(M) G
are
chosen
Differential Geometric
86
on
Singular Spaces
[162] that Eq. 2.6.2 defines a Poisson Moreover, they have shown that for every orbit type (H) and
SJAMAAR-LERMAN have shown in
PROOF:
bracket
Objects
on
IS' (Mo).
f , g E C1 (Mo) the relation
If) holds, where 1', 'I(H) is sequently the Poisson
911(mo)(Jt)
bracket
hence every
2.6.2 the bivector A
C,'(Mo)
on
A(H)
A(H),
and
Eq.
ffl(mo)(Hp 91(mo)(R)I(H)
the Poisson bracket of the
bivector
a
:--
is
from
comes
a
symplectic manifold (Mo)(H). Congiven stratawise by a regular Poisson symplectic form. By Proposition 2.2.6
composed of the A(H) must be smooth. Thus (MO, A) is and, as all strata are symplectic, even symplectic stratified.
Poisson stratified space
n
2.6.6 Remark In the article
[137)
ORTEGA-RATiu succeeded to
generalize singular
reduction to Poisson manifolds. 2.6.7
Example (cf. [109, Sec. 11) Consider the canonical SO(2)-action T*R' R: cotangent bundle M
it to the
on
V and lift
=
=
sin (p
0
0
X1
cos
X2
sin q)
&1
0
0
cos
&2
0
0
sin (p
p
-
Cos
Y
:21
0
0
sin Cos
&2
Y
simple computation that the SO(2)-action is symplectic with respect dxl A d&1 + dx2 A d&2. A moment map is symplectic form w J momentum the x2 &i. As 0 is a singular value of x, &2 (q, p) angular given by One calculates easily that smooth. be cannot reduced the Mo Z/ SO(2) J, space One checks by
a
to the canonical
=
=
-
=
Z
=
J-'(0)
is the union Of
Z(1)
=
Z(SO(2))
fO}
=
and
f(X1)X2)&1)&2)
00 1 Xl&2-X2&1
=01-
f0} and Consequently Mo is the disjoint union of the strata (MO)(SO(2)) R? \ 101, where an appropriate (symplectic) isomorphism is given by =
(Mo)(1)
X
W \ {0}
3)
(0) 0
X)
X
-i
SO (2)
-
E
&
(Mo) (1).
(2.6.4)
on Mo, which is in fact decisive for the geometric properties of M0, look like? A homogeneous Hilbert-basis for the SO(2)-invariant polynomials on T*R2 is given by the four functions
How does the smooth structure
Pi T
As J vanishes
on
(P1 (V)) P2 (V), T(V)),
Z, V
=% =
2 1
X22
_
X21 +,X22 +
we
E
+
obtain
R4. Now
(&21 + &2), 2 1
1
+
a
2, 2
proper
one
proves
P2
and
=
J
Xl&l + X2&2) =
XI&2
X2&2-
Mo -- R3, SO(2)v embedding L that modulo the vanishing ideal easily
2.6 Poisson structures
of Z the relation -r' of
lies in the
=
87
p21
+
XCone
p22 is satisfied. (1J 1) IJ2)'Y3) E
In other words this
R31 jj23
means
that the image
Lj21 +.y22 1. According
to Eq.. 2.6.4 By the theorem Of SCHWARz 4.4.3 and the definition of the smooth structure on Mo the algebra C!'(Mo) coincides canonically with the algebra of smooth functions on R3 restricted to Xc.,,,. L(MO). Hence Mo and X..,,,, are isomorphic as stratified spaces with smooth structure. t
even
iM
L
=
cone
XCone has
=
to hold.
=
2.6.8 Remark In connection with
symplectic stratified spaces the moduli spaces of on a according to ATIYAH-BOTT [5] are of broad interest. These moduli spaces can be interpreted as Marsden-Weinstein quotients of infinite dimensional symplectic spaces. By GOLDMAN [61] and KARSHON [97] they are symplectic, carry a canonical functional structure (cf. KARSHON [97] and flat connections
HUEBSCHMANN
finally
are
HUEBSCHMANN
important examples for symplectic stratified
spaces.
It is
possible to coin the notion of a deformation quantization A la BAYEN et symplectic 'stratified spaces and even for Poisson stratified spaces. So let X Poisson stratified and A the corresponding bivector field. Consider the linear space
2.6.9
be
[931), which turns out to be a smooth structure in our sense, and (CURUPRASAD-HUEBSCHMANN-JEFFREY-WEINSTEIN [74] and [91, 92]). Thus moduli of flat connections provide further interesting,
stratified
nontrivial and
al.
Riemann surface
[7]
for
:= C-(X)[[/Nll of formal power series in one parameter A and e'(X). By a formal differential deformation quantization of X or
A
with coefficients in a
star
product
we then understand nothing else than an associative law of composition such that the following axioms are satisfied:
(DQ1)
(DQ2)
The composition
*
is
f,
g E
x
A
on
X
-4
A
R[[X11-bilinear.
There exist bidifferential operators bk for all
A
:
e-00(X)
e00(X)
X
-4
e'(X)
such that
e00(X) ;kk bk(f, 9)-
f* g kEN
(I)Q )
For all
f,
g E
C,'(X)
the relations
f*g =fg+O(A) hold, where Ix denotes the
and
lx*f =f*lx =f,
constant function
(DQ4)
The bidifferential operator
By the
work Of DEWILDE-LECOOMTE
b,
is
equal
on
X with value 1.
to the Poisson bivector A.
[48, 49]
and of FEDOSOV
that for every symplectic manifold there exists a star product. result of KONTSEVICH [102] even every Poisson manifold has tization.
But
[53, 54] According a
one
to
knows
a newer
deformation quan-
concerning symplectic stratified spaces there do not exist up to now general products on such spaces. Nevertheless it seems rather promising to extend FEDOSOV'S geometric scheme for deformation quantization to symplectic stratified spaces which possess only orbifold singularities. (see results about the existence of star
4.4.10).
Differential Geometric
88
Objects
on
Singular Spaces
A rather encouraging ansatz to prove even for symplectic stratified spaces they have a deformation quantization lies in the method of KONTSEVICH, more precisely in his formality theorem which was first stated in [103] and then proved in [102]. Among other things the formality theorem implies that every Poisson manifold has a star product (see as Well VORONOV [181]). In the following we will explain the formality theorem and how it could be used for a proof that every symplectic stratified space can be deformation quantized. Hereby we will have to sketch the arguments as a detailed exposition would be beyond the scope of this work. The essential ingredients for the formality theorem are two differential graded Lie algebras, namely the Lie superalgebra T*(M) of antisymmetric polyvector fields on a manifold M and the Lie-superalgebra D*(M) of polydifferential Hochschild cochains of the algebra C'(M). Let me explain these ingredients somewhat more precisely. By a differential graded Lie algebra one understands a Lie algebra in the tensor category of complexes of vector spaces, in other words an object of the form 2.6.10
that
I
0=( 0"
__j
gk+l
d:g
,
k
--4
ok+1
,
d2
0.
=
kEZ
The Lie
superalgebra T(M)
has components
T k(M) the differential d
:=
mappings
of the form
fo
fk
&
C'(M, A k+'TM),
supplied
hand,
with the
k >
-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) <
&
(7ts (x)) 1.
Analogously we define [e, 51, 1 F-, 5 [ and I F-, 6 [. If in some cases we want to stress t's and Ts' are subsets of X, then we write TF-cX, and T'sCXT', s T1,X s s
that
the sets
3.1.2 Lemma After
(1)
For every open 5
(2)
:
S
--->
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
<
5-,m,l
R o
It E
inf
R
-
E
such that
x
E
S and
E
m
locally finite covering (Vn)nEN of ts. Let V be the neighborhood V a
S
=
I ij
i s_' (x)
E
with
J with
by
in X
UnEN V71
e
compact K 1
Ps ([0, 5]) --1
Ps (ij)
=
t
1.
lk :' is lower semicontinuous and vanishes nowhere by < p.
C S
For the
and 5
> 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 <
p
(1).
it remains x
E K the
is compact. But this follows from the fact that
and contained in the compact set
(fts, Ps) : t'S
for all
CK,5
Therefore the even
surjective.
This proves the claim.
Now
we
0
N, of
continuous function with T'S C V. Furthermore let
=
to show that for every =
a
family (x,,),,,Ej,
a
W, hence (1) follows.
c
locally compact, there exists V,, with compact closure Vn- C
e :
C
of S and
As X is
of S and
find
can
=
open
X
open sets
C S and
U-,,o
open set
ITV ,o
points of S as a locally finite
set
Then there exists
choose
us
Vn
such that
Hence
a
achieve after transition to
we can
and ps (y)
converging
To this end choose for every
then
N
E
n
neighborhood of S in X. We denote the restrictions of 7rs and ps to 'ks by 0S. By the results proven above the sets Wx,n := ftil (U,,n) n P.-1 10, 6x,n I form of neighborhoods of x in X. We show that the triple (Ts, fts, Ps) satisfies (1).
is an
fts and
open
But this contradicts the fact that in the
neighborhoods
a
is
an
sets
in X and for every
x
=
ts
in
%
converges to somey E
=
is
neighborhood Y, in X with compact [ form Wx,,, := vx n7ril (U,.,,,) n ps-'[0, X, if U,,,,, C S runs through a basis of neighbormonotone decreasing sequence of positive numbers
Suppose
0.
(_Y11)nE1q liMn,,,. 7tS (Y n) 7rs (-y) -y
of
(5x,n)nEN
=
V of
neighborhood that
S
E
X
Ts. We claim that the
C
Theory
have all
El
prerequisites
3.1.3 Definition Let
m
E
to introduce the notion of
N'
U
foo}
and S
of S in M of class Cm then is
a
a
tubular
neighborhood.
C-submanifold of M.
A tubular
triple T (E, e, y), where 7rE : E -4 S is a E!m-vector bundle over S with scalar product -q, F_ : S -- R>0 a Cm-map (or even a lower semicontinuous function), and Y a Cm-diffeomorphism from TS'CE IV ( El PEM :::::::: JIV112 11 (V) V) < F (7r(V)) I to an open neighborhood T of s, the so-called
neighborhood
a
=
::::::::
:_
3.1 7bbular
93
neighborhoods T, such that the diagram
total space of
T'
(3-1-1)
ScE
I '*" M is embedded
Hereby S
commutes.
into
canonically
T'cE S
E
C
its
as
section.
zero
-
Sometimes
formally
not
If f
M
:
also call the total space T
we
tubular
a
though
even
this is
correct.
N is a C'-mapping, then one compatible with f, if f' 71 fIT-
--)
of class C'
neighborhood
calls
tubular
a
T of S c M
neighborhood
_-
(E, e, y) induces via the projection 7tE : E -4 S Every tubular neighborhood T : TS -4 S, called projection, by 7rs 7rE (p-'. Moreover, where one obtains the so-called tubular function ps : TS -4 R of T by ps p1E ps'(0). pE is the distance function on E defined above. Obviously we then have S Additionally, the tubular function is submersive exactly over the T \ S. =
a
continuous retraction 7ts
=
-
=
-
=
n
m
=
with &
=
+ k via the first
typical
I and y
Then the
m
coordinates. Let E be the trivial vector bundle
fiber Rk and the Euclidean scaler :
TR'-
-4
Tn In
triple (Rn, the
given by
:=
e,
TR'-
=
JX
(IJ, V)
=
o) comprises
measuring
(XJ'.
.
.
,
Y-n)
R'
X2
M +1
+
a
X
V
<
11. of class :=
M
Xn)
PR- is
point
x
(X1,
given by
let
to
tubular
F-4
Rm
Moreover,
mapping idpn
I JIVII
Rn,
over
neighborhood of Wn in W neighborhood Tn. Its projection 7rn
a
::'--
the Euclidean distance of _4
E
orthogonal projection (xi,
The tubular function pnM
to R1.
product
fiber metric.
as
R" the restriction of the identical
We call it the standard tubular
C'. is
into the Euclidean space
Example Consider the natural embedding of RI
3.1.4
x, 0
-
-
-
,
0)
7rR-
from Rn
the square of the function
E Rn to the
subspace Rm, hence by
+ _X2n-
Example If h: (M', S') -4 (M, S) is a diffeomorphism mapping S onto S' and (WE, F_ hls,, 117-1 y) is F-, o) a tubular neighborhood of S in M, then h*T a tubular neighborhood of S' in W. Analogously one defines h-,,T' for every tubular neighborhood T' of S' in M'. 3.1.5
T
=
(E,
3.1.6
=
Classical tubular
neighborhood
theorem Let
-
E
Tn
N"
U
-
{ool,
mannian manifold of class em+2 with Riemannian metric R of class m+ a
M
a
Rie-
and S
c* M
W0,
such
1, eTn+2 -submanifold. Furthermore denote by N the subbundle of TIRM orthogonal ,
with respect to i to TS in TIRM. Then there exists that over TS'CN the restriction (p := expIT' : T'CN S
el-function
a
--)
M of the
E :
S
-4
exponential
function
SCN
is well-defined and such that the
triple (N,
E,
y) comprises
a
tubular
neighborhood
of
S in M of class C'.
A tubular
neighborhood (N, E, y) like in the classical tubular neighborhood by t. We denote it often in the form (N, F,, expIN)-
will be called induced
theorem
94
Control Let W
PROOF:
an
mapping (7r, exp) : W
open
M
--
V:= W n N. Then V is
the
an
Theory
neighborhood of the zero section of TM, such that the M comprises a C'-diffeomorphism onto its image. Set open neighborhood of S in N, where S is identified with
x
section in N. Consider the restriction expiv : V --i M. Now, for every and every tangent vector v (vh, vv) E To.,TM TM ED T-,M the relation zero
x
E
M
=
Vh
To. expv
+ Vv
is true, hence expIv must be submersive after possibly shrinking V to a somewhat smaller neighborhood of the zero section. Hereby one can choose V such that
exp-1 (S)
n V
S.
dim V the map expIv is not only submersive, but By dim M which coincides over S with the identical map. By the diffeomorphism lemma there exists an following open neighborhood t C V of S such that expit is a its onto diffeomorphism image. By Lemma 3.1.2 there exists a continuous and even a C'-function e : S --+ R" such that TS'CN C T. Consequently w comprises also
=
=
local
a
=
a
diffeomorphism
onto its
image,
hence the last component of
a
expITS'CN
tubular
neighborhood
of S.
El
M, N be manifolds and and S C N a submanifold possibly with boundary. assumptions f : N -i M comprises a el-function such that the restriction f1s is an embedding and for every point x E S the tangent map T-j is bijective then there exists an open neighborhood T of S in N such that f maps T difleomorphically to an open subset of M. 3.1.7 Lemma Let
If under these
PROOF:
We
GODEMENT
[601
like in LANG
proceed on
[108, IV.5]
and
use
an
argument given by
page 150.
simplicity we identify the image of S under f again with S. Now let (Nj)jEj locally finite covering of S by in N open subsets Nj such that the restrictions fj fjNj : Nj -- Mj are diffeomorphisms onto open sets Mj c M, and such that every Nj has nonempty intersection with S. Afterwards choose a covering (Yj)jEj of S subordinate to (Mj)jEj by in M open subsets Yj such that the relations Yj n S :A 0 For
be
a
and
Vj-
C
Mj
hold. Let
?j : Mj
Nj
--
be the function inverse to
and Y C M the
fj
points -y E Uj Yj Tj (y) ?i (ij), if -U lies in the intersection VYj- n VYj-. S Y. show that Y is We even a neighborhood of S. To this end choose Obviously C S. Then there exist an arbitrary point ij E Yj, such that 'Y lies exactly in the Yj,, closed hulls Yj-,, 1 k. One can now find an open neighborhood Y., C 1, Uj Yj of
set of all
such that
=
-
-
-
,
=
-
-
-
,
-Lj such that
Y,,
Tj, (ij).
fj,
C
Mj,
for I
=
1,
-
-
-
,
Hence there exists
V,j
N and
an
c
f
open
-I
(Yj)
Y., n Vj-
=
0 for j :A ji,
neighborhood V,
nn
-
-
-
,
jN. Obviously
of 1j in N such that
Nj,.
1<1
Now set E
f (V.,) n
Y..
Then
' .. n Yj, n Yj, the relation fjk
of S.
is =
an
of -Li in Uj Yj and for all Consequently Y is a neighborhood which assigns to every point Ij E Y n Yj
open
fj, ( )
Moreover, the function T : Y -4 N Tj(-U) is well-defined. By definition T
the value
neighborhood
holds.
is left inverse to the restriction of f to
3.2 Cut T
:=
and maximal tubular
point distance
f`(Y),
hence
fIT
is
injective. On
95
neighborhoods
the other hand T is
neighborhood
a
of S in N.
Thus the claim follows.
13
Let T (E, F,, (p) be a tubular neighborhood of S in M, where we suppose that equipped with a metric connection. Then the following propositions hold; the corresponding proofs follow immediately from the definition of a tubular neighborhood and the corresponding properties of the metric vector bundle E. 3.1.8
=
E is
(1) Identify E with the vertical bundle of TE over TIE o comprises
an
The tubular
E
--
N,
To.,, o.v-,
1-4
v,
isomorphism of vector bundles from
which is normal to TS
(2)
:
S c TE. Then the restricted
S in
-4
neighborhood
TISM
that
means
E to
TIsM
T generates for every
x
a
S
vector bundle N
=
point
E
x
E T
a
is the
image
of the horizontal space of
isomorphy H,,M with T,(,,_)S curve p (tv), t E [0, 11.
the canonical
along
(3)
the
is
T o (E)
canonical decom-
=
T,,(x)S. Hereby HxM
:=
TS E) N.
position of the tangent space T,,M into the vertical subspace VM _ v y-'(x) and the horizontal subspace H,,M which is canonically to
mapping
Tvy(E,,(x)),
isomorphic TvE under Tv o;
given by parallel transport of T,(x)S
arbitrary Riemannian metric on T, ii the scalar product on E, and for every x E T let Px TxM --) TxM be the projection onto KxM along VxM. Finally let Qx idTxm Px. Then there exists after possibly shrinking T Let 0 be
an
=
Lx(v w)=il7rs(x)(Tx o-l.Qx.v,Txy-l.Q.x.w)+O(Px.v P.x.w)I a
Riemannian metric
L
on
T such that (p
=
exp
-
function of the Levi-Civita connection of R.
bundle N is
li-orthogonal
means
Hereby exp Moreover, the
TlEy.
exponential
to TS that
XET,V,WET.,M
in other words
N-x=fVET.,Ml t(v,w)=OforallwETxSl, (4)
The Hessian of ps has rank
The statement
structed
(3)
according
can
be
codimmS
interpreted
as
=
3.2
Cut
-
dim S
neighborhood
that every tubular neighborhood T where N is the normal bundle and exp the
appropriate Riemannian
dim M
over
xES.
S.
saying that the tubular neighborhoods
to the classical tubular
means
an
is the vector
metric defined
can
theorem
are
be written in the form
exponential function exp on a neighborhood of S.
con-
universal. This
(N,
E,
eXPIN),
with respect to
point distance and maximal tubular neigh-
borhoods neighborhood theorem provides an important existence result for neighborhoods, but we do not know, how "large" such a tubular neighborhood
The classical tubular
tubular
96
Control
can
be, whether there give
order relation
an answer
(N,
el,
and
expIN)
M. Then
we can
neighborhoods and so on. It is the goal questions. But first we have to define an
to such
the set of tubular
on
"maximal" tubular
or
maximal tubular
are
of this section to
T2
neighborhoods, so that we can speak of "larger" neighborhoods. Consider two tubular neighborhoods T, (N, F-2, expIN) of S, both induced by a Riemannian metric L on
=
=
compare their total spaces
inclusion. If T, C T2
if el !
T, and T2 with respect to the set theoretic F-2, then we write T, :5 T2. The relation
equivalently obviously comprises an order relation on induced by L. With the help of several quite or
<
in this section to construct in
case
3.2.1
that
Theory
(M, L)
is
Denote
by t
a
Riemannian manifold.
E TM and x E M
V
neighborhoods of S geometric tools we will succeed
nontrivial
maximal element with respect to the order relation <
a
complete for
the set of all tubular
the escape time of exp, in direction v,
the supremum. of all t E V' such that exptv is defined. Furthermore, SN the sphere bundle of N. Then for every v (:-: S,,N with x E S the focal
means
denote by point distance ef (v)
and the cut
in direction
ef,s (v)
:=
:=
is
v
sup
given by
{t E
[0, t
point distance in direction
e,(v)
:=
e,,s(v)
:=
ker(Tv expIN) v
[0, t [ I 6,(expIN (tV)) S)
Hereby SR means the geodesic distance with respect one point x, then we define
Ti(X)
Ti(x)
and call
functions ef 3.2.2
:
the SN
-4
K o
0 <
V1
:=
every
e,,s(y)
Moreover, the relation Tj(y) We
[101,
GENBERG
Thm.
inf
je,,j_,)(v) I v
injectivity radius of
Proposition For
PROOF:
:=
<
U
V
tool,
E
> 0
exp at e,
:
and
E
to R. In
[0, tj
and Tj
E S one
x
0 <
tj S consists of
only
S,,.Ml
Vo
--
=
case
Altogether
x.
SN
S.,,N with
ef,s(v)
s E
by
{t E
sup
0 for all
==
we
M
:
thus obtain three -4
Vo.
has
e,,s(v)
:5 ef,j,
holds for every 'y E M.
only sketch the proof; for Thm. 1.12.13 & Thm.
details
more
2.5-151
or
we
refer the reader to KLIN-
BISHOP-CRITTENDEN
[16, Chap. 11,
5, Cor. 21.
As the
exponential function exp-, is a local diffeomorphism mapping a sufficiently origin of T,,M to a strong convex neighborhood of x, the relations
small ball around the e, (v) >
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 <
following
considerations
we
tV+
and -y
:
I
=
[0, tj]
want to construct
So it remains to show e,,s (v) :5 inequality; the proof
the second
a
case
the
tf
:=
M the
help of S-Jacobi-fields [16,
ef,,,,}(v)
<
t.
Let tj be
a
geodesic t " exp, tv. In the piecewise continuously differentiable
-4
3.2 Cut
point distance
and maximal tubular
97
neighborhoods
geodesic -y such that for sufficiently small I s I the points x and -y(ti) and have a smaller length F(s, -) -y., than -y. As T exp,, has a nonvanishing kernel in the point tfv, there exists a nontrivial Y (tf) 0. As Y is nontrivial, 0 and Y (0) J acobi-field Y along -y with L(Y (t), - (t)) there exists a point to between 0 and tf with Y(to) =,4 0. Let Z be a differentiable 0 for 0 < t < to, Z(tf) vector field along -y with Z(t) -VY(tf) 74 0 and Z(ti) 0.
variation F variation
:
] e, e [ x I
curves
M of the
--)
connect the
=
we
set for
some il > 0
which will be determined later
e,
[
F_ x
1
-4
Xj generates path
-y,
E (s)
the energy
F(s, -)
=
=
<
tf,
if tf < t < ti.
piecewise continuously differentiable variation F
a
M of -y, in other words there exists
consider for each
if 0 < t
Y(t)
0
whereY*(t)=
X,,:=Y*(t)+71Z(t) The vector field
=
=
=
Then
=
=
=
E (-ys)
=
I 2
an
F with
t
Xj (t)
Now
:
we
integral
f (- s (t), - .s (t))
,(T(t))
dt
and calculate the second variation of E: D 2E (-y) (X,,
Xn)
=
=
Via
J, (VXn, VXj) D 2E(,Y) (y*,
partial integration
that the second
one
is
one
,
(t)
dt
-
f, (R Xn, Xj)
,
(t)
dt
(3.2.2)
2 y*) + 2TID 2E(,Y) (y*, Z) +,n2D -E (,y) (Z' Z).
observes that the first term
on
the
right
side vanishes and
given by
2ij(limt/tfVY*(t)-limt,\,tf VY*(t)'Z(tf)) (1LY(tf))=_211 IVy(tf)12. 2 Hence, for il sufficiently small we thus have D E (,y) (X71, XTI < 0. Using the fact that the length of y is given by IyI2 2E(-y) III we obtain for sufficiently small s =
I,ySI2g that
means
well, there
ys has
a
<
2E('y ,) III
<
Xn(tj) 0 holds as length smaller than IyI. Hence, as X,(O) a path shorter than -y connecting x -y(O) with -y(ti). Thus =
=
must exist
=
has shorter distance to S than ti. But this
-y(ti)
IyI2g,
2E(,y) III
means
ec,s(v)
:5
ef,{-,}(v)
which proves n
the claim.
lower semicontinuous. If
3.2.3
and Tj
is
ef, e, and Tj
a
Proposition The functions ef, ec complete Riemannian manifold, then
PROOF: ness
TM, us
are even
(M, L)
continuous.
by the well-known theorems on the existence and uniqueequations the escape time t+ has to be a lower semiSN (see e.g. [2, Lem. 10.51). Consequently W'ax := ftv E
First recall that
of solutions of differential
continuous function
TM
are
Iv
E
and
SM and 0
(7r, exp)
denote in the
:
on
< t <
W"
t,+}
--)
M
c x
following by [x, -y]
TM is
an
open
neighborhood
M is well-defined. For
the segment
fexp (tw) I
x
w
of the
and V =
zero
section of
sufficiently
exp, -'(V) .
close let
and t c
[0, 11}
98
Control Theory
connecting proof.
and -y. After these agreements
x
on
the notation
we can now
start with
the
Let exists
suppose that ef is not lower sernicontinuous in the
us
0 and
point
v.
Then there
sequence (Vk)kEN of unit vectors vA; E N converging to v such that t linlk-4oo tk with tk := ef (Vk) exists and such that t < ef (v). Let wk E Tt,v,N be a unit vector in the kernel of Tt kVk explN. After transition to an appropriate some
>
e
a
:'--'
subsequence (Wk)kEN converges to a nonvanishing vector W E TtvN. By continuity exponential function w must be in the kernel of TtvexpIN in contradiction to
of the ef (v)
:5
t < ef
Next
we
(v).
prove
Hence ef is lower semicontinuous in
by
pose that there exists
e,(v).
v.
contradiction that ec is lower sernicontinuous in a
sequence of vk
converging to v
such that t,,,)
=
v.
So let
limk,,,
us
sup-
e, (vk) <
We denote
by x the footpoint of v and abbreviate: tA; ::'-- ec(Vk). Then we emcomplete hull M with respect to the geodesic distance; obviously M then is a length space (though in general not a Riemannian manifold). Under these prerequisites there exists by the theorem of BUSEMANN 2.4.13 resp. by the theorem of HOPF-RINow 2.4.15 for every k a positive 5k < -1 and a rectifiable path 2T parametrized by arc length and having the following properties: Tk : 10) Sk] bed M into its
(1)
Yk (0) E S and Tk (1)
(2)
The curveYk minimizes the distance
=
exp ((tA; +
bk)Vk)
frOMYk(l)
6 [L (Yk (1))
(3)
The
length
sA;
=
17kl[i
where
,
S)
to S that
means
ITA; I
=
is smaller than tk +
denotes the closure of S in
5A;.
The
curves -yA; thenfulfill linik-4oo Tk (Sk)= exp (t(,,)v). After transition to subsequences (Sk)kEN then converges by the theorem of BUSEMANN 2.4.13 to some soo < too and ('Yk)kEN uniformly to a rectifiable path -y : [0, sj -i M with -y(O) E S and -y(s,,o) exp (t,,ov) We now consider the mapping (7T, exp) : WmI --- M x M. As e, (v) < ef,,-,, (v), the map (7r, exp) is of maximal rank over the segment [0, t j v C TM and injective. Therefore by Lemma 3.1.7 (7r, exp) maps a relatively compact open neighborhood -
W C Wm' of
[0, t( I v diffeomorphically
shrinking
W
around
and U
x
one can an
suppose that
open
onto
an
(7r, exp) (W)
neighborhood
open set in M
has the form B
of the segment
x
x
M. After
U, where
possibly
B is
a
ball
[x, exp(t(,Ov)]. Furthermore
we
that U is connected and that exp (tv) for 1 (e,- + t,,.) :5 t < t does not lie 2 U. As the restricted exponential map n:= expIN : N n Wmax -- M is of maximal
can assume in
rank
over [0, t,,O] v and injective, n maps by Lemma 3.1.7 an open neighborhood W' of [0, t,,,] v in N n Wm' diffeomorphically onto an open neighborhood U' of the segment [x, exp (t,,.v)]. After shrinking W' and W appropriately one can achieve U U1. Suppose for a moment that the image im-y does not lie completely in U. Un=
der this assumption let sc be the minimum of all s' < s,,,, such that -y([s', sj) c exp, ([0, e,- (v) [ v) n U. As -y does not completely lie in U, we have s, > 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) <
such that
s,,O, hence
than the
-y(s,)
=
exp(t,v).
In
case
by soo < too the restricted segment connecting x and exp(tcv).
3.2 Cut
point distance and maximal tubular neighborhoods
99
impossible. Consequently t, < t,,,, and Y(s,) strongly convex ball B C U around -y(s,) such that y intersects the boundaxy of B in a point not lying on exp ([0, e, (v) v) n U. Let 0 < SDB < sc such that y(saI3) E aB \ exp([O, e,(v) [ v). Let further "Y[0, 9] -4 M with SH + 6,,(,y(saj3),-y(s,)) be the curve composed by -yl[o,,,.] and the segment [- (saB), -y(sj. By t < e,(v) and the definition of e,(v) the restricted path '11[s.,soo] has length > too t, consequently the length 9 of - is smaller or equal to t,. By construction the vectors 'y(9) and Ts, exp-,.(v) are not collinear. Hence there exist poi nt s This would
imply t,
which is
e,(v)
>
Moreover, there exists
lies in U.
a
::--:
-
exp(tv)
and
ment
[- y- (s'), exp(t'v)]
Next consider the
-v(O)
=
-y(O)
9,
E
s'
saB <
< s,
is shorter than the
curve -v
I-vl,,, As
(where
in B
'y-(s')
+
<
<
curves
I [ y- (9), exp(tv)] J,,
<
`y'j[o,s,]
t'
<
< e,
e,(v))
and
(v). (W")IIEIq
such that the seg-
['y-(), exp(t'v)]. [- (s% exp(tv)]. Then and
I [x, exp(tv)]
point exp(tv) has shorter distance
the
the contradiction e, (v)
the
composed by
and t, < t'
path composed of ;Y-1[s,,g]
t'.
to S than V. This entails
Hence im y C U.
neighborhoods of the Obviously one can choose the W,,+, segment [0, tc,,,Iv form the where a basis of neighborhoods x that such B,, W,, (7r, exp) (W,,) B,, U, of basis the and around balls a of x U,, neighborhoods of the segment consisting open each of the neighborhoods lie in must the above argument imy [x, exp(t,,.v)]. By of geometric curves, converges U,,. As the sequence (-Yk)kElq, regarded as a sequence by the theorem of BUSEMANN 2.4.13 uniformly to -y, there exists for every n some Next
we
set
Wo
=
W and let
c TM such that
run
through
W,, for all
C
basis of
a
n.
=
kn
E N such that
imyk
Un for all k
C
>
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 <
as a
points
and
e,(v),
<
collinearity
minfe,(v), e,(w)}
Hence
v
then
would entail 6
and
with exp tw.
=
,
L(v, w) (z, S) :5
6
,
(z, ij)
=
t
,y:[O,t+5]-)M) is not differentiable in t. t + 5 <
e,(v).
-I would hold
=
not collinear. Therefore the
w are
e,(v)l.
injective. Suppose this were not the case. S, two vectors v E S,N and w E Sy N as
there exists 6 > 0 with t + 5 <
collinear,
w were
0 < t <
x, ij E
exp tv
As t
S,,N and
E
v
embedding.
Then there exist two distinct well
I
N
Hence
z)
<
as
Let
z
=
otherwise
exp(t x
=
+
ij
.
6)v.
6, which contradicts t + 6 following path connecting Lj
-
exp
sw
if
s <
t,
exp,
sv
if
s
>
t,
jyj ,
=
If
v
Therefore < ec
(v).
and
z
t + 5 follows which contradicts
e,(v).
By
Therefore explTmax is injective. CN has to be immersive. By definition of ef and Proposition 3.2.2 the map explTma M1 reasons of dimension the image exp, (Tm' ) is open in M. Putting all this together
we
obtain the claim.
ScN
3.2.5
Proposition
0
Let
(M, L)
be
a
complete Riemannian manifold and
S
a
closed
submanifold. Then Tm' is maximal with respect to C among all open neighborhoods ScN T of the zero section of N such that eXPIT : T -i M is an open embedding and such that for every From the 3.2.6
v C-
T the segment
Proposition
we
Corollary Define
[0, 1]v
obtain the
followin&main
eml: S
Vo by
-4
em'(x) Then Tm'
=
M induced
by
PROOF:
(N, em', expIN)
an
=
inf
v
G
suppose Tm' ScN
us
open
neighborhood
SN and 5 > 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 <
=
Therefore the set
> 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 <
projection endomorphism of RI with p2 are not possible, simply P is a
submanifold of class C' and P
a
&',
valued section of class
that
means
P,,. Then the pair (P, S)
=
X
for all
or, if any
called curvature moderate of order
x
E T
:
T
let P, be
an
misunderstandings m, if the following
holds:
(CMI)
For every
point
x
components that
E RI the set germ of
S at x has finitely many connected neighborhood V C Rn such that for intersection S n B has only finitely many
there exists
means
every ball B C V around
the
x
a
connected components. situated
of S.
(CM2)
T is
(CM3)
For every point of aS there exists a neighborhood V C Rn as well as constants c (=- N and C > 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
<
C
(1
+
x
E
VnT.
(3-3.1)
d(x, aS)c
If P is curvature moderate of every order, then we say say that P is curvature moderate of order oo or briefly that P is curvature moderate. In other words P is curvature
moderate if and
only
if the components of P
are
Whitney functions
T
on
tempered
relative aS of class C'. A C"-submanifold S C
moderate
(of
order
such that the Tn" S curvature moderate
m),
Rn, where
now we
allow
if there exists
M
E
N>O Ulool, is called curvature
situated open
a regularly neighborhood TS corresponding projection valued mapping PS : TS - End(Rn)
(of
order
c-
is
in).
6 C direct calculation that for every C'-diffeomorphism H : 0 Rn with 0 C Rn open and aS n 0 =,4 0 the manifold S n 0 is curvature moderate One checks
by
up to order m, if and sense to define
moderate
(of
class C' such
a
only
if this holds for
stratum S of
a
H(S
n
0)
C Rn as
as
curvature
m), if there exists a covering of X by singular charts x : U -i Rn of that x(S n U) is curvature moderate (of order m). In particular it is now
order
clear what to understand
by
a
curvature moderate submanifold S of
The stratum S resp. the stratified space X is called curvature every stratum of X is curvature moderate. 3.3.3
well. Hence it makes
stratified space X with smooth structure
Example Subanalytic sets with their In particular all algebraic
vature moderate.
coarsest
varieties
Whitney are
a
manifold M.
moderate,
if S resp. if
stratification
are cur-
curvature moderate.
This
3.3 Curvature moderate submanifolds
103
can be proved only with some larger technical expense. It can be derived by unpublished work of PARUS114SKI [1401 or with the help of Newton-Puiseux-expansions as they have been used in MOSTOWSKI [130].
result
3.3.4
Example
In the
following
spiral
are
not curvature moderate.
will consider several situations where
we
one can
naturally find
pro-
valued sections.
jection 3.3.5
The slow and fast
Let f
Example
:
R'
N be
--
N submersive. Then there exists
C'+'-mapping between
a
an
open
neighborhood
manifolds and
f1s
:
S
T of S in R' such that the
mapping Pf : T -4 End(R7) projection onto the kernel ker TJ is of class C'. We call the pair (f, S) or, if any misunderstandings are not possible, only f curvature moderate, if the projection valued section Pf is curvature moderate; finally we call f strongly curvature moderate over S, if additionally S and after possibly shrinking T even the mapping PfS : T -- End(W) which assigns to every x E T the
restricted map which
assigns
projection In the
f1s
:
S
--
fIT
:
T
N is
-i
to every
x
C-
onto the kernel of more
N is
general
P,f
submersion. Hence the
is curvature moderate.
Ps,,,
-
M
that f
case
pubmersive
a
T the Euclidean
call f
we
N is defined
--
(strongly)
curvature
over
a
manifold M and
moderate, if there
exists
a
covering of DS by differentiable charts x : U -- Rn of M such that every one of the mappings f x-1 : x(U) -- N is (strongly) curvature moderate. In all of the last definitions we should have mentioned explicitly the order m; by reasons of linguistic aesthetics we have abstained from this. By the same reason we will often not mention in the following the order m, if the context makes clear which -
order is meant.
following lemma provides projection valued sections.
The
some
3.3.6 Lemma Let S C Rn be
and
a:
T
-4
End(Rn)
a
a
us assume
the
means
how to generate curvature moderate
submanifold which is curvature moderate of order defined
Cm-mapping
of S such that the components of a
let
further
over a
tempered
are
regularly
situated
M
neighborhood Additionally
relative aS of class C'.
following:
(1)
The rank rk a,
(2)
Each of the operators ax E
(3)
Denote
by X,
x
the
E T is constant
over
End(Rn)
eigenvalue
of ax
T.
is normal.
smallest
having
nonvanishing
absolute value.
Then there exists for every point of aS a neighborhood V in Rn constant c E N and a C > 0 such that the following estimate holds:
AxJ
>
C
d(x, aS)',
x
c
as
well
as a
V n T.
Define for every x E T the endomorphism pa Pk ,,,. as the projection onto ker ax X is pa T Then im a -4 : projection valued section and curvature End(Rn) ax. along =
moderate of order
m.
104
Control
PROOF:
To
function g
:
there exists
A a
the formulation of this and further
ease
Theory
proofs we say that a continuous from Z, where z n A 0, if
R with A C R1 behaves bounded away d E N such that --)
sup I g (x) I
d(x, Z)
'
=
< oo.
XEA
As
we can
End (R),
switch without loss of
x 1-4
(where 4
a,.4
generality from the mapping adjoint, operator) an d
denotes the
image and the same kernel like selfadjoint and positive semidefinite. same
be the smooth closed
In
around the
curve
one can assume
a,,
particular 7x-,
>
to aa*
as
a.4
:
T
has the
that a, is for every x E T 0 then follows. Now let
of C with y., (t)
origin
a
=
!A., e2ntt, 2
t E
[0, 11
Then it is well-known that
Pker a,
2ni
fYX
dz, z
-
a,,
hence
aip ker ax M
Now choose for Then
we
a
have for
point of aS x
E
a
V n T the
llaiPkerax 11
1
f
D a., Z
neighborhood V in following estimate
1 SUP 27r t,,[0,1]
-
R'
a.,
according
to the
assumption.
11,Mt)11 JJDAxJJ 11,YX (t) aX112 -
4
A.
ei dz. Z
ax
-
4
JJDAxJJ
:5 C
1
(,+ d(x, M)c
JJDAxJJ.
possibly shrinking V we can achieve that V is compact and that relatively compact in aS. Hence by the temperedness of the components of After
V n aS is a
the map
(aiPkera)1vnT
restricted to V n T must be bounded away from aS. Analogously one shows that the higher derivatives a'Pker,, restricted to V n T are bounded away from aS.
Altogether
one
thus shows that the components Of
Pkera
are
tempered relative aS
of class e,'. This proves the claim.
11
3.3.7
Proposition A submanifold S C R1 of class C'+' is curvature moderate of order m, if there exists a regularly situated open neighborhood TS C Tm' such that S the components of the Euclidean projection 7rs : Ts -- S are tempered relative aS of class C+'. For the
[59,
proof we need the lemma below which comprises
Lem.
3.3.8 Lemma Let L be
T '
a
a
generalization of the result
III]. a
Riemannian metric defined
tubular
map of the
neighborhood of S induced by projection 7& of Tt' in the form T.7&
=
E,
-
P,
R.
Then
X
E
on an
open
one can
T",
neighborhood S
and
represent the tangent
(3.3.2)
3.3 Curvature moderate submanifolds
where P
:
V'
End(R)
--
automorphism
of T,(,)S
Ni
S
:
L-orthogonal
S,
and
T,"(x)nl"
X "
F. the
=
(id
+
ni
P711
T,,.(.,,) N i_1
-
(id
ni denotes the smooth function
exp-'(x) (x), ei
-4
x
P,,.(.,)) ei
-
which is
and exp, the
function with respect to t.
exponential
After
PROOF:
section
R' denotes the vector field N i (70'(x))
-4
to
projection valued
given by
F,,
Hereby
is the
105
abbreviating 7& by following way:
7r we
expand
x
with the
help of the exponential
function exp in the
x
=
exp 7T(%)
(Ei=nl
7,(lx
exp, (X) (x), ei
N i (7r( X))
Replace
possible indeed, as exp,,(lx) (x) is L-orthogonal to S at the footpoint 7t(X). in this equation x by y (t) tv + x, V E Rn, and differentiate with respect to
t at the
point 0, then
This is
=
one
obtains
n
-1)(.))v,Ni(7r(x)) v=T,,7r.v+Eni(x)TNi.T,,,7r.v+(T(exP7t(--')x)v+T(exP71(X Now let the operator 0 and
P,(.x).T.,,7r.v
=
P7,(,.)
T,,,7r.v
act
on
this
equation. Then
one
obtains
by P,(
X
Ni(7r(x)).
).Ni( 7r (x))
the relation n
P,(x).v Hence G-, is
an
=
G,.T,x7r.v
with
Gx
automorphism of T,(-,)S,
=
and
ni (x)
id +
F-,
=
G%-1
P.,(,,) T,(,x.)Ni. -
is well-defined. Thus the claim
follows. PROOF T
:=
M
OF THE
PROPOSITION: Now
we
denote
by
7r
the
projection
Note that in the Euclidean case, which is the
TS.
case
7ts and abbreviate
we
consider at the
x + v. Its inverse is given by moment, the exponential function is given by exp, v 1 x, where x, IJ E TS, v E Rn. Hence Fx has the form -y exp, -y =
=
-
F-,
=
Gx
1
with
Gx
=
id +
E
(7r(x)
-
x,
ei) P,(.x) T,(-,)N i. -
(3.3.3)
i=O
Let P lie in
us
suppose first that S is curvature moderate that
means
the components of
Choose for every point of aS a compact neighborhood V C R' Thus it remains to show that all restricted partial to Definition 3.3.2.
M'(aS;T).
according
WT701vnT with Jai < m are bounded away from aS over VnT. Obviously product of functions bounded away from aS is again bounded away from aS. Now, the components of the function PIS are bounded, hence bounded away from aS. Therefore, this holds for T7rIS PIS as well. From Equation (3.3.3) one derives derivatives
the
==
106
Control
easily
that T
JIG.,,11
:
2'
can
be restricted to
and
is bounded away from aS
J,xJ
as
well.
away from aS. But this entails that
As S is curvature
moderate,
the
that that
(Dni)JvnT holds for
same
and
DPivnT
JIG-,11
regularly situated neighborhood of S such E Ts. Together with (3.3.2) this implies
a
11F.,.11 :! 2 hold for all x derivative (T7r)ivnT is bounded
the first
Theory
DT,(.)Ni, hence DGivnT is bounded away from aS. By Fx Gx-' and Equation (3.3.2) then entails that all partial derivatives (a'T7r)ivnT with
> .1 2
=
I have this
well.
By induction one moves to higher derivatives an analogous argument that these are bounded away from a S over V n T. Hence the components of 7t lie in M1+1 (a S; T). Now let the components of 7r E (!'+' (T) be tempered relative a S of class 'Ell+'. By the representation P,, T,(.,,)7t, x E Ts and a repeated use of the chain and LEIBNIZ rule one proves that the projection valued mapping P has to be curvature moderate of order m. This proves the proposition. 1-:1 =
property
(&xT70ivnT with JcxJ
as
and shows
> 1
by
=
The notion of
respect
to tubular
a
curvature moderate submanifold of R1 has been defined with
neighborhoods
induced
the Euclidean metric.
by
But it makes
and will later prove to be necessary for the extension theory of smooth functions that one has available even a notion of curvature moderate Riemannian metrics and sense
of curvature moderate tubular
neighborhoods.
3.3.9 Definition Let S C R' and R
regularly situated neighborhood T derstandings are not possible, only R a
conditions
are
a
Riemannian metric of class C' defined
of S. We call the
pair (R, S),
on
if any misun-
or
curvature moderate of order m, if the
following
satisfied:
(CM4)
The components Rij R(ei, ej) : T -4 R of [L with respect to the canonical basis of R' are tempered relative aS of class C1.
(CM5)
For every
=
C E
point of DS there exists
N and
d(x, DS)' C
Hereby 11 by p,,.
a
C
>
0 such that the
neighborhood V C R' as well following estimate is true:
a
1
11VII
:5
11VII[L.
:5 C
I +
d(x, a S)c
denotes the Euclidean
norm on
) JIVII, Rn and
as a
constant
XEVnT, VER.
11
the
norm
induced
In the
more general case that S is a submanifold of M and R a Riemannian metric neighborhood T of S in M the metric L is called curvature moderate of order m, there exists a covering of aS by differentiable charts x : U -4 Rn of M such that
on a
if
X*( L[Tnu) 3.3.10
and T
is curvature moderate of order
Proposition Let M E N'O, S C R7 be curvature moderate of order M + 2 regularly situated open neighborhood of S. Then the following statements
a
hold for any connection V
(1)
Tn-
If t is
a
on
the
Riemannian metric
tangent bundle ofT.
on
T and curvature moderate of order
is the Levi-Civita connection of R, then the Christoffel
with respect to the canonical basis
relative aS of class C1.
symbols
ofR1) comprise functions
M
rij,21
which
+ 1 and if V
of V are
(formed
tempered
3.3 Curvature moderate submanifolds
(2)
In
the Christoffel
case
symbols rij
107
of V
are
tempered relative
as of class
VTI,
the
mappings
(x, V)
V =-)
-4
(exp.-,'(x), ej)
-4
(expv, ej)
E R
and W D
v
E R
comprise functions which are tempered relative as of class C'. Hereby exp is the exponential map with respect to V, V C T x T C R2' an appropriate regularly situated neighborhood of S, W is a regularly situated neighborhood of S in TT c R2', and S will be canonically identified with the diagonal of (S x S) n V resp. the zero
section of TS n W.
For the
PROOF:
proof of (1)
we
function of class (!'+' such that Then the function 'F1
:
T
first
regard
, is for every
End (R) with
-4
t
:
T
pT!,
=
End (Rn)
-i
E T
x
selfadjoint
idR.
as a
and
matrix valued
positive definite.
is well-defined and smooth.
It satisfies x E
The component functions
on
T
Moreover, by assumption
we
have for every
belonging to
T.
t are tempered relative as of class e+'. sufficiently small ball B around a point of
as I sup 11 11=1
where
C
E N and C > 0
IlTi.-VII are
:5 C
1 +
d(x, aS)c
)
x E
,
B n
T,
appropriate. Hence the restriction (aj'p)jBnT has to be partial derivatives tempered relative
bounded away from as. Analogously one shows that even the higher (a a-FL-) IBnT with I ocl < m + 2 are bounded away from a S. Hence 4 is as of class eln+'.
As it is well-known the Christoffel
symbols
of the Levi-Civita
connection have the form 1
+ 2
Thus
by the considerations above the Christoffel symbols
as of class e'. This proves
have to be
tempered relative
(1).
exponetial map is tempered. Denote by t+ exponential map like in 3.2.1, and let W'ax fV E TTJ q > 11 be the maximal domain of exp. With the help of Theorem 1.7.11 we show first that W" is a regularly situated neighborhood of S in R2n. Now for every vector v E W C TT _ T x Rn the curve -yv(t) exp-jtv), t E [0, 11 satisfies the initial value Now
for
V
we
come
to the
proof
E TT the escape time
that the
V
of the
=
problem
, vk (t)
+
E
r 13
(,yv (t)), Vi (t) V(t)
=
0,
,
YV (0)
=
X,
V(O)
=
V.
Ili
By assumption on the Christoffel symbols and by Theorem 1.7.11 there exists a regularly situated tubular neighborhood i C R x R2n of JO} x S x 10}, such that the
108
Control
mapping,ylt T
R1, (t,v) t-- 'y,(t) is tempered relative fOJ x as x fOJ of class CITI. positive continuous mappings 5, z, z' on S such that
--+
Hence there exist
5(x) for all
! D
d(x, aS)d,
E(x)
appropriate
constants c,
E S and
x
Theory
Bl(,,,)(0)
B' F- (X
x
d(x, as)',
! C
)(x)
c',
and
e'(x)
d C- N and
C, C',
Be',(x)(0)
x
C
T,
!
C'd(x, as)"
D > 0 and such that
X E
S.
Define
U (B,'(-X) (x) n7ts-'(x))
W:=
x
&
,)(0). B'I,.,,(. 12
XES
pair (t, VJ
Then for every
[-1, 11
E
W the
x
(5 (x)t
point
1
--1-v 6(X)
,yl[-,,,]>,w and exp1w are tempered relative fOl x as x fO} resp. as A similar argument shows that an appropriate restriction of hence
(2)
3.3.11
)
x
lies in
fO}
exp-1
t,
hence
of class C-. is
tempered,
follows.
13
Let S and T be like in the
Corollary
metric which is curvature moderate of class
T ',` induced
neighborhood
exp(Tm' ScN )
by
L is
proposition and let R be a Riemannian Cl+' on T. Then the maximal tubular
regularly
situated to S that
means
VL,max
with
IV E W1
Tm'
ScN
is
X
aregularly
X
situated
x
E
S,
expV E
T, 6,(x, S)
=
Ilvil..
of S.
neighborhood
will
Next
we
to
Riemannian metric
examine, under which assumptions projection valued sections associated are tempered or in other words under which assumptions these projection valued sections are curvature moderate. To this end we study in the propositions below local orthogonal systems with respect to R. a
3.3.12
Let
Proposition
moderate of order
m E
2, T Euclidean projection Ps : T m+
moderate. Let L be m
a
fjl (1)
on
((Pj)jEi
Rn
\
bounded
(2)
by
End(Rn)
submanifold which is curvature
neighborhood
over
of S such that the
T and such that
PS is curvature
T which is curvature moderate of order
B,.(z) C Rn be an open ball of radius F- < 1 around a B2,(z) n as is compact. Then there exist two countable (fi1)iEJ,1<1
=
=
following holds:
Moreover,
the number
N(x)
a
smooth
partition of unity
of indices i E J with
x
C- supp
q5j
is
4n.
There exist constants c,, E N and estimates
is defined on
a
open
have compact support and comprise
(pj
B n T.
regularly situated
--4
K such that the
The maps over
and
S c Rn be
Riemannian metric
+ 1. IPurthermore let B
point of DS such that families
a
N",
are
11aO1q)i(X)11
C,
0 such that for all
>
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-
<
x
C
'
11VII
C Now choose for
D,
m.
By construction of
are convex.
supp
4)j
has
n T n B. Next choose
We
now
the
(j
in 1.7.9
restrict the index
nonvanishing intersection with linear mappings Oj E GL(Rn)
Oien) is an orthogonal basis of Rn with respect to the scalar (Ojel, product L,,, (Oj edimS+l) Oj en) spans the vertical space T(p-,(x) (p (E,,,(x)) and such 1 holds. For x sufficiently close to xj the following that 110je,11 110jenli I n are then well-defined: 1, recursively fixed vectors fjI (x),
such that
*
*,
,
*
*
'
,
=
...
=
=
=
fjl(x)
=
PxOjel,
fj(,+,) (x)
=
POj e1+1
-
E k=1
fjk (X)) fik (X) fjk (X))
tj-x (PxOj e 1+1 Lx (fjk (X))
110
Control
Now
we
satisfy
will show that the
fjk (x)
well-defined for
are
n B n T and
C-
supp, j
(j
n B n T.
x
Theory they
the estimates
1 1
2n
To this end
>
-
< 1 +
P,,v for
norm
E supp
X
'
2n
estimate the
we
help of (3.3.11)
11PXV11
11filNil
<
-
x
E supp
4)j
n B n T and
v
11T-YPV11 11X
Xj 11
I 11PXjV11 11PXjV11
E Rn
with the
(3.3.10):
and
-
11P.V
PjV11
-
11PjV11
sup
-
-
YE[Xj,xl I
>
(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
<
show with the as
help
of
[xj, x] lies in supp (Pj n B n T. This entails (3.3.13) holds for all 1 < . Then we and (3.3.11), (3.3.10) + 1 (3.3.14) that (3.3.13) holds for I
>
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-'
<
Cj.
(3-3-15)
3.3 Curvature moderate submanifolds
We will show that such
an
every fixed 1 there exists
a
ill
estimate holds for the other
polynomial
function T,
:
fj,
as
R('+')+'-'
well. --
Recall that for
R' such that
fjl N T1
=
As
(P-,Oj
fjl (x),
el,
-
-
,
fj(,-,) (X),
satisfies the estimate
fjl
curvature moderate
and
-
C,1
>
over
(3.3.13)
.
.,
with constants
induction
S,
,.
R(fji, fji)
R(fj(1-1), fj(1-1))
,R11,
'
'
'
,
independent of the j and
Nn
)
as
L is
-
I shows the existence of constants cm, E N
by
0 such that
JJafj1(x)JJ d(x, aS)'-'
sup xEsupp
<
C,1.
(3-3.16)
4)jnBnT
By the the orthogonalization scheme of Gram-Schmidt and the definition of fjl the right hand side of (3.3.9) is the R-orthogonal projection P.X! onto the horizontal space of the tubular neighborhood TR indeed. Moreover, (3.3.12) and the results proved so far entail that the component functions of P ' induce Whitney functions on T which M are tempered relative aS of class E!'. Altogether the claim follows.
a C'-submanifold, T (E, e, o) a tubular neighS, 7r: T --> S the corresponding projection and P : T - End(R) the continuous mapping which assigns to every x E T the projection onto the horizontal space of the tubular neighborhood along the vertical space. Moreover, let Q : T -- End(R be the projection valued section with Q., idR. P., for every X E T. Then the tubular neighborhood T is called curvature moderate of order m, if the following axioms
3.3.13 Definition Let S C R' be
=
borhood of
=
are
satisfied:
(CM6)
The
projection
order
(CM7)
by given by
w)
Then
case
--)
:
T
End(R)
-4
=
il the scalar
il,(,.)(T,.
product
on
E and
is curvature moderate of
a
by
R the Riemannian metric
o-l.Q,,.v,T,, o-l.Q,.w)+(P,..v,P,,.w),
L is curvature moderate of order
that S is
T is curvature U
valued section P
m.
Denote
p., (v,
In
-
on
T
xET,v,WER.
in.
M, we say that the tubular neighborhood covering of aS by differentiable charts x : tubular neighborhoods x,,(Tisnu) of x(S n U) are
submanifold of a manifold
moderate, if there
exists
R1 of M of the form that the
a
curvature moderate.
A curvature moderate tubular
neighborhood
T of S has the
following
property:
(CM8)
The function p: T
-)
R is
tempered relative DS of class C!1.
additional
112
Control
3.3.14
Proposition
moderate of order L
a
m
Let +
M
E
M0,
3, let T be
Riemannian metric
on
a
S c M be
regularly
a
Theory
submanifold which is curvature
situated open
neighborhood of S
T which is curvature moderate of order
m
+ 3.
and
Then
the maximal tubular neighborhood induced by R according to the classical tubular neighborhood theorem is curvature moderate of order m. Moreover, in case M Rn the following holds: =
(CM9)
The components of the
projection
7rg of T9
are
tempered relative aS
of class
&n.
As the statement is
PROOF: that M
Rn.
a
local
one we can assume
without loss of
generality
Under this
assumption let T be the tubular neighborhood of S with respect to the Euclidean scalar product, 7r: T -4 S the corresponding projection onto S and P : T -- End(Rn) the projection valued section according to 3.3.1. After possibly shrinking T the projection 7& : T -4 S of T4 and the mapping PR : T -- End(Rn) which assigns to every X E T the projection onto the horizontal space of TR along the =
vertical space are well-defined over T. CT and S remain regularly situated.
By (5) of Proposition valued functions P"ei
are
By Corollary
3.3.11
we can
achieve
hereby that
3.3.12 we already know that the components of the vector tempered relative aS of class C+', hence (CM6) is satisfied.
By Lemma 3.3.8 and the argument given in the proof of Proposition 3.3.7 it is that the projection 7& induces Whitney functions over T which are
immediately clear tempered relative
aS. This proves
(CM9).
Next recall
Proposition 3.3.10 and check that the components of exp-' comprise relative aS tempered functions of class C+'. Now use the already proven axiom (CM6) and the assumption that R is curvature moderate and check that the Riemannian metric L' on T given by
K(v,w)=[i,,,,(x)(T-,exp,-nR'(x)*Q x'.v)T,,exp-'(x).Qx i.w)+(Pg.V,P '.w), 7rg
has to be curvature moderate of order
3.4
Geometric
m.
X
This -finishes the
implications
of the
xET,
proof.
Whitney
condi-
tions In this section
will introduce
some geometric results about tubular neighborhoods Whitney conditions. Hereby we will often need a pair of disjoint submanifolds of the manifold M. To simplify notation (R, S) will always mean in this section such a pair of disjoint submanifolds. we
of submanifolds
3.4.1
satisfying
the
Proposition (Cf. MATHER [122, Lem 7.3]) Let TR be a tubular neighborhood pair (R, S) satisfies the Whitney condition (A), then there exists a
of R in M. If the
smooth function 5
:
R
-4
Xo such that
(7CR)isnT,5,
R
:
s n
TR5
-4
R
3.4 Geometric is
implications of the Whitney conditions Whitney (B)
submersion. If even
a
satisfied, then
is
(7tR, PR)lsnT6R
113
T5R
s n
:
-4
R
choose 5 such that
one can
R
x
is submersive. First let
PROOF:
abbreviate:
us
to show that for every
E R n
x
7rR and p
7r :=
9 there exists
By neighborhood
a
is submersive. But this is
7t1snu resp. (7t, Asnu prove the claim for the
that M
case
=
Rn, R
a
:=
pR.
U of X in M such that
local statement, hence it suffices to
R' with
=
Lemma 3.1.2 it suffices
n
+ k and that TR is
m
=
n
the standard tubular
neighborhood Tm. (A) case. Suppose the claim does not hold. Then there exists a sequence of elements ljj of snT' converging to x and a sequence of unit vectors vj of TR with Vj E (T.,j7t(T,,,S))J-. After transition to an appropriate subsequence (vj)jEN converges to a unit vector v E T.,,R and T,,,S -4 -r. By Whitney (A) the relation 1 T,R C r holds, hence v E T7r(r) C (TR)-L follows. This contradicts the fact that v We first consider the
M
is
a
unit vector of TR. Now
we come
of S n Tn M is
fj
secants
case.
Suppose there
exists
(T,,, p) IT-Ujs
such that
x
a
line
a
means
for all
i larger than
jo.
This is
dim S
-
(Tpiuns)
Suppose
transition to
>
Kj
dim S
m.
1
Define
subsequence
a
converges to
-
there exists
now
that dim Tl,,7r(Ej) <
spaces
=
Yj
7r(ljj)
m.
-
a
Of
r
a
-4
sequence
(LJj)jEN
and f like above. we
condition
a
U n S and possesses the fiber dimension
can
vector space K which
By Whitney's
neighborhood Furthermore, the kernel
R is submersive.
over
(*IJj)jEN
0 is satisfied for all vj with
Hence there exists
assumption.
x
=
1.
contradiction to the
a
such that Pisnu : S n U is well-defined ker
U C T' of M bundle E
of elements
fj
appropriate jo there exist unit vectors yj and nonvanishing projection
an
T.,,S with nonvanishing orthogonal projection onto f onto fj. As fj C (kerT,,,p).L, the relation T.,jp(vj) :A >
(Iji)iEN
kerT.,,p. After transition to an appropriate subsequence of T,,,S converges to a subspace -r and the sequence of f, which is orthogonal to Rm as well. By Whitney (B) f C T Must
in
j
sequence
0. Then the secant
=
sequence of spaces
to
hold, that
to
to Rm and
orthogonal
(.Yj)jEN the
(B)
to the
converging
C
Let
assume
SnU
Kj
converging
to
ker(T%7r)JE..,
=
such
After
that the sequence of vector
by assumption
must have dimension
the relation K + f + TR C
(B)
.
x
r
holds.
By definition the vector spaces K, f and T,,R intersect pairwise onl y in the origin, dim S dim f dim T,,R 1 hence the contradiction dim. K < dimr m follows. -
Therefore,
after
the
shrinking all'y E S
to be satisfied for
=
-
neighborhood
U of
n U. This entails
our
x
-
-
the relation dim T,,7r(Ej)
second claim.
=
M
has El
the inverse of the
preceding proposition holds as precisely by TROTMAN in his article [172] about geometric versions of Whitney-regularity. It is shown in [172] that the pair (R, S) satisfies Whitney's condition (A) at x E R if and only if for every chart 3.4.2 Remark In
a
certain
sense
well. The inverse statement has been considered
x:
U
the
--
R1 around
x
of class C' such that RnU is
more
mapped
to
an
open set of Rm c Rn
projection 7,x
:
TR
n S --
R,
x
-4
(x-1
o
7rn M
.
X) (X),
TR:=
x-'(T'), M
114
Control
is submersive. On the other hand condition
chart
x:
U
--)
Rn around
(7,', px) : TRn
s
x
R
-4
x
of class IS' the
R'O,
--4
x
(B)
is
Theory
true, if and only if for every such
mapping
(7e(x), px(x)),
px(.X)
:=
(pn X) (,X), .
M
is submersive.
Interestingly enough it does not suffice in either cases to consider only Finally, the result by TROTMAN implies immediately that the conditions (A) and (B) are both C'-invariant. Independent proofs of the result of TROTMAN have been given by HAJTO [76] and charts of class V.
PERKAL 3.4.3
[142].
Corollary (BEKKA [8])
Bekka's condition PROOF:
(C)
If the
is satisfied at
pair (R, S) satisfies Whitney (B)
at
x C-
R,
then
x.
Without loss of
Rn. Moreover, let 7T and generality we can suppose M proof of the proposition. By the argument given in the proposition there neighborhood U of x in R" such that (7t, P) isnu : S n U i R x R is submersive. =
p like in the
exists
a
---
Now let
(IJk)kEq be a sequence in S n U converging to x such that the sequence (ker T-Yk (Pisnu)) kEN converges to a subspace A c Rn. After transition to a subsequence one can achieve that the sequence (fk)kEN of connecting lines fk -*:::::::Yk7r(ijk) converges to line f perpendicular to TR, and that (TlAkEri converges to a subspace 'r c Rn. By Whitney (B) and (A) T,,R ED f c r holds. On the other hand A (D t T is true as well. The second sum hereby follows from Whitney (B) and the fact that according to the proof of the proposition the sum of the projection Of Ek to T,,,S and the subspace kerT.Yk(Pisnu) is equal to T,,,S. But as the line f is perpendicular to both subspaces =
T,,R and A, the relation T.,R
C A must
hold.
Thisproves
the claim.
El
Corollary (BEKKA [8]) If the pair (R, S) satisfies the conditions (A)+(5) R, then there exists a neighborhood U of x such that the mappings
3.4.4 x
c
(7rR)isnu:SnU---- R are
submersive.
PROOF: sive.
By
By (5)
Moreover, (R, S)
3.4.1
one can
we
already
and
(PR)isnu:SnU-)R
then satisfies Bekka's condition
(C)
know that there exists U such that
achieve after
at
shrinking U that ker(TJpls)
at
X.
(70isnu
is submer-
has codimension I in TS
for every y E U n S, as Ps,.y(-y 7r(-y)) is perpendicular to ker(TJpls) T.,S n ker(Tp). But this implies that p1s is submersive. For the proof of (C) let ('Uk)kEN be a sequence of points of S n U converging to x such that the sequence of kernels converges to a subspace r C R. After transition to a subsequence the sequence of vector =
-
spaces span IPSV, (1J-k
T, R T-,R
C TED f
=
c r must
7T(IJk))l converges to a line f C R. By Whitney (A) we have liMk,,,. Tj, S. On the other hand f is perpendicular to r and T'R, hence -
be true. This proves the claim.
3.4.5 Lemma Let
R, S
C R'
with R c
aS, let 7rR be the projection and PR the neighborhood of R, and let finally Ts be the Denote by P': TR -4 End(W) the projection
tubular function of the Euclidean tubular
Euclidean tubular
neighborhood of S.
El
3.4 Geometric
implications of the Whitney
conditions
valued section onto the kernel bundle of T7rR, and onto the horizontal bundle of Ts. we
understand the
id]Rn
(1)
-
(2)
(3)
e >
& >
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 <
o
1PIT6
=
S C r:
In
particular this implies the following
(of class
and
PJTSS5CE
two relations to hold:
ftRI o(T5CE)) PRI p(T5CE)-
7rRj p(VCE) S
S
PRjW(T5Cr) S
of vector bundles
morphism
min(E, Z)
S
_
As the dimension of the fiber of E is as
well and
comprises
a
equal to codimMS, the pair (iP-1, 6) is well-defined morphism of tubular neighborhoods from T to T. Therefore
say in this situation that T and
we
f
are
isomorphic,
and denote it
by
the
following
symbols: T
By
a
T
T
or
T
briefly
-
C" - isotopy from M to N
one understands a 0'1 -homotopy H : M x [0, 11 -- N mappings Ht : M -4 N, t E [0, 11 are embeddings of class (!'. M be a submanifold, Z C aS be locally closed and T an open neighbor-
such that all Let S c
hood of S such that S is closed in T. leaves S
If h
:
T
-4
M denotes
an
embedding
which
invariant, that means if NS ids, then we say that h is tempered relative Z of class C', if the following holds: there exists a covering of aS by sets V open in M together with C'-charts x : U -4 RI of M such that V n T C h:-1 (U) and such that the mappings V n T D x 1--4 x(h(x)) comprise functions which are tempered relative Z n V of class C!'. If H : T x [0, 11 -- M is a V" - isotopy leaving S invariant, that =
Ht(x)
x holds for all x E S, then we say that H is tempered relative Z of embedding Ht : T --) M is tempered relative Z of class E!1. Given an embedding h : T -- M the support of h is the closure of the set of all points x E M with h(x) :A x. Analogously we define the support of an isotopy H : M x [0, 1] --) M as the closure of the set of all points x E M with H (x, t) :A x for some t E [0, 11. If finally f : M -i N is a e-'-mapping, then a C'-mapping h: M -- M resp. a e,'Tlf resp. f Ht f homotopy H : M x [0, 11 ---) M is called compatible with f if f h holds for all t E [0, 11
means
class
if
C',
=
if every
-
=
,
=
-
-
Now
we
have all
ingredients
needed for the two main theorems about tubular
of tubular
neighborhoods
neighborhoods. 3.5.2
Uniqueness
manifold of M of class C,+2 and f
f1s
S
:
M
N is submersive. Further let
-)
N
a
Let
M E
N",
S C M be
a
sub-
em+2 -mapping such that the restriction
To and T, be two tubular neighborhoods of S compatible with f and let * : Tolu -4 T, ju be an isomorphism of tubular neighborhoods of class E!' over the open subset U C S. Finally let A, Z C S with A C U be two relatively S closed subsets and V an open neighborhood of Z in M. Then the following statements hold: :
--
in M of class E!' which
are
3.5 Existence and
(TU1)
uniqueness theorems
There exists
C'-isotopy
an
H
119
neighborhood T C M [0, 11 -- M which leaves
of S with S closed in T and
open
T
a
S
invariant, is compatible with f and has support in V such that the tubular neighborhoods W (TolAUZ) and TIJAUZ are isomorphic. Hereby h is the embedding Hi. The isomorphism * : h,, (TolAUZ) --+ T1JAUZ can be constructed such that IPJA JA is satisfied.
(d
(TU2)
MATHER
Under the
:
x
[122, Prop. 6.11)
assumption that V is
a regularly neighborhoods Ti, i
S and the two tubular order H PROOF:
(TU1)
and
and
m
finally
0, 1
of Z, that
neighborhood
are
curvature moderate of
that f is curvature moderate of order
be constructed such that H is
can
situated =
tempered relative
m
+ 1 the
isotopy
aS of class C'.
We will prove the claim in two steps. In the first one we show the properties R". In (TU2) for the local case that means under the assumption M =
the second step
we
1. STEP Let M
will reduce the
general
to the local
case
one.
Rn and S C Rn be
Further let 7ri, i a submanifold. 0, 1 be projection of Ti and 7t the projection of the tubular neighborhood T (E, E, 0) of S induced by the Euclidean scalar product. Denote by Pi and P the projection valued section associated to the tubular neighborhoods Ti and T. Vector bundle isomorphisms from Eo resp. El to E are now given by =
=
the
=
&i,.,: Ei,.,, Let
E0
-4
E.,,,
--
El be the isomorphism. of
composition of &0 and
&11.
-i
v
XE S.
P.,.T pi.v,
vector bundles of class
C'n-1 constructed by
Over U the isomorphism. & coincides
by assumption with neighborhood U' C U of A appropriately we can achieve that & is even a C'-vector bundle isomorphism. By the polar decomposition A.2.1 there exists for every x E S a unique positive definite operator E1,x --- El, such that Z ' Eo,, -- Ei,x is unitary, hence ip : E0 -- El is an isomorphism, of vector bundles with scalar product. Furthermore the bundle map (1 t)& + tip E0 -4 El is well-defined for t E [0, 1] and comprises an isomorphism of vector bundles which coincides over U with 1 . But this implies the existence of an open neighborhood T of S (in which S is closed) such that for every t E [0, 11 *: &1U
=
*. By changing & outside
a
-
-
Gt is well-defined and
:
T
-4
R",
comprises
x
is
an
where
now
-
=
and
ids
compatible with f,
we
f (Gt (x))
possibly shrinking
morphically
onto
-
t)& + t*) yo-') (Y-) -
onto its
Gt1u,
=
U' is chosen to be of the form U'
appropriate C'-function with T'0
assumed to be
After
(yi ((I
diffeomorphism
a
Gt1s holds,
-i
an
T
open
we can
C T.
:
T
idu, 1
:=
7to- (A)
n
T50 and 5
:
S
-4
As the tubular neighborhoods Ti
=
f (x),
X
E
T.
W0 are
(3.5.1)
achieve that Go maps the
--
then
have
neighborhood i
Gt,o
image. Obviously
R",
of S
x F-
.
Then the
Gt(GO'(x))
neighborhood mapping
T diffeo-
120
Control Theory
is well-defined and
diffeomorphism
a
onto its
image. By
construction
Gt,o
has the
following shape:
Gt,o(x) As
=
( pi ((1 -
t)id + t -')
-
-
yi
1) (x),
X
G
T.
-, is positive definite, the derivative DGt,o(x) is for every x E S diagonalizable only positive eigenvalues. For later purposes we will keep this fact in mind. Next consider the projection valued C111+1-mapping Pf : 'k --> End(R) which we
with
obtain
according V,w
defines
a
=
Pf (D (Pfw).v)
connection V
on
Qf
3.3.5 out of f. Set
Example
to
+
=
X
Qf (D (Qfw).v),
T. Let exp be the
idR.
-
X
x
E
'k.
X" (i)
W E
V,
Pf,
corresponding exponential
Then
(3.5.2) function.
By
definition of V the map exp is of class (!' and
f (exp tv)
==
f (x)
(3-5-3)
[0, 11 and sufficiently small V E ker Tf PfRn, x E T. possibly shrinking T we can suppose that exp,-'(,y) is defined for all x E S and -y E T n f-(f (x)) and that the geodesic -y (t) exp(t expx 1 (ij)) lies in 'k. Now choose a smooth function K : rk -- [0, 11, with the'following properties. For all X of a sufficiently small open neighborhood of S in T let K(X) K(7rl(x)). The support Of K is contained in V, and over an open neighborhood V' c V of Z the equality KJV/ I must hold for all t cz
=
X
After
=
=
=
holds. Then define:
Ft,o(x) From
now on we
=
exp
(K(x) expX'(Gt,o(x)),
have to consider both
cases
E
X
of the claim
T,
t E
[0, 1].
(3-5.4)
separately.
(TUI): In this case we show first that after appropriately shrinking T and mappings Ft,o : rk -4 R' comprise diffeomorphisms onto their image. To this end it suffices by Lemma 3.1.7 to prove that for every x E S the derivatives DFt,o(x) are ids the derivative DFt,o(x) acts identically on tangent vectors bijective. As Ft,ols V E T-,S. If on the other hand v E ker T,,7r,, then ad
the
=
D Ft,o (x).v
As mentioned
=
(I
-
K
(X)) V + K (x) D G t,o (x).v
above, the operator DGt,o(x) has only positive eigenvalues, hence the shrinking T finally
first derivative is invertible. After further H
:
T
[0, 11
x
--)
M, (x, t)
--)
(Ft,o Go) (x)
(3-5.6)
-
is well-defined and
comprises a e,'-isotopy from T to M which is compatible with f. Hereby, compatibility of H with f follows from (3.5.1) and (3.5.3). The support Of K lies in V, hence supp H C V has to be true. AS K can be chosen such that 1, the relation Ht(x) Gt(x) is true for x E V' after possible shrinking the KIVi of V' hence Z, neighborhood the
=
=
1
pl
-
h(x)
1 =
pl
-
G, (x)
=
*
-
(pol (x),
X
E
V.
3.5
and
Existepce
On the other
hand,
tubular the
case
thus obtain that
we
=
=
x over
the set U' defined
above, hence
(p-11-h(x)=q)-11(x)=*-yo1(x),
and
neighborhoods. By M
Gt(x)
have
we
Ht(x)=x Altogether,
121
uniqueness theorems
*JAUZ
:
construction
h.,(To)[AUZ
-4
T1JAUZ
X
is
an
E
U'.
isomorphism
which is the last part of
*[A
(TU1)
of for
R.
(TU2): We consider the just constructed isotopy H and will show that the mappings Gt,,, K and exp involved in the construction are tempered relative aS of class C"'. By the representation (3.5.6) of the isotopy H the claim that H is tempered ad
relative aS of class C' then follows The Christoffel
symbols
rilj
=
immediately.
(ek, V,,ej)
because Pf is curvature moderate of order
m
are
tempered
relative aS of class
ell,
+ I and because the vector fields
V, ej shrinking
by Equation (3.5.2). By Proposition 3.3.10 one can achieve after exp-1 are tempered relative aS of class C'. Furthermore it is easy to check using Lemma 1.7.10 that the function K can be constructed such that it is tempered relative aS of class C". Note that in all of these constructions T always remains a regularly situated neighborhood of S. By H (x, t) Ft,o (Go (x)) and (3.5.4) it therefore remains to show for the proof of the claim that the components of the embeddings Go : T -- M and Gt,o : T -4 T To achieve this let us represent the vector are tempered relative aS of class Cm. bundle isomorphisms & : Eo --i E, and : El -El in matrix form with the help of local orthonormal frames. From this representation one can immediately read off the temperedness of the matrix components, hence of Go and Gt,o. Without loss of 0, 1, with a subvector generality we can identify each one of the vector bundles Ei, i bundle of T1sM normal to TS; hereby the identification is given like in 3.1.8 (1) by the restricted tangential map TIE,yi : Ei -4 TM. Next we fix two Riemannian metrics are
defined
T that the functions exp and
=
=
over
T:
(v, w)
=
(T, (p i
ii
T,,q)i
l.Qi,,,.w) + (Pi,.v, Pi,_.w),
x
E
T,
V,
E
w
R,
(3.5.7) where i1i denotes the scalar
tionally
let
us
product
on
Ei and Qi,, the projection idp,.
-
Addi-
Pi,-,.
set gF,
point of aS a ball B with sufficiently small radius < 1. AcProposition 3.3.12 the three Riemannian metrics Li, i E JO, 1, El then generate locally finite countable families (4)j)jEi and (fj'1)jEi,1<1<,, consisting of functions resp. sections over T such that (4)j), (f, ,) satisfy 3.3.12 (1) to (5). In particular every n-tuple (f, ,, f, n) thus comprises for every fixed j a Li-orthogonal frame over the set ( j n B n T. We will construct with respect to this orthogonal frame matrix valued mappings ajl : T -- End(Rn), i 0, 1, which represent the vector bundle isomorphisms &i over supp (pj n B n T. More precisely we set for X E supp 4)j n B n T Now choose for any
cording
to
-
-
-
,
=
aj ,kl N
( fEj(k+dimS)(X))fJ (1+dimS)(X) I Ifj( k+dim S) N I I 9F I I fi(I+dim i S) N I I 'i
'
and extend the thus defined matrix valued functions tubular
k,1=1,...
=
neighborhood
T.
By
3.3.12
(4)
a,
n-dimS,
to On-functions defined
on
and because the Riemannian metrics VD,
the Ll
122
Control
and ttE
and
curvature moderate of order in
are
> 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,
<
(3-5.8)
I +
a.
for
E supp
x
d, dm, D, Dm
the
Wi
:
T
--
End (R) with the property that
n B n T and such that after
q5j
estimates
following
< D
are
1 +
Wi (x)
possibly enlarging
the
true:
x
d(x, aS)d
I
I +
: Dm
d(x, aS)d-
write the vector bundle
and -dj and where
v
E
EO,-,
and
x
jej
n B n
T,
)
< in.
I
isomorphism, &
in the
following
form
using
the
(V) fj0 (k+dim S) N) fj (I+dim S) N 1 IIfj(k+dimS)(X)IIIJO 11 fIj(I+dimS)(X)II91
PO
0
Y,
)j (X)
(pj
E S:
n-dim S
&XV
E supp
(3.5.9)
1
jjam-d (x)II
C
< Tn.
1
jj-d,(x)jj
<
D
we can
MI
d(x, aS)d-
construct Cm-functions
d(x, aS)d
Now
G N
T,
1
jja'a, (x)II hence
d, d,
1
0,
1
11 a3 (x) 11
<
D
is inverse to
Theory
ajI, 1m N a , mk N
l,m,k=O
L
(3.5.10) This
representation, Proposition 3.3.12 and the estimates (3-5.8) and (3.5.9) entail n the functions T E) x F-4 &no(x)QO,7ro(x)ek E Rn are tempered 1, relative aS of class &n. Recall the fact (see 3.1.8 (3)) that the map Yj is given by the exponential function of the Levi-Civita connection of Lj. Using Proposition 3.3.10 it that for k
=
-
-
-
,
then follows that
G,:T--4R7, is
x -4yj-&nO(-,,)-yO1(x)
tempered relative aS. Next
we
have to prove the T
temperedness relative aS
--4
W,
x
-4
of the functions
7, (.,,) Q 1,7,1 (.,,) ek
-
By the polar decomposition A.2.1 the relation analyticity of the square root
X
E
S,
holds hence
by
the
1
aI
(Z:,,, () Q 1,7q () ek) (X) 11 :5i
IIaI(&7,1(-)Q1,n (-)ek)(X)jj ,
A similar estimate follows for every
higher partial
k,
1
=
1,
-
derivative.
n.
More
(3.5.11) precisely
(
7, (.) Q 1,7, (.) ek) (X) is given by a sum of quotients which contain in the nominator higher partial derivatives of (.) Q ,,, (.) (multiplied by combinatorial factors) and in the denominator powers of I (x) 11, but both only up to highest order I OCI. By (3.5.8), a ly-
(3.5.9) and D'
and the >
representation (3.5.10) there exist
on
the other hand constants d' E N
0 with
d(x, aS)d' D'
<
jj -,Jj
<
D'
1 +
x
d(x, aS)d'
E T.
3.5 Existence and
uniqueness theorems
Together
with 3.5.11 and the
functions
on
DS.
Exactly
T
Gt,o (x) tempered
is
temperedness
given by 4.,(.)Q,,,,,(.)ek
like for G,
one
=
123
of
this
-1.)Q1,,,1(.)ek (
and
inequality entails that tempered relative
have to be
7EJ
thus concludes that
( pi ((I -
-
t)id + t -')
-
wi
relative aS of class C'. This finishes the of the
1) (x),
E
x
proof of (TU2)
will be carried out
by
T, in the local
case.
first
embedding applying the just proven case. More precisely one considers the tubular neighborhood Tm (Em, em, pm) of M in R" induced by the Euclidean scalar product. Then the f(x) mapping f can be extended canonically to a map on Tm such that f (7rm (x)) for all x E Tm. Furthermore we then switch from the vector bundles Ei, i 0, 1 to Rn U := EieEmls and define, after possibly changing the ei, embeddings (p : T'i Sc 2.
STEP The
proof
the manifold M
as
a
general
case
closed submanifold in Euclidean space R' and then
=
=
1
2
neighborhoods T (V, ei, Y induces a vector bundle isomorphism of S in Rn. Moreover, E01U --i Ellu by (v,w) i--) ( (v),w), and ' : Tolu -- T11U becomes an isomorphism. of tubular neighborhoods. Collecting all these data one recognizes that by To, T1 and f the assumptions for the local case in the 1. Step are satisfied. Moreover, this holds for the curvature moderate case (TU2) as well. Firstly we thus obtain a unitary vector bundle isomorphism *' : E0 -4 El, where it is clear by construction that the restriction of *' to EmIs is equal to the identity map, and that ip' maps the bundle E0 to El. Secondly we obtain an isotopy H' : T' x [0, 11 -4 Rn, the components of which are defined by (3.5.4) and where T' C Tm is an appropriate tubular neighborhood of S in R. Obviously, the function K contained in (3.5.4) can be chosen such that K(X) for all x E 'k', where i' C Tm is a further tubular neighborhood of K(7rM(X)) S in R. The exponential function appearing in (3.5.4) as well will be defined with respect to the following connection: by
p
(v, w)
=
w
+ yi (v). Then
we
obtain two tubular
=
=
(V,w)(x)=
Pmf(D(Pfuw).v)+Qfu(D(Qfjw).v)+Qm,,j(D(Qm,,jw).v), 'Y
V
V
V,
W
E
Xu'+' (i'),
X
E
T,
ii ='gm (X).
Hereby QM means the projection onto the vertical bundle of the tubular neighborhood TM, Pf for X E M means the Euclidean projection from T.,,M onto kerTf and Q,f Pf This definition of V guarantees that'exp v lies in M the endomorphism idT,m for v E T.,.M and that exp.-,,'(Ij) E T_,,M holds for x, -y E M sufficiently close. In the curvature moderate case the Christoffel symbols of V are curvature moderate. Now one concludes by the definition of the tubular neighborhoods T and the shown H'(7tm (x, t)) is true for properties Of &, K and H' that the relation 7rm (H'(x, t)) all x E T' and t E [0, 11. At this point we can fix the "unslashed" objects: the vector bundle isomorphism : E0 -4 E, is given by * (v) 7tE (*'(V, 0)), V E Eo, where canonical The the is : ED El EmIs El projection. isotopy H : T x [0, 11 --) M 7rE, El is obtained by T := T' n M and H(x, t) := H(x, t) for x E T, t E [0, 11. By construction ip and H satisfy (TU1). All constructions in the 2. Step do not influence a possible curvature moderate behavior, hence in the curvature moderate case (TU2) holds as well. This finishes the proof. D X
-
X
.
=
=
=
124
Control
Theory
Existence of tubular
3.5.3
neighborhoods Let M E N", S be a submanifold N a em+2 -mapping submersive over S 1 irther let C U be closed relative S, and To a tubular neighborhood flu. Under these prerequisites the following statements
of M of class ISM+2 and f: M
--
-
U C S be open relative
S,
of U in M
with
compatible
A
hold:
(TU3)
There exists
tubular
a
TolA- If7fo-'(W)
then T and the
S, EJA
(TU4)
n S
=
F-01A
If To is
:--
U' for
a
regularly situated neighborhood
a
moderate of order
m
+ 1 and if f is
U'
of order m+ 1, then the tubular neighborhood T moderate of order ad
PROOF:
(TU3):
C'+'
on
M.
After
be chosen such that
can
strongly
can
To
are
curvature
curvature moderate
be chosen to be curvature
For the first part of the claim choose
T
restricting
mapping Pf
-
m.
an open neighborhood T of arbitrary Riemannian metric 0 of class appropriately one can achieve that the projection
S in M such that S is closed in T and choose
valued
TIA
U of A which is closed in
c
of A, if both S and
submersion
a
with f such that
compatible
neighborhood
isomorphism (*, 6) : TIA -4 T01A MATHER [122, Prop. 6.2])
(Cf
5-
T of S
neighborhood
=
T
:
End(TM)
--
an
onto the kernel bundle of Tf is of class
ISM+';
hereby projection Pf is assumed to be selfadi oint with respect to 0, By the tubular neighborhood T' (E0, 0, (p') of S induced by 0 one obtains a further projection the
X
=
IS'+'-mapping
valued
mapping assigns of V.
P'
to every
T
:
-4
End(TM) (after possibly shrinking
E T the
X
The last
projection orthogonal projection onto
valued
0-orthogonal projection section is given by Pf,l
the kernel bundle of Pf
Pf,0 is well-defined and of class IS+'. If one
now
Qf,o X
is
A__limpo X
A
:
T
in
complementary
there exists for every
x
T,,M
E T
a
to im
X
=
X
second Riemannian metric
p,x (v,
w)
L
=
as
sets
T
:
imQE),
End(TM),
--)
=
fls
S
:
the 0-
-
X
N is submersive.
--
Q1
where
X
as
this
PO. After shrinking T further Qf,0 Pf PXfI" then the image
positive definite mapping Ax
idlimpo and Ax(imQf,O) End(TM) can be chosen I
P',
again);
-
X
of
T
onto the horizontal space
By
Hence
such that
P0. Obviously, idT,, m following one obtains a
=
-
X
C'-section.
End(TM)
E
X
the
T:
on
0 (Axv,
A.,w)
v,
W E
T-,M,
x
T.
E
With respect to L the spaces im P0 and im Qf,' are orthogonal to each other. By N X % S we now understand the L-orthogonal bundle of TS in TlsM, by exp, the exponential function of the Levi-Civita connection of R, and by Tmax the neighborhood of the ScN zero
V
E
section of N
N,
lies in
n T" ScN ,
as
x
E
Qf,O)Ty(t)M. ,Y(t
defined in Section 3.2. Hence
S the tangent vector
neighborhood
by definition
to the
curve
-y (t)
of R for every vector =
exp (tv), t E
[0, 11
Therefore f (exp (tv))
holds for all t E
(t)
=
f (x),
of S in M which is
the restriction of explTmSCN
.
Let
Nx
n
Tmax, ScN
X
E
S,
triple T (N, eml, p) comprises a tubular compatible with f, where y is defined like in 3.2 as V C t := exp (T" ) be an open neighborhood of A ScN
[0, 11 Consequently .
v E
the
=
125
3.6 7bbes and control data
I such that V lies in 7to- (u) ni. By the uniqueness theorem for tubular neighborhoods there exists after possibly shrinking 't an embedding h : 'k -- M with support in V
such that claim in Now
his
idIs
=
(hJ)JA
and
-
ToIA.
Hence T:=
KJ
satisfies the first part of the
(TU3). to the second
we come
part. Under the assumption that U'
neighborhood of A in S satisfying the claim 7q1(u,) neighborhoods V, W and W' of A in S with
n s
C U is a
closed
U' choose further closed
=
A c V'c V c U" c U'c W'c W c W" c Wc U.
According to the first there exists a tubular neighborhood f the assumption on and an isomorphism f1w, -(*,g) Tolw,. By
=
(f, Z, Cp)
U'
one
of S in M
finds
an
open
1
neighborhood 0 of S\U'in M with 7to-'(V) nO 0; for instance set 0 := M\7to- (W). By assumption on the neighborhoods V, U', W and W' it is possible to construct a 'C'n-mapping e : S -4 R"' such that =
F-JU < EO, Now
we
F-JA
=
F-01A)
F-,W\V:5
8,w\v,
F-ls\v:5 Zls\v
T' --- M as the mapping which Sct and which coincides over (S \ V) n
define
p
:
over
i'\w
and
the set
s
C
n.-1 (W)
0.
n TI
s ct
is
with the restriction of
0. equal to po 7tt-1 T'ct s By construction one checks easily that this y is well-defined and injective. Hence the (f, F,, (p) together with the isomorphism (*IA, F-JA) fulfills the second part triple T =
(TU3). (TU4): As soon as the assumptions in (TU4) are given, the projection valued sections PO (as S, 0 are curvature moderate) and P1,1 (as f is strongly curvature moderate) have to be curvature moderate of order m + 1; in particular the common of the claim ad
domain T of P' and Pf,' is
a regularly situated neighborhood of S. Moreover, as 0 is M, the Riemannian metric i has to be curvature moderate of order m + 1. Consequently the tubular neighborhood T induced by i is curvature moderate of order m. By the fact that To is a regularly situated neighborhood of A and by (TU2) in the uniqueness theorem one can achieve that the above chosen hJ is embedding h : T -- M is tempered relative aS of class C'. Hence T a
Riemannian metric
on
=
curvature moderate of order m, thus satisfies in the curvature moderate
case
(TU4).
This proves the claim.
Tubes and control data
3.6
be a locally compact stratified compatible with 8 we understand
(X, 8)
Let
3.6.1
tube of S
By
a
the
following conditions:
(TBI) Ts
is
an
ns
:
Ts
triple Ts
neighborhood of S in X such that TS'R := Ts n R :A 0 implies R > 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
<
. ' jwn
n
means
one can
such that
(Ts)sEs
(2)
have the
Br(x) shrink
holds.
following
properties:
(3)'
For every KR
:
With the
R
-4
pair of
strata R > 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
<
5,1u.
for all R with
5RIU,,
<
set
us
U,, & U,
E
X
(3)'.
Theory
n
U,, :A 0 1
and
5
=
1:
d,,
p,,.
nEN
Note that dn
(5R)JUn to (3)'
for all
> 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
<
<
Un :A 0. According this implies for all X E T5 n T KR that for every n E N with X E Un the S R relation x E Th n T KR is true. Consequently7rR(x) E Ts, which by (CT2) entails that S R Now shrink Ts to T5 and for all R > S ps(7rR(x)) ps(x) < 6(x), hence 7rR(X) E T5. S S shrink TRto T KR This gives the induction step for S= Sk+1 Since for R every stratum S m
n
n
=
.
.
of X
only finitely
many strata R > 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
Then there exists Then 7tR (x) E
We postpone the
proof
as
F-
:
R>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-
<
(4). Suppose
es.
TS
E
K-y
n TR
C
TR.
of the
U.
the
proves the claim.
p. 143. This will not lead to any
property a
-5R
Ts, ifij
Thus, by definition
3.9,
use
C
Kj n TR C
(6)
of proper control data.
stratum S the set W
1
=
TsE- n 7cs- (U)
compact path connected neighborhood of
smooth function with
compact by
R" with T RKR
until Section we
and7tR' (uj)
Ts
hence 7rR (x)
will show that for every
Finally path connected, where --
E
until then
is
S
R
:
5R
7rR(x)
=5R
O,iflj
=
holds for all
U,R(.,,)
circular arguments
nTS
smooth KR
a
relation has to be true
we
-5
K
7tR-'(Uj)
such that
This
X
in S and
immediately gives (5).
there exists
a
point
First
ij E W which cannot
be connected with
x by a continuous path. Let W. be the set of all points z E W be connected with ij by a continuous path in W. Then W,, is closed in W, hence compact and by assumption on ij has empty intersection with U. Therefore
which
can
I ps (z) I
W-y I> 0, hence there exists a converging sequence (Zj)jEN Of d. Let z E W., be theliMit Of (Zj)jErj and R > S limj,,,,, ps(zj) the stratum of z. As (7rS,R, PS,R) : Ts,R --4 S x 10, oo [ is submersive and R locally path connected, there exists a point z' E R n W., with p(z) < d and7rs(z) 7-rs(z). But d
inf
=
z
E
elements of W., with
=
=
this contradicts the
minimality
The last part of the
proof immediately entails the following.
3.6.8
Up
d, hence
W is
Corollary Every controllable stratified
to
ones
of
connected.
space is
locally path
connected.
do not know whether there exist nontrivial control data besides the by collars of manifolds with boundary. The following theorem an
now we
induced
answer
path
to this
gives
question.
131
3.6 Tibes and control data
submersion f
which
X
Tn
curvature moderate
strongly
(dim Etx (S))2- (dim S)' + 1,
+
(Ts)SES
manifold M there exist normal control data
a
on
X
7.1])
MATHER [122, Prop. M' and X be Euclidean embeddable.
E
moderate and f m
M to
-4
compatible with f. (d
are
Let
:
space X and every smooth stratified
Whitney stratified
3.6.9 Theorem For every
If X is
strongly
curvature
every stratum S both of order
over
then the normal control data
(TS) SE8
be chosen
can
to be curvature moderate of order Tn.
In the first part
PROOF:
subspace
will consider the
we
that X is
case
of R' and then extend it in the second part to the
case
a
of
Whitney stratified arbitrary Whitney
stratified spaces. 1. PART First
suppose that X is
we
a
Whitney stratified subspace of
Let X1 be the k-skeleton of X that
differentiable manifold N.
R'
or
of
a
the union of
means
X, and Sk the family of all strata of dimension < k. 0. Now it will be shown by induction on k that 0 and 8'
all strata of dimension < k of Moreover let
us
set
X`
=
=
compatible
for all Xk there exist control data the curvature moderate Let
Ts
=
us
suppose that for
(Ts, ?Ts, ps)
satisfied.
with f which
curvature moderate in
are
case. some
k E N
we
have
a
(Ts)SESk-1 of normal tubes (CT1), (CT2) and (CT5) are
system
in X such that the control conditions
3.4.1
By Proposition
suppose that for all R < S with
we can
R, S
E
8 k-1 the
mapping
(7tR, Pp,)IsnTR
:
S n TR
-4
R
x
R
assumptions of the curvature moderate case assume adTs, S E Sk-1 is curvature moderate of order ditionally of the submersions Tn + (dim EtX (S))2- (dim S)2; then by assumption on X every one of order moderate m + Etx curvature is (S))2 (dim S)2 (dim strongly (7rR, PR) isnTR 0. over S. Finally suppose that R and S are comparable if and only if TR n Ts is submersive.
Under the
that every
of the tubes
one
-
As any two strata of X of equal dimension k are not comparable, the following can be performed separately for every stratum of dimension k. So let S
constructions be
stratum with dim S
a
In his
proof
=
k.
of the claim JOHN MATHER constructs tubular
two steps. For every 1 < k let
Let
S,
=
U,
n S. In the first
U,
neighborhoods TS
in
be the union of all TR with R < S and dim R > 1.
step MATHER defines
a
tube T,
=
(TI, 7rj, pl)
the tubes TR of strata R < S
1.
of S, possibly
in X
have
Hereby, only in the boundary of finitely many strata, shrinked tubes are the corresponding only finitely many times. In a second step the tube to extended then a will be tube To (Ts, 7rs, ps) of S in X. In the following we
by
a
decreasing
induction
been shrunk. But
as
on
every stratum lies
original argument of MATHER by the curvature moderate case and show that under the corresponding assumptions all constructions can be performed supplement
the
to be curvature moderate.
1. STEP For I
T1+1
=
k
we
have
0
Sk
so
has been constructed and that the
If R <
S, dim R
(CT)I+l
>
1 +
1,
X
E
T1+1
n
in this
following
TR and
o
=
=
finished. Now suppose that
commutation relations
7rl+l(x)
PR'7 1+1(X) 7T1+1 (X)
7IR
case we are
E
TR,
PR(X)) 7rR (X)
-
then
are
satisfied:
132
Control
After
possibly shrinking T1+1 Q < S of dimension
stratum
the curvature moderate
moderate of order
m
one can assume
>
1 such that
(dim Etx(S))'
+
X
that for every x E T1+1 there exists a E TQ and 7r,+,(X) E TQ. Moreover, in
that every
case we assume
Theory
of the tubes
one
T1+1
is curvature
(dim(S))2 +2(1+1).
-
As for any two different 1-dimensional strata R, R' < S the relation TR n TRI 0 is true, it suffices to construct the tube T, seperately over each one of the sets TR n S. In other words of which to
want to construct
we
is
TRnS1+1
a
tubular
isomorphic to the
commutation relations
are
satisfied: for every
x
R
where 7IR S of
7-rSR.
7rTR. After
:=
satisfied for all
already
T SRof
neighborhood
PR
'
7rS (X)
7rR
o
7jR (X) S
Rn E TS
TR with
=
commutation relations
are
where 711+1 is used instead
,
to
U
TR \
following
TR
E
7rR(X))
shrinking TR appropriately these E T1+1 n TR with 7r1+1 (x) E TR TR
7rRs(x)
PR (X))
X
To realize this shrink the tube
TR n S the restriction
restriction of T1+1 and such that the
7rQ-1 (TQ \ TR))
JQJ R
and denote the thus obtained tube
Q
R be
<
stratum with dim
again by TR.
Q < Q
a
7ri+l(x) E TR n TQ, hence R 7rQ(x) E TR, hence the claimed PR
*
7Q+1 (X)
7CR
o
7T1+1 (X)
Next recall that
=
> 1
S.
<
For the chosen
such that
the
By
PR
'
7rQ
"
7C1+1 (X)
PR
7IR
o
7tQ
o
7C1+1 (X)
7rR
:
T1+1 let afterwards
711+1(x)
E
TQ.
just performed shrinking
commutation relations
by assumption (71R) PR)IsnTR
E
x
and
TQ
E
x
'
TCQ (X)
snTR
---
have
true:
are
7tQ (X)
o
Then
we
i
PR(X)) 7 R (X)
=
R
x
R is submersive.
On
the other hand the control conditions is
(CTI) mean nothing else than that the tube TSR compatible with the mapping (PR, 7rR) : sl+i n TR --4 R x R. Now we shrink each one of the tubular neighborhoods TQ for Q < S, dim Q > I to
tubular
neighborhood T'Q
such that the closure of T' n S lies in
S1+1 and such that Q T1+1 is a regularly situated neighborhood of T'Q n S. If one defines S1+1 analogously to S1+1, then the closure of S1+, lies in SJ+1, hence by the existence theorem for tubular neighborhoods there exists a tubular neighborhood TRwhich satisfies the control conditions (CT1+1) and which is isomorphic to T1+1 over S a
in the curvature moderate
case
I S 1+1, In the curvature moderate
case
(71R) PR)isnTiz
is
strongly
curvature moderate of
order
m
+
hence
(dim Etx (R) )2
by the
of order
m
1,
-
(dim(R) )2
existence and
+
> 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 <
_
uniqueness theorem T SRcan be chosen
(dim EtX (S))2- (dim(S) )2 there exists
(dim Etx (S) )2
m
+
(dim EtX (S)) 2
-
(dim(S))2
in
133
3.6 Tibes and control data
Namely
x
E
choose To
7ro(x)
and
TQ
f
o
small that for all
so
x
To is compatible
the tube
To there exists
E
Q
some
<
S with
But then
TQ.
E
possibly shrinking To,
after
(CTo) implies that
2. STEP
with f.
7ro (X)
f
=
7TQ
o
o
7to (X)
=
f
7tQ (X)
o
=
f (X).
By applying the existence theorem for tubular neighborhoods there exists analogously Step after appropriately shrinking To a tubular neighborhood Ts of S in X which is compatible with f and which satisfies the control conditions. After possibly shrinking Ts again we can assume that the system (TS)SESk satisfies the above induction hypothesis that means the induction step is finished. As for every fixed stratum S and sufficiently large k the tube TS will not be shrinked anymore by the transition to Sk+', we thus finally obtain control data for X with the desired properties.
to the 1.
2. PART After the claim has been shown for the
dable
we now come
to the
general
charts xj
atlas of
an singular embedding with respect to
need
open in Rni and the
the a
certain
closed in
are
xj(Kj'+,)
spaces
the
j
Oj.
Rni, j
C
(Kj)jEN
In the
altogether help of a
smooth
N which is
of X.
following
on
X
we
inductively the
Hereby
Oj
are
will construct for
we
are
compatible
in
partition of unity and after possibly
smooth and submersive functions
Oj
E
C Rni control data which
this end first construct with the
shrinking
that X is Euclidean embed-
will induce the desired control data for X. To
and which
sense
Oj
-4
compact exhaustion
a
xj(Kj'+,)
Whitney stratified
KjO+1
:
case
For the construction of control data
case.
fj
Oj
:
M such that for all
--
E N
(I a) fj
-
xj (x)
(1b) fj+l (X)
=
f (x), if
fj 7tnn,'+' (X)
=
are
by
Vs
tubular
=
I
if
proved,
consisting of tubes induced
if
-
Now the claim is
and
Kj'+,
E
x
X
Xj+1 (Kj)
E
one can
-
provide
for every chart xj control data MS)SES Kj'+,) in Rni such that they are
around xj (S n
(Vs, 7rS7, p s)
oj) S
neighborhoods (Vs, e s,
and such that the
following conditions
satisfied: E S n
(2a)
For all
(2b)
7rnj+'(T4+1 snKi)
(2c)
For all
x
=
j
PC
Now
T3s+,'K
7rnj
o
For one
x
(TIS)SES
E
o
xj +1 (x)
holds true where
T3s,K,
'7tn.,*+' j
has
one
j
n, +I
3
(2d)
E
x
j+1
the relation F-S
Kj
-
=
es
T',K, S
-
xj (x) holds.
(7rs) -1 (xi (S n Kj))
7eS,'+l (X)
7eS
-
(x) 7tnn.,'.+' j
T's
the
equality fj
7es (x)
-
properties
are
nj+l nj, where Zl mj priate restriction of (y, -
the sets S n
p3s+l (x)
If
(TO)SES S
to the 1. Part.
then
already given, projections 7['n' " one
pulls
via the canonical
(f3s'+', 11s', qoj+') S
i's
and
f (x) holds.
according (TO)sEs S
constructs control data
with the desired
=
n
=
j.
(X)
neighborhoods (Els, els, pls) hoods
for 1 >
Kj'+,
1
of
xj+,(S
n
Kj+,),
where
f's+1
to
back the tubular
to tubular
=
sisnK,*+,
neighbor-
E) R'j with
and where CcpiS comprises an approS isnK,-+, holds, identified in our notation have we canonically Hereby idR-j). F_
=
with their
j
.
7t
nj+l nj
images under xj respectively xj+,.
Control
134
Theory
Recall the arguments
given in the 1. Part. Using the uniqueness and existence neighborhoods several times one now checks that the restrictions V,'+1 conditions (2a) to can be extended to tubular neighborhoods Tc+1 S S 1snKj satisfying (2d). We will not give the details of this somewhat lengthy consideration but mention El again that it can be performed exactly like in the 1. Part. theorems for tubular
Controlled vector fields and
3.7
integrability
By a flow on a metric space X one usually understands a continuous mapping X, (x, t) 1-4 -y(x, t) -y,#) which is defined on an open subset j C X x R and which has the following properties: 3.7.1
j
-y
-
(FI)
=
For every
J,
:=
j
n
E X there exist
x
({xl
(F3)
If t E J, and
j
--
(x, t)
E
:
X and
is
equal
x
on
-
E
S
:
J,,(t)
X is
j
-4
an
X
then t +
Now let X C M be
as
a
a
t,+
:5
oo
such that
j-, follows and
-y,,.(t +
S)
the relation -y :! _ C
1
and
-y(x, t)
S). "y- for =
two flows
- (x, t)
for all
submanifold of M and V: X
-4 TM a vector field tangent to Lipschitz condition. According to the classical theorem of of solutions of an ordinary differential equation there exists
- _'(t) for X
Allowing :
X
then
such that is
0 <
local
(IF)
S,
<
holds true.
by defining being equivalent to j
-4
E
S E
x
t,-
It,-, t,+
ordered set
uniqueness and existence a unique maximal flow -y: j
x
=
<
-oo
J.
X and which satisfies
from V
t,-,, t:,t with
to the interval
.(0)
For every
-y
R)
E X the relation -y,,
(F2)
The set of flows
x
--
an
V(-Y,,.(t))
in detail. For the
E
X and t E j-,
-y-, (0)
and
=
stratified subset of the manifold
x.
M, while
with V (x) c
a
is satisfied. In other words
locally integrable. JOHN case
of
a
so-called controlled vector
corresponding
=
x
one requires TxS for every stratum S and all mapping ask the question under which conditions on X and V a flow -Y exists
TM that it is
one can
(IF)
arbitrary
X such that for all
theorem in
we
want to know under which conditions V
MATHER has treated this
question in his articles [122, 1231 Whitney stratified subset X C M he could show that every field on X has a maximal integral flow. We will prove the this section, but before let us introduce some necessary
notation. 3.7.2 Definition
field V
:
X
such that
--
for.every pair
(CT8) If
(MATHER [122, 9])
TX is called
Let X be
weakly controlled, if
controlled stratified space. A vector
a
there exist control data
of strata S < R
T7 S,R' VITs,R
:--'
V' 7rS,R-
additionally
(CT9)
Tps,R VITs,R -
=
0
(Ts)SE'S
of X
3.7 Controlled vector fields and is
satisfied, then V is called controlled A weakly controlled vector field V
Ts
:
(with respect
stratum S is called radial
135
integrability
to
TX
-)
on a
tubular
if V is controlled
S),
neighborhood of the Ts \ S and if the
over
relations
VIS
(11131) (11132)
Tps VITs\S
=
Tps VITs\S
=
-
(RB3)
--i at
-
a
0
(11131) only
hold true. If besides
is true with
=
-h-1 at
mapping h: TS -i R", then V is called conformally additionally a C+'-structure and if the control structure
radial.
stratified
If X carries
on
X is
curvature moderate of order m, then we call a (weakly) controlled vector field V curvature moderate of order m, if there exist a covering of X by singular charts x :
U
(Ts)SE8 curvature moderate of order m such that for these (T135), (CT8), (and if applicable (CT9)) are satisfied and if the
Rn and control data
-4
objects the axioms following conditions hold
true:
For every stratum S the vector field Vsx : US -4 Rn, x which is defined on a tubular neighborhood of x(S n U) is
(CM10)
The fundamental existence result for controlled vector fields is the Its first part goes back
to MATHER
again
3.7.3 Theorem Let X be a
a
controlled vector field V: X
Tf
additionally
+ 1
over
every
X carries
PROOF:
1. PART
V'
m
Xk
By -i
byV[Xk
Vk'
construction of
Vk
Vk
on
M
a
controlled submersion :
M
--)
TM
a
W
-
f.
on
strongly
strongly
some
curvature
curvature moderate of order
choose V curvature moderate of order
k MATHER constructs in
Xk such that
Vk
.
over
[122]
a
Xk the
satisfying
controlled
vector field
Afterwards he defines V
controlled vector field
m.
:
the claim.
X
-4
TX
For the
gives proceed like in [122] and first choose some control data js)SE's Hereby we will use in the following that the skeleton Xk inherits a
control structure from X in If k
--
we
compatible with f. a
=
the k-skeleton
on
which
V
one can
induction
an
TXk
-
+ 1 and if f is
Vk satisfies relation (CT10) and Vk+1 lxk =
X
e,+'-structure and possesses
a
stratum, then
vector field
:
:
TX such that
--
moderate control data of order m
[122, Prop. 9.1].
controlled space and f
(CT10) If
relative
following theorem.
Then there exists for every smooth -vector field W
manifold M.
V(x-1(7eS(x))
tempered
of class E!m.
x(aS n U)
in
--4
a
canonical way.
us suppose we are given for k > 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 <
TS,
S,
T7 R VS(X)
=
Vk
(CT9)R
TpR, Vs(x)
=
O
=
W of (z).
I
(CT10)s
Tf
Xk+1
:
controlled vector field finish the
we are
and
z
E
given
a
S the
following
smooth
satisfied:
(CT8)R
Now define Vk+1
S
F12
S n TR
x E
__
on
Vs(z)
-
TXk+1 by Vk+1
7rR (X)
o
Vk and Vk+ 1
Vs. Then we obtain a Is Xk+1 which satisfies the induction hypothesis. This would IXA:
=
proof.
Hence it remains to construct vector fields
Vs with the desired properties. By the perform the construction of Vs separately for stratum. Now let ij be a point of S. Then define for every + I)-dimensional every (k stratum R < S a neighborhood U-,,R of ii in S by assumptions
the control data
on
we can
SnTR
U-LJ'R
S and set
Uj
nUR,,j.
:=
if IJ E
TRI
if -y
TR5
:==
Then the
e
\T
/2
family (U1J)1JES comprises
an
open
covering
of S.
R<S
On the other hand if
then
8,,
is
totally
one
denotes
ordered
by S.
by the
the set of all strata R
incidence
relation,
as
<
S such that -y E TR
the control data have been
chosen such that any two strata having nonempty intersection are comparable. Assume that for a point y E S the set 8V is nonempty. Then there exists a
largest
element one
field
can
Vj
R,,
in
find, :
Moreover,
Uu
%. By after
z
Tf But if the
if
S-u
=
controlled
-
condition
Vs(z)
Tf
=
-
(CT8)R.,
S n
and
TRy
(CT9)R.,,
(CT10)s holds,
T7rRu Vs(z)
--
R',
x
R" is submersive,
-
=
Tf.
are
a
smooth vector
satisfied for all
x
E
U.U.
as
Vk.
7rR'J
(Z)
=
W.
f(Z).
-y does not lie in any
after
unity ((Pj)jEN subordinate Uj one can finally
supp yj C
:
neighborhood U. appropriately,
TU,,, U,, the
E
(7rR.., pR,,)
one of the tubes TR with R < S, or in other words possibly shrinking U,,, choose the vector field V,, : Uj --) TUj for all z E Uz the condition (CT10)s is satisfied. By assumption f is a submersion, hence such a V,, exists indeed. After the choice of a partition
0, then,
such that
of
point
the
shrinking
such that
--i
for
the fact that
to the
define
Vs (X)
=
covering (U1J)'YEs and of points -Uj Vs as follows:
E iEN
pj
V.Yj (X),
X
E S.
E
S with
(3-7.2)
3.7 Controlled vector fields and
integrability
137
We show that this VS satisfies the desired control conditions. the relation then This
(CTIO)s,
as
it is satisfied
by construction of U., the implies R < Rj, where Rj
fact that
V,,
If
U...
over
E
X
Obviously, VS satisfies T"/2 for some R < S,
relation ljj E TR follows for all -yj with x E Uj := UVj' := R,,,, and 7TR,(x) E TR. Hence by the above proven
satisfies conditions
Vj,
by
all
(CT8)R,
the
(CT9)R,
and
have
following equalities
to be true:
E
T7tR VS (X) ,
(Pj (X)
T7CR
-
,
VVj (X)
T
=
(pj (x) T7rR V' 7rR, (x)
E
-
E
=
-
-
Q1 %EUj)
T71R
*
T7TRj Vyj (X) ,
(pj (x)
-
V'
-
7rR
-
7rR, (x)
UIXEUj}
Oj(X) Vk
E
=
*
-U I XEUj}
UI'XEUjl =
Oj (X)
.
o
7CR (X)
=
Vk 7TR( X)) o
fjl%EUj}
E
Tpp,-Vs(x)=
E (pj(x)'-TPR*T7CRjoVijj(X)
pj(x)-TPR'V-y,(*X)=
ji I XEUj}
-UIXEUj}
pj (x)
TPR 'Vk 7rR, (X)
-
0
=
*
-
fjl%EUjl
This shows the first part of the claim. 2. PART We
moderate structions
only
proof that
sketch the
the vector field V
case
already
can
under the
have been used several times.
It is essential for the curvature moderate like in Lemma 1.7.9 with K
unity (Wj)jEN proved by using the fact that
neighborhood TS
=
that
case
one can
aS. That this is
partition of
chose the
possible
indeed
can
that in the
V.,j uniquely by requiring
case
0 the relation
S.,
I
V,jj (x) Eq.
x
3.7.2 then is
3.7.4 a
( ker T-,(7rR, PR)) holds for all -L V1., (z) E ( ker Tzf ) is true for all
E
relation
are
Corollary Let
additionally
X be
Ts
a
--
X carries
a
--4
First choose
R" is
a
and that in the
U,,,.
case
The vector field
in every chart.
S..,
VS
=
0 the
defined
by 13
controlled space. Then there exists for every stratum S TX.
C+'-structure and possesses some control data which m. + 1, then one can choose V curvature moderate of
=
0 and
VjTs\S
By the last fields
on a
control data for X.
some
controlled submersion. Then
controlled vector field
the
i
m.
PROOF:
TS \ S
Vis
E
z
curvature moderate of order
order
a
Uj
E
tempered relative aS of class C'
radial vector field V: If
be
every tubular neighborhood TR is a regularly situated of R. With the help of a curvature moderate Riemannian metric over
determines
one
assumptions of the curvature moderate, as similar con-
be chosen curvature
TS \ S hence V
two results
we
--4
preceding theorem there exists -1. Now define V by TpS V at =
-
we come
vector fields.
-
radial vector field.
know that there exist
stratified space. Therefore
integration of controlled
a
Then the restricted map ps
the
TX such that
comprises
now
by
back to
our
enough controlled vector original matter of concern,
138
Control
3.7.5 Definition A flow -y
(x, t) y (x, t) E all
E
J the fact that
S
If V
-
X
:
j
:
TX is
--
a
controlled space
(Cf
stratified vector field
stratified, if for already implies that
on
X, then
one
exists
If V is
a
For all
E X
X
with t+ X
T11-}X 10'tx+ I : jX} is
PROOF:
is
a
ness
stratified E
S
are
controlled vector field
of
on a
[0, t.+ [ --
>
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)
<
< t <
:
j
--
<
S,
stratum
t2
Itl t2l
<
t-+,-
we
=
K
a
need the
-ys to be true. Hence
:=
lij
following
result.
compact neighborhood of x in
For V and
Set U
-
' Ijn(sxR)
uniqueness of -Y.
open and -y continuous
S and t1, t2 (=- R such that txF_ <
=Ys)
the relation proves
of
E
(Ts)sEs choose e > 0 with 7rs (ij) E K, ps (-Lj) < mEj. 2
Ts I
X associated to V fulfills for all -Y C- U and all t E R with the following control conditions:
min(t,+j, t2)
'y (-U,
t)
E =
(CT12)
7rS ('Y (1j,
t))
(CT13)
PS (-Y W,
t))
=
Ts, 'Y (7rS (1j), PS M
-
t),
3.7 Controlled vector fields and
PROOF TR is
139
integrability
LEMMA: Let the point ij E U be an element of the stratum R > 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.
<
the
By
small
=
=
proper.
s
7tS ('Y (IJ, to +
and
by
a
S))
=
7tS (T ('Y (IJ,
=
T ('Y(7rS (1j),
analogous argument
one
THEOREM
OF
(CT13)
and
(CT11), (CT12)
PROOF
to), S)
=
'Y (7rS (T(IJ,
=
11 (7TS (1j),
to)), S) S),
to +
similar calculation PS (-Y (,Lj, to +
Hence
to), S))
are
s))
-
satisfied for 0 < t
shows the claim for
3.7.6,
PS (-Y)
=
CONTINUED:
negative
Before
<
min(t,,,t2).
By
an
El
t.
showing J
to be open
andy
to be
(IF)ma.,. Assume that this is not the case. Then there exists ij E X with q < oo (or t > -oo), a compact set K C X and an increasing (resp. decreasing) sequence (Sj)j Erq C1 t , t,+j [ such that Y-, (sj) E K and limj,,,,, sj t ). We consider only the first case sj -) QY; t.+, (resp. limj,,,,, sj the second one can be handled analogously. After selection of a subsequence -Y'J (sj) continuous
will prove that -y satisfies
we
=
=
converges to ij E R. If R
flow of
-yj
:=
VR,
S,
we
hence
we
Let S be the stratum of
x
and R the stratum with
contradiction to the fact that YR is the maximal integral must have S < R. Now, if j is chosen sufficiently large, the points a
U from Lemma
neighborhood
3.7.7, and for all k
E N the
relation holds: 0 < Sj+k
Then
E K.
x
obtain
all lie in the
yj(sj)
following
element
an
=
-
Sj
<
t
-
and
sj
0 <- Sj+1c
-
Sj
<
t2-
by (CT13) 0 <
PS(1Jj)
=
=
PS(Y(lJj) Sj+k Sj)) PS(T(Y(IJ) Sj)) Sj+k PS (lJj+k) PS (((*LJ) Sj+k)) =
-
-
Sj))
=
-
But this
ps(x)
=
Next that
tx-
means
that the sequence
0. Hence we
<
(IF),,,ax
(x, S)
will construct for
tl
< S <
t2
<
(PS('Yj%EN
cannot converge to 0 which contradicts
is true.
tx+.
Let
E
j
an
neighborhood
open
us assume
that
t,+,
<
t2 for
neighborhood of x according to Lemma 3.7.7. Then elements sj E [0, t,+j [ converging to t,+,. Let ljj -y (-tj, tj) is
a
=
(CT13)
we
then have liM 7rS (Ijj)
=
'Y (7TS ('Y),
j--400
lia PS Wj)
j
400
=
PS (-Y)
tj
in
J. Let tj, t2 be such ij E U, where U
some
there exists for all
a
sequence of
j. By (CT 12) and
140
Control
As
(7ts, ps)
Ts
:
-4
which contradicts
[0,
es [ is proper,
(IF).a.,.
Thus
(-Lij)j Eiq
t,+J
must have
a
Theory
converging subsequence
! t2 follows and analogously
:5 ti, which
t
altogether gives UX]tl,t2[C J, hence J is open. As (CT12) and (CT13) hold as well for all ('y, t) E U X I t1) t2 1, the map y must be continuous at (x, s) hence on the whole domain
J.
The theorem will be
shown, if we can finally prove that every stratified integral 'y- of V which satisfies condition (IF)ma., coincides with -y. But this is obvious, as J n (S x R) -- S has to be maximal by (IF) max, every one of the restrictions i/ls : J1s flow
=
hence coincides with ys.
3.7.8
Let f
Corollary
X
:
-4
M be
a
proper controlled
trolled space X and the manifold M. V Tf
X V
TX and W
--
W
=
-
f the
:
M
TM which
---
following relation Jv
where
(f
=
MR)
x
Extension theorems
In this section
we
spaces, which
was
con-
satisfy
the control condition
(CTIO)
holds: -1
(Jw))
Jw denotes the domain of the flow of W and JV the
3.8
between the
mapping
Then for any two controlled vector fields
on
will continue the extension
of V.
one
controlled spaces
theory for
smooth functions
on
stratified
introduced in Section 1.7. The main result in 1.7 allows to extend
jets on the closure 9 of a stratum to a Whitney function on X, but it does not provide general criteria to decide, when a smooth function on S which falls off fast enough near the boundary can be extended to a smooth function on X. For our extension theory we need an intrinsic notion for the fall off behavior of a certain
smooth function f E
(!OO(S)
stratum S
the
aS. The
boundary
corresponddirectly to smooth as for such functions f E e,'(S) the higher derivatives a'f are in general not at our disposal. To obtain an appropriate notion of flatness at the boundary, we will extend the definition of FERRAROTTI-WILSON [59, Sec. I] for special singular subspaces in R' to the case of arbitrary (A)-stratified spaces. definition 1.7.1 for
on a
ing functions,
Whitney
3.8.1 Definition Let
L be
a
smooth Riemannian metric
and
M
E NU fool. A C'-function f
flat
on
the
boundary of order
c
:
lim
norm
V is Levi-Civita connection
of
a
L. If f is
tensor with
respect
S
X",
E
-Oj ES
Hereby
near
functions cannot be transferred
to
--)
R
on a
on an
(A)-stratified space X,
stratum S of X is called
if for all natural k <
11"Milt. -
m
geometrically
and all ij c aS
0
6['(X, as)c on
L, and
tensor fields induced
6,,
is the
geometrically flat on the boundary geometrically flat at the boundary.
of
by
L
on
distance
geodesic arbitrary order,
we
S, 11
on
-
11 '
is the
S induced
by
will say that f is
3.8 Extension theorems
on
controlled spaces
141
V'f df and Vkf f, V'f V(Vk-1f) for k > 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 <
Vk)
=
VV2T7r'V3) m
E (Vj 9, Tjk), I<j
where
Tjk
is
products of
an
j-times
covariant tensor field which
vector fields of the form
Vvi,
-
-
-
can
be written
VviT7r.Vi,,
as
with 1 < k
the
-
j.
sum
of tensor
142
Control From
now on we assume
elements el,
-
-
,
VvlT7r.V2(x)
the relation
C',
that every
one
of the vector fields Vi is
e,,. As S is curvature moderate of order
-
=
m
and
7r
P7r(x)D(T7r.V2)(x).Vj(x) implies
one
Theory
of the basis
tempered of class that
Vv,T7r.V2
is
bounded away from as (recall the notion "bounded away from Z" which had been introduced in the proof of 3.3.6 on p. 104). Inductively we thus obtain that the vector fields
Vvi,
every
point of as
Cj
...
I <
Vvj,T7r.Vj,,
by assumption
(RA1) some
on
C
>
g is
j, I
<
W C Rn
I+
Cj
together
with constants
x
wn s.
c-
(3.8.2)
d(x, aS)cj
geometrically
flat of
p. 50 holds true, there exists for
sufficiently large
order and
as
the axiom
later to be determined constant d E N
a
0 such that
IIV'g(x)ll for all
all bounded away from as hence there exists for
and Cj E N such that
> 0
IlTjk(X)II As
m are
relatively compact neighborhood
a
< k.
But this
<
Cd(x, aS)Cj+CS-(T,I)+d+l the existence of
implies
(=-
x
Wn
appropriate
S,
constants
1
> 0
such
that
JID k(g 7r)(X)II .
1 d(x, as) c-(r,l)+d+l
<
S
x
(E
wn s.
prerequisites of the generalized lemma of Hest6n s are satisfied, hence G is Whitney function on S. In particular the following estimates then hold for every a E N1 with I al :! in, if x, ij Econverge to some z E S:
So the a
(RTG)(') But
we are
not
(Ix
o
(3.8.3)
-
yet finished with the proof. It remains
smooth function f such
=
:
0
-4
R such that f
=
g
-
to show that
neighborhood
7r on a
find
one can
a
of S. To construct
f recall that according to the assumption on S the tubular neighborhood regularly situated neighborhood of S. Hence there exists a smooth function I on Y : R, \ as -4 [0, 11 such that y is tempered relative aS'of class C', that y a neighborhood of S, and that y vanishes on a neighborhood of CTS \ as. Now we define f : R' --i R by
TS is
an a
=
f
y(-Lj) g(7r(-y)),
if -y E
0,
if -y E Rn
T,
\
T.
Choosing d resp. c sufficiently large, then by the temperedness (P the thus defined comprises a (!'-function on R' indeed, hence one on X As (p is identical to I on neighborhood of S, the map f therefore fulfills the claim of the extension theorem.
f a
M
3.8.4 Remark FERRAROTTI-WILSON have proven in extension result for
tiability class
m >
singular subspaces 2.
of R' with
a
[59,
Thm
111-2.1
an
analogous
dense top stratum and differen-
isotopy lemma
3.9 Thom's -first
Thom's first
3.9
stratified spaces H
:
trivial,
Y
M
x
lemma
isotopy
3.9.1 Definition A stratified
manifold M is called
143
mapping f
if there exists
X
:
a
X such that f
-4
--
M from
stratified space X to
a
stratified space Y and -
H(U, x)
One says that f is locally trivial, if there exists a covering of M such that all restricted maps flf-i(u) : f-'(U) -4 U are trivial. Thom's first
3.9.2
manifold and f
X
:
a
a
a
local
one
smooth
a
trivial.
M
case
R1
=
v for all isomorphism of stratified spaces H : X0 x R1 --) X such that f H (1j, v) E X0 and v E R1. Hereby Xo := f-'(0) is the fiber of f over 0 according to Example
an
-Lj
locally
it suffices to construct in the
M.
E
open sets U
controlled space, M
proper controlled submersion. Then f is
As the statement is
PROOF:
lemma Let X be
isotopy M
--
by
x
a,
of
isomorphism
for all -y E Y and
x
=
an
=
-
3.6.6.
Consider the coordinate vector fields el on R1 and choose en av. aVI ..... En on X according to Theorem 3.7.3 such that Tf Ek El,
controlled vector fields f for k
-
As the vector fields ek
globally integrable, Corollary 3.7.8 entails that the vector fields Ek are globally integrable as well that means for every k the domain Jk of the maximal integral flow Yk of Ek is equal to X x R. Hence we can define for all ij E X0 and v (vj, Vn) E Rn: ek
-
=
1,
-
-
-
,
n.
=
-
-
are
-
H(IJ,V) =/n(/n-l(... (Ti (X) V1)) On the other hand G (x)
('In (Yn-1 (*
-`
As for all
we can
c
x
*
*
define
a
(Ti (X) -vi))
mapping ...
G(x)
lies in Xo -
:
X
-)
X0
-Vn)) f(x)))
x
Rn indeed that
H (y, v)
As all theYk
=
are
v
t))
means
=
(Vi)
Rn
by
...
)
Vn)
:=
f(X),
X
f (x) + tek,
E
X.
flows, G-
H
=
(3.9.1)
G is well-defined.
holds for 'Lj E X0 and
G and H have to be stratified
v
help of
the first
Moreover, by (3.9.1)
the
E Rn.
idxxR. holds
true
as
well
mappings. Therefore H is the desired properties. This proves the isotopy lemma. With the
x
X
f (_Yk (X,
relation f
)
G
)vn)-
...
a
as
H. G
stratified
idX. Moreover, isomorphism with =
El
isotopy lemma the proof of property (6)
in
Prop.
3.6.7
now
is obvious.
PROOF
OF
PROP. 3.6.7
(6):
For all proper control data
(TS)SES
of X and every
stratum S the
mapping (7rS, ps) : Ts \ S -410, es [ is a proper submersion, hence locally trivial by the isotopy lemma. In particular, this implies that for every stratum R > S the restricted map (7tS,R, PS,R) : Ts,R --1]0, F-s[ is locally trivial, hence a differentiable fibration.
3.9.3
fibers.
Corollary Every Whitney stratified space is locally trivial with
n
cones as
typical
144
Control Theory The space X is controllable
PROOF: is
a
appropriately), As
by
Theorem 3.6.9 and
proper submersion for every stratum S
a
hence the claim follows
consequence of local
triviality
(where
(7rs, ps) : TS \ S -- ] 0, F-S [ (Ts)SES are chosen
the control data
immediately
from the first
isotopy
lemma.
D
obtain the theorem that for every controllable
we
stratified space the closed hull of a stratum can be resolved in a certain sense by a manifold-with-corners. This result originates in the work Of VERONA [177] and will be
proved
in the
following.
(VERONA [177, Prop. 2-6]) Let X be a controllable stratified space of 0 0, V X and such that X' is a manifold of dimension d. Then there exist a (d-1) -dimensional manifold Q (without boundary), some proper control data (TS)SES of X, and a proper continuous mapping H Q x [0, 11 -4 X with 3.9.4 Theorem
finite dimension such that M
the
=
following properties:
(1) H(Q x10, Q
(2) H(Q (3)
The
(4)
X
f0j)
C X' and
If Lj E
Q
and
x
is
=
a
smooth
H (-U,
H( , t)
E
0) a
is
a
point of the
mapping
stratum
TR and 7rR(H( , t))
H
S,
then there exist
stratum R < S such that for every
3.9.5 Remark VERONA has claimed in
ified space X the
embedding.
of M in X.
neighborhood
of ij and
Q
the relations
a
ax.
=
image of H
hood U C
is
HlQx]o,,[
=
7rR(H( , 0))
[178, Prop. 1.3] that as a C'-mapping.
be chosen
can
a
neighbor[0, 1
E U and t E
hold true.
for every Whitney stratBut in general this does
hold, because otherwise every Whitney space would have a locally finite volume example constructed by FERRAROTTI [55, 58] of a Whitney
not
in contradiction to the
stratified space which does not have set the
subanalytic
[86],
see
mapping
H
can
also BIERSTONE-MILMAN
PROOF:
Let 0 <
strata of
do
<
di
<
locally
a
finite volume. But note that for every el, even of class C"' (HIRONAKA
be chosen of class
[15]). <
...
dk
=
d be the sequence of dimensions of
and k the thus defined dimension
By M =A 0 we have (Ts)SES Finally let S be the union of all strata of smallest dimension do. According to Proposition 3.6.7 4) := (7rs,x-, ps,x-) : TsnXO -i10, es[then is a differentiable fibration and S' := 4) -'( IE-1) k > 1.
with
of
Further let
F, :=
[e]).
X,
I es
is
a
(d
be
-
l)-dimensional
Without loss of
generality
T S =T S16's holds true and that
agreements
we now
proceed
submanifold of X'
we
can assume
(7ts, ps)
in several
depth of X.
proper control data of X.
some
:
TS
-4
(see
3. 1.1 for the definition
after
possibly shrinking es that [0, 16F-s [ is proper. After these
steps.
1. STEP Some
helpful smooth mappings. [0, 11 x [0, 11 -- [0, 11 x [0, 11 be a function smoothing the corner according to Lemma C.4.1, and X : R -4 [0, 11 a further smooth function such that X(s) 0 for I for s > 1. The smooth curve [0, 11 -4 R2, s < 0, x'(s) > 0 for 0 < s < I and x(s) Let (p
:
=
=
3.9 Thom's first s
-4
x((s
-
isotopy lemma
!)') p(s, 1) 2
145
will be abbreviated
the letter c, its components
by
by
cl and
C2-
2. STEP The
First choose
flow of
integral
radial vector field.
a
radial vector field V
a
on
X such that
a
(P. (V (X)) Moreover,
require VIS
we
VIX.. Then ps(y-, (t))
at'
Now let -y
0.
=
J
7ts (x)
=
=
{(X, t)
(xo
E
y is smooth
Obviously
G
=
t + PS
extend the
one can
:
i
on
T's S
n
x
Ts)
n
(x-
integral
flow there exists
X' be the
integral
N
flow of V'
-
integral flow
to
a
continuous
we
have
a on
x
-
on
ps (x)
:5
t <
2F-S (7ts (x))
-
mapping
ps (x)J.
The function
R).
(x, t)
-4
'y.
((t
-
1)
ps (x))
2 [. Moreover, by (T" S \ S) x] 0,
smooth function 5
:
X'
--4
the
properties of the
Xo which satisfies
=t,
ifx=G(,Lj,t)withijES'andtE]0,3/2[,
!t,
ifx=G(-Lj,t)withijES'andtE]3/2,2[,
! 2,
if
5(X)
induction
a
RI
x
[0, 2 [--4 X,
then is continuous and smooth
by
-4
X with
J
Thus
J'
=
by -y, (- ps (x))
--4
:
for every x E X' the integral curve 'y, satisfies the differential equation -1 with initial value ps(x), hence PS (Y. (t))
Thus
Ts.
E
X
-
further
x
G(S'x]0,2[).
ingredient fop
the
proof of the
theorem which
we
will
now
lead
k.
3. STEP: Proof of the claim for k
=
1.
By the properties of the integral flow -y the manifolds Q := S' and the mapping H := Glscx[0,1[ satisfy the above conditions (1) to (4). This gives the claim for k 1. 4. STEP: Begin of the induction step, construction of an integral flow. Let us suppose next that the theorem is true for all spaces of dimension depth < k, and that X is a space of dimension depth k + 1. With S from the 1. Step let X' X \ S As the following constructions can be performed separately for every connected component of S we can assume without loss of generality that S is connected. By induction hypothesis there exists a (d l)-dimensional manifold Q' and a continuous map H' : Q' x [0, 2 [-4 X' such that HI'Q, x [0,11 satisfies the claim with respect to X' and .
-
such that we
in
is a diffeomorphism onto its image. In the following constructions glue together the manifolds Q' and S' as well as the mappings H, and G way such that the resulting objects have the desired properties. The difficulty lies in smoothing the corner which results from intersecting H'(Q'x r), 0 < r < I
H,'Q,XIO,2[
want to a
now
and S'.
146
Control To achieve this
R'O,
by (4)
first show that
we
there exists
a
Theory
smooth function -r'
:
Q'
-)
==3 F-S /2
!T'(-y) H'(-y, 0) E TS 2 0)) as well as ps (H(ij, t)) ps (H'(ij, 0)) hold true. But then the submanifolds H([r'l) and S' of X' are transversal for all 0 < s < 1, and their intersection B := Hl([T']) n Sf comprises a (d 2)T'
such that for all
<
3
H'(y, t)
the relations
E
(y, t)
Ts and
E
Q'
x
1 [ with t <
[0,
and
-
7rs (H(-y,
t))
choose
further smooth function
=
7ts (H'(-y,
-
dimensional manifold. Now
we
a
X'
T :
--->
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<
the radial vector field V: X
-TX from the 2.
=
Now let
us
bring in
flow 'y
J'
-4
and
x
by
U, where I0
:
2,
if
Y. Essential for the
commute,
as
following
by assumption
on
--
is
Al. 2
Step
and its
integral
the observation that the flows 'Y the definition of V and W the two
now
H' and by
vector fields V and W commute.
5. STEP: Induction step Now
we can
define
one
detailed
continued, construction of Q. the union Q Q1 U Q2 U Q3 of
as
=
1
I H (1j, W(Aj) 3
=
Q2
=
=
fx E
are
'y E
and 5
(H'(,y, irl (ij)))
!
3
IH"(G(b', C2(S)),C1(S)) I
s
E]o,
SE/81 I r(x) !
fx E G(S'
X
I!3 I) I T(X)
Q comprises
submanifolds and
proof of
Q'
the sets
b E B and
realizes that
components
C X'
Q1
Q3 Then
Q
this fact
functions H" and G and
use
6
j
-
=
even a
as
d
-
> 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 <
m
jjVkf(,X)Jj L 3.9.8
Example By
the
already
<
C
I +
x
5 ,(X, as)c
mentioned article
every stratum of the canonical stratification of
tion of class C!'.
subanalytic
3.10
set
The
(at
reason
least
[86]
E V.
(3.9.6)
of HIRONAKA there exists for
set a tempered resoluaccording to HIRONAKA there exists for every bimeromorphic resolution of singularities. a
subanalytic
is that
locally)
a
Cone spaces
According to Corollary 3.9.3 every Whitney space X is locally trivial with cones as typical fiber that means there exists a covering of X by open sets U such that U can be mapped by a stratified homomorphism, k onto a cartesian product of the form (SnU) x CL, where S is a stratum of X and L a compact Whitney space. But in general one cannot choose k as a diffeomorphism between spaces with a smooth structure as is already shown by the example of Neil's parabola. Though Neil's parabola XNej, C R, CSO c R, there does not exist a is stratified homeomorphic to the edge X.cj ,,,. diffeomorphism around the origin of W, which (locally around the origin) maps Neil's parabola onto Xd,,,,, as the legs of X,,ei, touch in higher order then the ones of the edge =
XEdp A special class of stratified spaces is given by such spaces, for which the homeomorphism k has additional regularity properties like for example that it can be chosen to be smooth or Lipschitz. Thus we obtain different categories of so-called cone spaces
148
Control
which
Theory
well suited for further
geometric-analytic considerations. In this section we by a recursive definition. But before we come to the details let us mention that by C"'(0) (resp. C'-(O)) with 0 C R' open we will understand the space of all real analytic functions (resp. of all Lipschitz functions) on 0. are
will introduce these
cone
3.10.1 Definition Let
spaces
N U 11-, oo, wl. A cone space of class C' and depth 0 w real analytic) countably many smooth (resp. for m connected manifolds together with the stratification the strata S of which are given by the union of connected components of equal dimension. A cone space of class em and depth d + 1, d E N is a stratified space X with smooth (resp. real analytic) structure such that for all X E X there exist a connected neighborhood U of x, a compact cone space L of class em and depth d and finally a stratified homeomorphism is the
topological
sum
m G
X of
=
k: U If
(S
-
n
U)
CL.
x
0 then it is
required additionally that L is embedded into a sphere via a fixed global singular chart I : L " S' and that k and k-1 can be chosen as mappings of class el. Hereby, the smooth structure on CL is the one induced by 1. In other words this means that the smooth structure is given by the global chart m
0
smooth
CL Sometimes A
cone
call I
we
:
W", [t, ij]
---
L
---)
S'
space of class Cm
F-4
t
-
I (y),
link chart of L and k
a or
briefly
a
[0, 11, 11
t E
a cone
E L.
chart.
em-cone space then is
a
stratified space with
smooth structure such that for every x E X there exists a neighborhood U and integer d G N such that U is a cone space of class (!' and depth d.
an
Example According to MOSTOWSKI [130] every complex analytic set X C C' a Lipschitz stratification. Now, for Lipschitz stratifications an isotopy lemma holds, where the local trivializations are bi -Lipschitz, hence complex analytic sets with a Lipschitz stratification according to MOSTOWSKI comprise nontrivial examples for e'- -cone spaces. Moreover, PARUSII SKI could show in his article [141] that every subanalytic set possesses a Lipschitz stratification, hence comprises with 3.10.2
has
so-called
this stratification 3.10.3
a
e'-
-
cone
space.
Example Every manifold-with-boundary
corners
is
a
cone
space of class e,'
as
somewhat
one
3.10.4 Theorem
satisfies in every PROOF:
Then X
=
S' of depth d, B
x
structure defined
Every cone space of class (!m with m > 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)
<
(K)
the set
M(H)
n
M(K)
is
M(H). is said to be the
orbit bundle. The orbits
lying
principal orbit t'ype of M, and M(H-) the M(H*) are called principal orbits.
in M)
4.3.4 Remark The
assumption that G\M is connected does not mean any restriction arbitrary G-manifold can be decomposed into the G-manifolds 7T-'(Z), where Z runs through the connected components of G\M. of
generality
PROOF
OF
be shown
THE
as
dim M
=
M has
only
since
an
THEOREM:
The existence of the
it is done in JKNICH
0 the orbit space one
orbit. Let
G\M now
[95,
principal orbit type in (1) will induction by dim M. For by assumption only of a single point, hence
Theorem
consists
2.1] by
M be n-dimensional. We first consider
a
slice bundle
159
Space
4.3 Stratification of the Orbit
G XG. SV-, assigned with Vx Of X G M and the sphere bundle SNx the claim is satisfied for induction metric. hypothesis By G,.-invariant respect Gx\SV,. is connected. If, however, it is SNx provided that the orbit space G\SNx
N,,
G XG.,,
=
to
=
a
=
must be the trivial representation. point of SNx would be (G,.). However, in any case it follows that with z > 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 <
union therefore forms
SJAMAAR is
a
[161,
subgroup
By
C
Without loss of
1.2.21].
Lem.
of H.
open and dense set
an
of all N' have to
it
generality
can
be assumed that K
virtue of the slice theorem it suffices to show that for M
a non-empty M(K) the closure Of M(K) contains the V". Be the H-module V endowed with an H-invariant submanifold M(H) G/H x the orthogonal space to V1. Then M has the representation metric, and let W be
of the form M
=
G xH V and =
M
=
(G
W)
XH
X
VH'
the
M(K) Since
W(K)
=
is invariant with
by assumption
to
M(K)
of which results in
use
G
XH
V(K) to
respect
=
(G
XH
W(K))
VH.
X
multiplication by non-vanishing
empty, the origin of W lies in
must not be
W(K).
scalars and It therefore
follows
M(H) which
was
to be
=
G/H
X
VH
C
x
x
VH El
proved. acting properly
To any G-manifold M with G
4.3.5
(G
to each
E M the germ
on
it
give a stratiM(G..) Usually 8
we can now
8,, of the
set
point assigning stratification by orbit types.. Due to theorem 4.3.2 8 is a stratification in the sense of definition 1.2.2, indeed, provided that we can furthermore show that the decomposition of M into the submanifolds M(H) is locally finite. Together with the slice theorem, this is, however, a direct consequence of the following lemma.
fication of M
x
-
is called the
4.3.6 Lemma Let H C G be
G-space
G
XH
V possesses
manifold M with PROOF:
claim is has
M
Let
trivial,
us
can
only finitely
by
case
an
H-module.
Then the
many orbit types. In
on
it possesses
show the second claim
many orbit
be covered
compact subgroup, and V
proper G-action
a
because in this
only finitely
a
by
particular, every compact only finitely many orbit types.
induction
M consists of
types. Now, let M be
virtue of the slice theorem
on
dim M. For. dim M
only finitely an
many
=
0 the
points, hence M
n-dimensional manifold. Since
by finitely
many open sets of the form
and H C G compact, it suffices to show the induction step for these G-manifolds. Since H is compact and V a H-module, there is an H-invariant G
xH
V with dim V
< n
160
Orbit
metric
on
V.
dimension
Let SV be the unit and is
< n,
a
sphere
with respect to this metric.
compact H-manifold,
Spaces
Then SV has
By the induction hypothesis the proof of corollary 4.2.8 the
moreover.
SV possesses only finitely many orbit types. Due to isotropy group of [(g,v)] E G xH SV equals gHvg-', i.e. the number of orbit types of SV, G x H SV and G x H (V \ {0}) agree. Compared to G x H (V \ {01), the space G xH V has at most the orbit type (H) in addition, hence the induction step follows. the proof of the second claim entails that the first claim needs to be true as well. M
Moreover,
4.3.7 Theorem The stratification group action makes M into
PROOF:
It
only
a
by orbit types of a G-manifold Whitney stratified space.
M with proper
remains to show that the
Whitney condition (B) is satisfied. (compact) isotropy groups of M, i.e., in other :C;, words, M(H) < M(K) may hold. Furthermore, let two sequences (Xk)kEN C M(K) and (IJk)kEN C M(H) be given, converging to a -y E M(H), where we additionally assume that in a smooth chart around -y the secants Ek XklJk converge to a straight line f, and the tangent spaces Tx,M(K) converge to a subspace r. Due to the slice theorem we can assume without loss of generality that To this
let K
end,
H C G be two
--"::
M=G
XHV=(G XHW)
VH
X
and
[(1, 0)],
a slice of H, V' denotes the subspace of H-invariant vectors, and orthogonal space with respect to an H-invariant scalar product on V. Let g be the Lie algebra of G, the one of H, and m the orthogonal space of c 0 with respect to an H-invariant scalar product on g. Via the exponential function on
where V denotes W
G
(V)'
=
we
the
obtain
a
natural smooth chart :
y
where U C M is of the sequences
VH and
U
V,
-4 m x
y ([(exp
&, v)])
v),
&
E M, V E
V,
suitable open neighborhood of -y. We may assume that all elements (XA;)kErq and (Yk)kGN lie in U. Recall now that M(K) = (G X H X
a
W(K))
G/H X VH. Since W(K) is invariant with respect to multiplication by non-vanishing scalars, it follows after a possible selection of subsequences
M(fj)
W,
V)
=
:=
lim k--4oo
Y(Xk) Y (Xk)
-
-
Y(IJk) Y (*Y k)
W(K)
E M X
With the representation Y (Xk) Wk) Vk) E after a possible selection of subsequences)
11W11 Using on
once
VH
M X
with
W(K)
x
VH
W(K) we
=
W(K)
then have
=
x
VH
cr
=
lim k-4oo
This
implies
in
particular f
=
span
(again,
W.
again the invariance of W(K) with respect
span w
{0}-
jjWkjj to
multiplication by scalars gives
the other hand m x
U
Wk
liM k--400
X
(&, w, v)
Tx,M(K)
C r, which
shows the claim.
4.3 Stratification of the Orbit 4.3.8 Theorem Let f
M
:
161
Space N be
-
G-equivariant smooth mapping
a
between the
proper way. Under the additional assumption M, that f is a stratified submersion with respect to the stratification 8 on M by orbit there exist G-equivariant control data (TS)SES compatible with f that means
N
manifolds
on
which G acts in
a
types,
for every stratum S the relation G
-
Ts
TS holds and
C
(KB14)
7tS (gX)
=
g 7tS (X),
(KB15)
ps(gx)
=
PSN'
provided
that
x C-
Ts and g
G.
C-
place G-equivariant versions of the existence and uniqueness theorem for tubular neighborhoods. Afterwards one proceeds in accordance with part 1 of the proof of Proposition 3.6.9 and constructs as described there a G-equivariant control data using as ingredients G-equivariant For the
PROOF:
proof
of the claim
one
needs in first
objects only. We already have got a G-equivariant neighborhood theorem; this is, in the end, actually
version of the classical tubular the slice theorem with the
help G-equivariant versions of the existence and uniqueness theorem for tubular neighborhoods hold as well. Since the formulation of the various steps of the proof is canonical though of which
one
somewhat
shows
(almost)
tedious, the proof
word-for-word
in Section 3.1 that the
as
M
is left to the reader.
Though we have just found a'natural stratification of M by orbit types, the G\M lacks so far. In the following considerations the result will be that the quotients G\M(H) possess a manifold structure in a natural way, where for the proof of this fact Proposition 4.2.10 plays an important role. The manifolds G\M(H) then
4.3.9 one
of
define the desired stratification of the orbit space. 4.3.10 Theorem Let H be
one
of the
isotropy
groups of
a
of the normalizer
proper G-action
NG(H)
on
M.
of H in G acts
NG(H)/H quotient group rH properly and freely from the left on MH, i.e. MH -4 rH\Mti becomes an opposite rH-principal bundle. Furthermore, the submanifold M(H) can be identified with the associated fiber bundle G/H rllx MH -- rH\M, by the G-equivariant diffeomorphism Then the
=
19H,X1 -49X.
'Y:G/Hr1tXMH--4M(H)) (cf.
BOREL
PROOF:
[19, 1]
and J. NICH
[95,
1.5])
Theorem
For each element g of the normalizer
NG(H)
point x C- MH gx lies MH- Since, by definition,the
and each
we NG(H) MH, again isotropy group of any point Of MH is equal to H, the N G (H)-action induces a left action of the quotient group rH on MH, which has to be free. The action is proper as well, for NG(H) and MH are closed subsets of G resp. M, and G acts properly on M
in
hence
have
an
action
X
MH
--
by assumption. This gives the first claim of the theorem. For the proof of the second one'first note that rH acts freely from the right on the homogeneous space G /H: g H -yH := g-yH is for g E G and 'y E N G (H) a well-defined product, actually. Consequently, the associated fiber bundle G/H P,,X MH consists -
162
Orbit Spaces
of all
equivalence classes [gH,x] with respect to the equivalence relation (gH,-yx) T E N G (H). Then, one immediately realizes by g (-yx) (gy)x that T is a well-defined differentiable mapping on G/H PX MH Since the G-left action commutes with the rH-right action, G/H r,,x Mjj becomes a differentiable G-space, and T a Gequivariant differentiable mapping. The fact thatT is surjective is obvious. It remains to show injectivity. However, from gx ij it follows y := g-' c- NG(H), hence This [g H, x] means nothing else but that T is inj ective. [gyH, -y-lxl [6 H, yl. El
-
(gyH, x),
=
-
=
=
4.3. 11
=
Corollary
Let M and G
Gx of the orbit space stratification of G\M. PROOF:
G\M
The claim is
decomposition
as
in the preceding theorem.
the germ of the set
G\M(G,,)
Assigning to each point rG.,\MG., one obtains a
direct consequence of Theorem 4.3.10 and the fact that the
a
of M into orbit types describes
a
stratification.
]Functional Structure
4.4 4.4.1
On the orbit space
G\M
of
a
proper G-action
of "smooth" functions. Its sectional spaces
are
one
each
U
C
G\M
E!G'\M(U)
has
defined in the
COGO\1\4(U)=IfEC!(U)Ifo7rEe,00(7C-'(U))II For
_
therefore
a
canonical sheaf COO
G\M
following
way:
UcG\Mopen.
is
canonically isomorphic to algebra of G-invariant smooth functions on 7t-'(U). By Theorem is fine. entails, among other things, that the sheaf C!' G\M open
Coo (n-1 (U)) G, the
4.2.4, (4)
this
In this section it will be shown that (!'
comes from a canonical smooth structure G\M the stratified space G\M, indeed; in other words it can be defined by a singular atlas in accordance with Section 1.3. To this end, the first and fundamental step is on
the
following classical theorem, attributed to D"ID HILBERT, but probably proven independently by MENAHEM SCHIFFER, too (Cf. WEYL [186, Chap. 8, Sec. 14] and BIERSTONE
[14]).
4.4.2 Theorem Let H be
representation on
V is
space of H.
a
compact Lie group, and V
Then the
algebra T(V)"
a
finite dimensional R-linear
of the H-invariant
polynomials
finitely generated.
A finite
generating system of T(V)' as in the theorem is usually called a Hilbert basis T(V). If the generating system consists only of homogeneous polynomials, the Hilbert basis is said to be homogeneous. A Hilbert basis is called minimal, if there is no generating system for T(Y)' with less elements. of
PROOF:
We denote
by T(V)
the
algebra
with HILBERT's basis theorem the ideal in
of
polynomials
T(V) generated by
on
V.
In accordance
the H-invariant
non-
polynomials is finitely generated. Therefore there exist H-invariant polynomials p 1, Pk generating the ideal. Without loss of generality we can assume that the polynomials pj are homogeneous, since each H-invariant polynomial can be decomposed into H-invariant components. Let do be the lowest polynomial degree that constant
-
-
-
)
163
4.4 F inctional Structure appears in the
generating system
Then every non-constant H-invariant do. We show by induction by the degree
P k.
p 1,
must have at least the
degree do that each element of T(Y)' is a polynomial in the pj. Let p be a homogeneous H-invariant polynomial of minimal degree do. Then there is a representation of p of
polynomial d >
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 *
*
)
<
one
obtains
a
representation
equation
of the form
over
H
(4.4.1),
given by
f
qj (V) and
both sides of this
Integrating
Tj(gv) d t(v),
v
E
V,
By the induction hypothesis every qj is a polynomial in as well. This gives the induction step and therefore
d holds.
hence this holds for p
P k,
0
completes the proof.
recall that the algebra CO'(M) of smooth functions on M possesses topology (see appendix C.1), and that via the pullback every smooth function f : M -4 N induces a continuous homomorphism f* : e"(N) -4 COO (M) of Fr6chet algebras. Since COO(M)" is a closed subalgebra of C'(M) with respect to this topology, e_',G M becomes a sheaf of Fr6chet algebras. The theorem following now can also be regarded as a topological quotient of the algebra of states that Cc' ,\ ,\M At this
a
point
we
natural Fr6chet
smooth functions construction of
a
4.4.3 Theorem
theorem and p
=
on some
Rk and represents the second important step towards the
smooth structure
on
G\M.
(SCHWARZ [156], MATHER [124]) Let H and V be as in Pk) be a Hilbert basis of T(V)H. Then (pi,
the
preceding
-
)
P* C' (W)
-4
C' (V)H,
f
"
f
*
(Pl)
*
''
)
Pk)
surjective topologically linear mapping between R6chet spaces and splits topologC' (R11). ically that means there is a topologically linear right inverse e : C' (V) the mapping Moreover,
is
a
ff : H\V induced
by
p is
HV
F-4
(P 1 (V),
continuous, injective and proper.
P k (V)))
164
Orbit Spaces
4.4.4 Remark The was
proof that p*
able to show that
p*
even
is
surjective comes from SCHWARZ. Then MATHER splits topologically. Moreover, he gave a simplified proof
for the result of SCHWARZ. PROOF:
Since the proof of the theorem is very tedious, we refer the reader to the [156, 124] or to the monograph of BIERSTONE [14]. ID
already
cited literature
Now, let Gx that
means
E
G\M be a point in the orbit space and U a "slice neighborhood" of x G-equivariantly diffeomorphic to a neighborhood of the zero section
it is
of G ) G,
V-,. Let
p
and
G\U
G,,\V,,
-4
(pl)
system p
=
following
way:
:
U
G XG,_ V.,, be the corresponding G-equivariant the canonical quotient map. After choosing a
---)
pk)
T(Vx.)Gx
for
we can
x:G\U--4W,
define
a
singular
embedding generating
chart around Gx in the
Gzi-4j --q(Gz).
Since the G-x-invariant functions separate the points of V, the map x is injective. Its is obvious; that x is also a homeomorphism onto its image results from the fact that f is proper. Since the respective components p and 0 are smooth, and the fiber bundle G XGx Vx -4 G/G., as well as M(li) -i G\M(H) for H C G possess local
continuity
sections, the restriction of x onto a stratum of the form G\(U n Consequently, we have with x a singular chart at hand, indeed, if it that each of the restrictions
M(H)) can
is smooth.
yet be shown
is immersive. In the
corresponding proof it XIG\(unm(,)) family of all such singular charts represents a singular atlas for G\M and that the smooth functions belonging to it are given by C' For the G\M* explicit proof of our claims we now need the following result which makes a statement will turn out that the
about the Zariski derivative of V
(see appendix B.3).
4.4.5 Lemma Let K and V be as above and let q = (qj, qj) be a minimal homogeneous Hilbert basis of T(V)". Then the Zariski derivative df -4 is an isomorphism
in the
origin that
means
Toz is
an
Toz (H \ V)
--->
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.
<
homotopy HS
for all Ij S E V. Then
V
:
we can
[0, 1]
--4
V such that
extend
Hs
to
x
a
smooth
homotopy H: V
where
B,(O)
and Ho (-Lj)
x
B,(O)
[0, 11
x
-4
V
x
B,(O),
is the open ball of radius
for all y E
T
(IJ S) IJ B) t)
around the
-4
(HS (IJ S), t
*
Id B))
origin of R'+'. Then H,
=
idwv,,
For the
following it is important that H is a homotopy relativeWV,T, hence Ht(Wv,,)C WVTfor all t E [0, 1] Moreover, Ht (R) C R holds for every stratum R and every t El 0, 11, as the strata :A S are given by S x 10, 1 [- , where runs through the strata of the link L. Now let 0C E f2k (Wv,,) be a closed k-form. According to Proposition 2.3.7 there exists a smooth form W E Q'(V x BT(O) with =
x
Wv,,.
-
k* w VO) il
=
=
oc.
As doc
Vk E
T,,R
0, the relation dw (vo
=
of
a
0 Vk)
0
=
0 holds for all
tangent
vectors
We define the form 11 E f2k-I (V x BT(O)) by is the operator from the above lemma, and set
stratum R C X.
KH*w, where K := KvXBT(O),l k*-Q E f2k-1 (Wv,,). Now we claim that dp
=
which entails exactness of the sheaf sequence
neighborhoods of %. explicitly looks like:
For the
proof
of
(5.2.3)
(5.2.3)
oc,
let
(5.2.2), us
as
the
Wv,,
form
a
basis of
write down how the form KH*dw
1
(KH* dw) (y) (vi
Vk)
=
fo, (dw) (H (-Lj, t)) (k (y, t) ij E V
x
0
B,(O),
TH.vi o
VW*
*
...
)Vk E
0
TH-Vk)
T,(V
x
dt,
Br(O)).
5.3 DeRham theorems
Hereby k tangent
is the
on
partial
map of H.
orbit spaces
derivative of
173
H(ij, t)
in direction of the variable t and TH the
Let R be the stratum of ij and the v,
footpoint ij. As H(ij, t) E R holds for t E]O, 1], we must Vk) 0 by the above considerations, hence k*(KH*dw) 0 (5.2.1) of Lemma 5.2.1 entails by H*w 0 =
tangent
=
vectors of R with
(KH*dw)(-y) (vi
have
0
...
0
0 holds true. Now relation
=
do
Wil
=
k*dKH*w
=
=
k*(dKH*w
+
KH*dw)
=
k*H*lw
=
oc.
El
This proves the claim.
5.2.3
Corollary The deRham cohomology of singular cohomology.
C'-cone space X
a
canonically
coin-
cides with its
For
PROOF:
an
arbitrary topological
space Y and every k E N let
Sk(Y)
be the
free Abelian group generated by the k-simplices in Y that means by the continuous maps 9: Sk -- Y, where Sk denotes the k-th standard simplex (see 1.1.14). Together
boundary operator a : Sk(Y) -- Sk(Y) one thus obtains the well-known k singular complex (S. (Y), a) of Y. By S (Y; R) we understand the vector space of all R. from to Together with the coboundary operator 5 that is homomorphisms Sk(y) the operator dual to a we thus obtain the singular cochain complex (S* (Y; R), 5). Its cohomology is the singular cohomology of Y. We now want to describe the singular cohomology of the cone space X sheaf theoretically. Hereby we will use constructions with the
given by GODEMENT [60, Ex. 3.9.1]. If V c U c X are open, then one has a canonical restriction morphism S"(U;R) __) S k(V;R), which commutes with the coboundary operator. Thus Sk( ; R) becomes a presheaf on X. Let Sk( ;R) be its associated sheaf, and (S* ( ; R), 6) the corresponding sheaf complex. As X is paracompact, the canonical morphism Sk (X; R) __ Sk (X; R) is surjective (see [601), and the cohomology of (8* (X; R), 6) coincides with the singular cohomology. Now observe that according to [60] 8k R) is for every k a soft sheaf, hence in particular acyclic with respect to the right derived functors of the global section functor r(x; ). As X is locally path connected and locally contractible, we thus obtain a r(X; )-acyclic resolution of the sheaf of locally constant real functions on X .
-
-
-
RX
_4
50( ;R) .
__
81( ;R) .
_
...
__
Sk( ;R)
preceding theorem (5.2.2) is a r(X; )-acyclic resolution of Rx as well, singular cohomology that means the cohomology of (8* (X; R), 5) coincides with the deRham. cohomology, i.e. with the cohomology of (Q*(X), d). The canonical morphism from (fl* (X), d) to (8* (X; R), 6) is obtained like for manifolds by integration of a k-form a over every singular chain a E Sk(X)But
by
the
-
hence the
5.3
DeRham theorems
In this section
we
orbit space of
a
orbit space.
on
orbit spaces
cohomology of the complex of basic forms on an canonically mirrors the singular cohomology of the
will show that the
proper G-action
Moreover,
we
will show that under certain conditions
on
the dimensions of
174
DeRham-Cohomology
the strata of the orbit space the sheaf
complex of
to the
For the
of
case
that the
proved
ones
We
G\M
us
with
Let M be
some
E
of the basic
complex
quasi isomorphic
[104]
has
already claimed and cohomology of
coincides with the
mention that the methods used
f1kasi, JU) b
simple explanations
G-manifold
a
be the canonical
space of basic k-forms 0C
is
by
KOSZUL
are
different to
given here.
begin
5.3.1
compact Lie group G KosZUL
a
cohomology
the orbit space. Let the
complex of differential forms
basic forms.
on
the basic
complex.
which the Lie group G acts
on
projection and for
7r-'(U).
be basic. In
on
This
particular
gives
rise to
then is
cx
properly, let 7r: M be the G\M let D'.,,(U) b sheaf fl'ab ,i, on G\M. Now let
every open U C a
differential form
a
on
n-'(U),
hence
form the differential doc. As for every g E G the relation (D g* doc d(D g* Lx is true and for every & E 9 by C ARTAN i&, doc 0, the derivative di&m oc + f, &m or. one can
=
=
of
a
basic form is
d) Hereby
we
again basic. Hence eoo
G\ M
__4
f1lasic b
=
-
a
further sheaf
fl2asic b
__j
,
canonically identify
Coo(q-1(U))G'
obtain
we
with the sheaf
on
G\M:
f1kasic b
(5-3.1)
having
the sectional spaces
,
E!G\M
complex
where U C X is open.
By application of the global section functor we then obtain the so-called basic complex (Qtasi,(G\M), d). We will determine its cohomology Hb*asi,(G\M) and call it the basic cohomology of G\M. By the universal property (Kk) of the space of Khhler differentials (see Appendix the following diagram commutes: B) d_`- flb'asic (G \ M)
C'(G\M)
dJ f2'(G\M) The
morphism h' then induces
W
(Q \m', d)
:
--
(fltasi, d).
that under certain on
the level of
5.3.2
be a
a
assumptions cohomology.
Equivariant Poincarg
differentiable manifold
on
in
a
functorial way a morphism of sheaf complexes goal of the following considerations to prove
It is the on
G\M
lemma
which
a
the
morphism
(d [185,
S.
23], [6,
G-invariant closed submanifold of M. Then there exists
there exists
a
(k
-
1)-form 0
over
do oc
is
basic, then
PROOF:
one can
As N is
a
finitely many) orbits,
choose
Thm.
an
C'C
an
with
isomorphism
61, [75, 2.9])
Lie group G acts properly and let
U of N such that for every G-invariant closed k-form
If
W leads to
L :
N
Let M
"
M be
invariant
neighborhood vanishing pullback L*IX
U such that
oclu and as a
PIN
=
basic form
0-
as
closed invariant submanifold of
well.
M,
it is the union of
hence it suffices to prove the existence of
a
0
(locally
with the desired
on
properties only for glue together these
invariant
an
175
orbit spaces
5.3 DeRham theorems
of
neighborhood
a
single
orbit C N. Then
invariant smooth
one can
of
an partition unity. So help N orbit We GX. is that an we can assume identify N with generality is the where the zero section of the bundle G XG. V%, G., isotropy group of a point consider the bundle x E N and Vx the slice at x. For this following homotopy:
of
forms with the
without loss of
H
Obviously
:
G
XG
V%
[0, 11
X
-)
G
H then commutes with the
G-action,
KH*oc is
from Lemma 5.2.1. Hence
(1(9) V)b t)
V-x)
XG
well
as
as
G-invariant
a
1(9) (1
-4
-
t)V)l
the operator K
(k
-
1)-form
-
KGxG,V-,l
:=
and satisfies the
relation
dp As
t)
then
we
=
=
-KH*da +
-
N, the equality PIN
0 for all Ij E
L*oc
=
=
oc.
0 is true.
Moreover, if
LX
is
basic,
have for every & E g
i&m 0 as
oc
=
-KH*i&, 0c
0,
=
H and K commute with the G-action. This proves the claim.
5.3.3
Corollary
complex of
The sheaf
basic differential forms
n
provides
a
line
reso-
of
G\M
lution
Coo
RG\m In
particular
in
a
G\ M
the basic
f1lasic b
__
f1kasic b
fl2asic b
__4
coincides with the
cohomology
singular cohomology
canonical way. is fine and every sheaf
As Coo
PROOF:
G\ M
k
f1basic must be fine. The quence to the
5.3.4
0%,,ic one
flk.,c b
is
a
COO
G\
M-module sheaf,
equivariant lemma Of POINCARI implies that the sheaf seby a standard argument analogous
is exact. The rest of the claim follows
5.2.3.
given for Corollary
Corollary
Let GX E
G\M
be
a
point of the orbit
space and
pi) (pi, by a exists a contractible neighborhood G\U in the domain of x, in which x(G\U) is closed, and a smooth homotopy R : 0 Hilbert basis p
chart around Gx induced
X:=
x(G\U)
PROOF:
all sheaves
such that
ffoix
idx and
=
Rjjx
for
=
=
x(Gx)
=
T(V.X. )Gx
an x
the
x
.
singular
Then there
open set 0 c
[0, 11
--)
R',
R1 relative
0.
homotopy H : G x Gx Vx x [0, 11 -4 G X G,, Vx which has been proof of the equivariant lemma of POINCARA is G-equivariant and the desired homotopy R as one shows by an application of the theorem of The smooth
constructed in the induces
0
SCHWARZ.
5.3.5 Theorem Let M be
the orbit space
G\M
of differential forms
RG\M
__4
a
G-manifold
on
which G acts
properly
and
assume
that
does not possess strata of codimension 1. Then the sheaf complex on
0
G\M gives
COG\M
___)
rise to
jQ1G \M
___4
a
fine resolution
02G \M
f1kG
\M
176
DeRham-Cohomology
morphism le : (Q m, d) d) comprises a quasi isomorphism, hence and the deRham cohomology of G\M coincide. cohomology Hb%,j,(G\M) both are Moreover, cohomologies canonically isomorphic to the singular cohomology of G\M. The
the basic
PROOF: we can
RG\M.
Obviously,
all the sheaves in the sequence
(besides RG\m)
are
fine.
So,
if
yet show the exactness of the sequence, then it comprises a fine resolution of As we already have proved this property for the basic complex and as h* is a
morphism of complexes of sheaves, le would then be a quasi isomorphism. Together Corollary 5.3.3 the rest of the claim would then follow as well. Hence we only
with
(Q \M, d).
need to prove the exactness of dimensions of the strata the
First recall that
relation holds
following
Der(C'(G\M), C'(G\M)) Now choose
X'(G\M).
on
the
(5.3-7)
G\U be a neighborhood like in Corollary 5.3.4, by a minimal Hilbert basis p for T(V_') G.., like in 5.3.4 and 0 C R1 an open neighborhood of 0 such that X := x(G\U) is closed in 0. If 0 C IS'(0) denotes the vanishing ideal of X, then by Proposition 2.2.8 two f1k (G\U), if and only if for all forms w, Tj E f1k (0) induce the same form E fl'(X) x:
V1,
G\U
-
-
-
,
point Gx
=
by assumption by Proposition 4.4.9:
a
---)
R1
Vk
E
Let
G\M.
c
singular
a
chart induced
Xgoo (0) W(Vl)'*
But
by (5.3.7) X'(0) 0
which
are
is
equal
Vk)
=
to the space
71(Vl)'*
X"0(0) X
to every stratum S of X. Let
tangent
us
Vk)of all smooth vector fields
keep this
on
0
result in mind for later
purposes.
flk(G\U)
Now let OCE such that
be closed. We want to construct
a
flk-1 (G\U)
form
Lx. The claimed exactness of the above sequence then follows immedido ately. Let R: 0 x [0, 1] -- R' be the homotopy relative X:= x(G\U) from Corollary 5.3.4. Using the operator K := K0,1 of Lemma 5.2.1 we obtain =
dKTT*
+
KTT* d
=
R,
-
Ro.
(5.3-8)
w E nk (0) with x*w oc. As p has been chosen as 0 holds by Lemma 4.4.5 and doc 0. Hence there dw(O) exists a form il E flk-1 (0) with d-q (0) w (0). Set Co w dil. As TT is a homotopy relative X 0, Eq. (5.3.8) entails that for every x E X and all x(G\U) and dCojX Vl)''*)Vk(= XX 00(0)
At this a
point choose
a
k-form
minimal Hilbert basis
=
=
=
=
=
-
=
dKFI* Cv). (Vi, =
CO-M)
*
*
*
CO-(Vl) Hereby P by claim.
we
-
-
-
,
svk)(X)
-
=
-
-
-
-
,
*
-
-
-
-
-
)
vk)('X)-
have used that x* (ij
(TT*o Co R*1 Co + (KR* dCo)). (VI, Vk) (x) K CJU-(Vl) )Vk)(0) (_I)k ((TT* dCv). (Vi, Vk)) N
Vk) (x)
-
KR*Co)
ff*Vi
E
E
X' X (0),
hence
W dw. (Vi,
flk-1 (G\U), the equality do
=
-
a
,
Vk) (x)
=
0.
Defining
follows. This proves the n
5.4 DeRham
cohomology
DeRharn
5.4
of
Whitney functions
177
cohornology of Whitney functions
It has been shown in the
preceding
sections that the deRham
cohomology on cone singular cohomology. For arbitrary Whitney spaces this need not be the case. As already mentioned, this fact has been shown by HERRERA, who gave in [821 an example of an analytic variety X such that the deRham cohomology with respect to analytic functions is larger than the singular cohomology of X. On the other hand one knows by the work of GROTHENDIECK [71] and HARTSHORNE [80] on algebraic deRham cohomology (see as well HERRERALIEBERMAN [83]) that one can calculate the cohomology of an algebraic or complex analytic variety (Y, Oy) by the deRham cohomology of the formal completion of the structure sheaf Oy. This means the following. The variety Y is embedded in some C' and inherits the structure sheaf Oy Oc./g, where 9 is the vanishing ideal. Instead of Oy one now regards for natural k the sheaves 0C-/J k and passes to the inductive limit by := li1q0(C./jk. This limit is called the formal completion of Oy. Starting spaces and orbit spaces coincides with the
=
Oy
from
one
ll w
complex 6 := Q*(Oy), applies the global section cohomology of the thus obtained complex. The resulting algebraic deRham cohomology and coincides by [71] and [80] with
forms the sheaf
functor and passes to the
cohomology is called cohomology of Y. Now the reader might
the
compare the
concept of formal completion in the algebraic
case
with the construction of the sheaf of
analogy
Whitney
functions in Section 1.5.
The
of the two constructions then becomes apparent, hence the
conjecture seems cohomology of Whitney functions on a Whitney space gives back the singular cohomology of X. Indeed this will be the case for a curvature moderate Whitney space X, as will be shown in the following. Let us remark that in the proof of this theorem we will use mainly analytic as well as geometric methods. reasonable that the deRham
First we have to explain in some more detail what to understand by the deRcohomology of Whitney functions on a stratified space X with a smooth structure C'. To this end we choose a covering of X by chart domains U (Uj, xj) and consider the corresponding sheaf F-' := E' of Whitney functions of class C'. According to X'U 5.4.1
ham
=
Section 13.3
by
ax"U
we
then form the sheaf
or more
Out of f1l
of Kdhler differentials. It will be denoted
briefly by al.
one can
construct for every k E N
a
ak
further sheaf
=
8x, ,u
:=
Mal.
Its sections will be called
Whitney k-forms on X. Together with the Kdhler derivative d : jak U -- Clk we thus obtain a sheaf complex (6*, d) which after application of X, X,U the global section functor gives the Whitney-deRham complex of (X, C'): 800 (X)
We will call
--4
81 (X)
the'cohomology H d,,(X)
--
-
-
-
of this
-'
8(X)
--
-
-
.
complex Whitney-deRham cohomology
of X. For the calculation of
H ,,,,,(X)
5.4.2 Lemma Under the
into
some
let
us
first show
assumption that the stratified space X can be embedded a global singular chart x and that the covering U is
Euclidean space R1 via
178
DeRham-Cohomology
equal
f2x'
the sheaf 8'
(X, x)
to
has been obtained
be identified
can
canonically
with the chart
by pullback
x
that
with 8" Oeoo f2k X
means
QX
Here by
.
X*444
PROOF:
is
a
To simplify notation we can assume without loss of generality that X locally closed stratified subspace of Rn and that x is the identical embedding.
It
suffices to prove the claim for k those smooth functions
Then, by the
1. Let
=
Rn the
on
extension theorem
0 be the sheaf on X consisting of the germs of partial derivatives of which vanish in every order. Of WHITNEY 800 CRc',On Ix /0 holds true. By the =
second exact fundamental sequence B.1.5
m2
k
32
5.4.3 Theorem Let X be
exact sheaf sequence
hence the claim follows.
0,
(A)-stratified
an
an
0-
f2;nlX
(DeR
to Lemma C.3.3
According
thus obtains
one
El
space, which for every
m
N`0 has
E
curvature moderate control data of order M and which possesses
resolutions of class C'.
Let U be
a
locally
finite
covering of
locally tempered by chart domains.
X
Then the sheaf sequence
Rx comprises
F,1
X'U
8kX, u
That all
-d-24
are
fine is
is fine and that the sheaves
the exactness at every of the
point
8k,U
X
a
(5.4.1)
X
consequence of the fact that
8kX U
are
ak,
-%
X U
Poincani lemma for
having for every
subspace
81,U
on
X.
by Proposition 1.5.4 0, only
all F_' -module sheaves. As do d X'u
has to be shown. But this is
a
=
consequence
lemma.
following
5.4.4
m
X'U
fine resolution of the sheaf of locally constant real functions
a
PROOF:
F_'
-4
m E
forms Let X C R' be
Whitney Mo
and which possesses for every
m
curvature moderate control data
locally tempered
(A)-stratified of order (T')SES S
an
resolutions of class Cm. Let EOO
be the sheaf of
point
Whitney functions with respect to the embedding X -4 R. If x is a of X and W a contractible open neighborhood of x then there exists for every
closed
Whitney
PROOF:
form
0C
E
ak (W)
a
P
to fix notation let
Mainly
E
8k-'(W)
us
fulfilling dp
=
oc.
recall first the operator
5.2.1 which will be needed several times in the
following.
Then let
Km,t us
from Lemma
fix
an Tn
E N !2
_ for T S', 7en S-,
simplify T, 7r, p, p'S', e'S let S be contractible a c U,, Finally relatively compact open neighborhood of x and 5 a positive real number with 5 < e(-y) for all -U E U'. After these preparations we set 0 := 7c1(U.,,) n p-'([O, 5[) and W:= Wu,,.,6:= 0 n X. Note that the Wu,,,5 are contractible and run through a basis of neighborhoods of x, if (U, 6) runs through all admissible (U., 5). Now we divide the proof in several steps. and fa
=
m
+ 2. To
and TR for T5' and R
notation
we
will write
e
so on.
.,
f2kM be the sheaf defined over 0 of k-forms w such that w and dw are of class Cm and let f2k be the image sheaf of f2' under the canonical epimorphism TTL M 1. STEP Let
k
71 -
eo /]R
I -
)
k
fjF7n/R _W
_
FM (De. W 0
f2k0 It is the goal of the 1. Step .
to show that for every
5.4 DeRham
closed
do
=
cohomology
Whitney
form
Oc E
To this end
w.
of
Whitney functions
ak (W)
there exists
first choose
we
k-form,
a
a
179
Whitney
form
E
in f1k having "" (0)
w
8k (W)
such that
image under
oc as
epimorphism 7rk. To simplify notation further we suppose without loss Step that X C 0 hence S U, that X possesses only finitely many strata and that all of these are compatible with S. Finally we can assume by the assumptions of the lemma and after possibly shrinking X that every stratum R of X possesses a tempered resolution fR : MR -4 R of class C". Under these simplifications let 0 < do < di < < dd be the sequence of dimensions of the strata of X. In k-1 do. Now we first construct a form Wo E fle- (0) particular we then have dim S the canonical
of
in the 1.
generality
=
...
=
such that w
As U is contractible and 0 : 0 x [0, 11 homotopy all'y E 0. We set wo
-
a
dwo
for
tubular
Ko,,H*w
Ko,,H*dw
=
suppose
a
we are
w
-
given forms
(dwo
+
-
constructions
following
-
-
have dw E OM (S;
can
E
be
For the construction of wj+i necessary tools.
provide
some
tubular
neighborhood TRCRn
and)
=
Wi E
wo,
=
E!"=
(5.4.2)
x
is
0 is true, hence
0) fjk+1 (0)
hence
.
M
E
fl'-' (0), i M
< d such that
am (Xdi; 0) ilk (0).
(5.4-3)
.
fulfilling
gm (Xdi+l; 0) f1kM (0).
(5.4.4)
.
performed separately
ponent of a dj+j -dimensional stratum R of X, that Xdi+l \ Xdj consists only of a connected
order fn
ido and Ho (1j)
For this wo the relation
equality (Ko,lH*dw)ls
we
dwi)
+
form Wi+j E flk-1 (0), M
w'- dwi+l As the
0
=
a
M
us
look for
we
ek_,,(O). eo
U, H,
C
=
.
w':= Then
E
of U in R' there exists
Om (S; 0) f1k (0). This entails (5.4.2).
E
Next let
neighborhood x [0, 11)
H the
satisfied, as by assumption on w and 0. By da by (5.2.1) (w dwo) Is
(5.4-.2)
.
0 such that H (U
H
-
gm (S; 0) f2kM (0).
E
we now
we can assume
com-
without loss of generality
stratum R.
have to make
By assumption
(E, e, y)
for every connected
on
of R in Rn
some
preparations and
must
the control data there exists
(which
a
is curvature moderate of
which induces TR- In the
following we identify TR with TRCRn. As TR regularly situated neighborhood of R, hence 1 over by Lemma 1.7.10 there exists a function 4) E M'(aR;Rn) such that 4) TF-R/2 and such that vanishes on a neighborhood of Rn \ (TR U aR). For every form q R jai (0) then 4) Co lies in ain (Xdi; 0) f1lm(O) as well, has support W E aTrt(Xdi; 0) in TR U aR and coincides with Co on a neighborhood of R. By Corollary 3.7.4 there exists our second tool, namely a with respect to S radial vector field V: X -4 Rn that is curvature
moderate, TR
has to be
a
=
-
.
-
.
means
(7t, P),,V As
(5.4.5)
a third tool we choose a collar k (kj, k2): M'R c MR --) NR X[O, 1 [ and a decreasR -4 [0, 11 which is identical to 1 on ing smooth function 00, 1/21 and which vanishes on [3/4, oo[. Denote by R]O,t], 0 < t < 1 the open set fRk-1 (NRX10A) C R. These data then induce a homotopy H x [0, 1] by
(X, t)
=
_4
fRk-l(kif X,
R
1(x),(t*(k2fR1(X))+(1_'P(k2fR1(X)))k2fRVN )
I
if
x
else.
E
R]0,1],
180
DeRham-Cohomology
Moreover,
R
H(x, t)
H(7rR(x), t)
=
can
be extended to
for
As fR
TR.
E
x
H
mapping
a
:
7rR is
,
x [0, 11 TR U by requiring tempered relative aR of class Vn,
H : TR x ] 0 1] has to b e tempered relative a R of class e'. By w' E gm (Xdi; 0) Q1 (0) this means that the form w +j (KTR,l H* W') lies in gm (Xdi; 0) flk-1 (0). y
.
According to the definition of H hence by Eq. (5.2.1) and dw'IR
the relation
Ho(R]o,1/2])
C aR
holds and H,
=
7rR,
0
=
dw!2+1
7r*WIIRO,,/2,)
(5.4.7)
R
JR]O, 1/21
E am(Xdi U R]o,1/2]; 0) f2,, (0). Next we consider the homotopy TR) (X)t) --i OR(t%l(x)). Then Fo 7rR and F, idTR are true. Using Eq. (5.2.1) again we have for Cui+l := (P (KTR,,F*w')
dwi'2+1 -7r*Rw'
so even
F: TR
-
M
[0 '11
X
=
=
-
d(bi+,IR As F is
tempered
dCoi+l [0, 11 --4
-
(5.4.5) ,y,
+
R is
and
E
gm(Xdi+l; 0) the
given by
integral
V is controlled
as
.
over
-
I
(5.4.8)
JR'
jm(Xdj; 0) f2-1(0),
lies in
-
and
M
true. A third
homotopy
G
:
TR
X
flow -y j -4 X of the radial vector field V. By X \ S the relation J., D [0, p (x) [ holds true and
continuous function
a
7r*RW
ilk,n (0) holds
precise definition of G
For the
WIJR
C"n, (bi+l
relative aR of class
7rR*w'
be extended to
can
7r(x).
w
=
J,,
on
choose
now
U
fp(x)l by setting -yx(p(x))
smooth function
a
K
:
R
-4
[0,
1
0 for x E R]o,1/4]. Then we set 'Y(X, K(X)P(X)) E R]0,1/2[, where K(X) G(x, t) ='Y(7rR(X), tK(X)P(X)) for (x, t) E TR x [0, 11. Hence Go 7rR follows as well, as Gj(TR) C R]0,1/2[ and GtIR]0,1/41 idR]o,1/41. Thus w '+ 1 := 4). (KTR,,G*(nR*w'- dwi'Z+,)) is a form of g, (X1j; 0) f2k-1 (0). Moreover, by Eq. (5.4.7) w ..-, '1 satisfies the relations
such that
=
=
z
.
dw
G *1 (7rRw'
2+ 1IR
anddw ' -dw 2+1
dwi'+JJR
.
dwi+,IR
W'JR we
7rmk-l(WO
:=
1.
one
COi+j
+ W,i+1
-
sets
(5.4.9)
now
W11 i+1)
so
-
d(wo
+'*'+ =
oc.
+
Wd)
*
*
Step
we
RX
+
Wd)
E
Om(W; 0) flm (O). *
E flk-1 (W) Whitney form. Then Step. 8'-module sheaves, they are in particular fine,
be the induced
are
obtain
___
80 (X)
a
80
fine resolution
__4
81
M
Thus, by the considerations complex M
*
This finishes the 1.
2. STEP As all the sheaves
by the
If
7rRW'IR
have
follows, hence do hence
%
far, in particular Equations (5.4.7) to (5.4.9), entail that additionally (5.4.4) hold true. Hence the inductive step has been
and
w
Let
dwi+llr
M
then the considerations
finished and
dwil+,)IR
+7rRw'E gm(Xdi+l; 0) f1k (0).
2+1
Wi+l
=
(7rRw'
in the
-4
8k
M
proof of Corollary
81 (X) M
___4
__4
M
.,.
__
5.2.3 the
8k (X) M
__
...
cohomology
of the
5.4 DeRham
of
cohomology
coincides with the
Whitney functions
singular cohomology
set W C X there exists for every closed E
8k-l(W)
with
M
3. STEP Now
dp
of X.
Hence, Whitney form
over
0C
any open contractible
E8k (W) Whitney M
form
Lx.
=
to prove the claim.
going
are
we
181
form of class CI
Let
a
E
8'(W)
be
closed
a
open and contractible set W C X.
over an Whitney According to the 2. Step there exists for every natural m > 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
<1
dVmjjKm,m-2
2-*
Now set
V= P2
+
E(P.
-
P,1
-
d-V.)-
m>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
<
we
obtain the
(9 aa(k)
a,(O) (9 ct,(I) 0
aETk+l k-I
sgn((T)(ma,(o)(DC cr(l)(2)Cta(k)+E(-1)1+lM(&Cta(O)(D**'(&CLa(l)(Daa(k) 1=0
(TETk+I
+
F-k(mb
0
(_I)k+1MCLU(1) ak)
(9
0 a2 (9
c
0 a,,(o) 0 CL(3-(2)
CLa(k)
E sgn(u)m 0 a,(o) (9 a,(,) (2)
+
0 aa(k)
IESk+I
k-1
+E(-1)1+1 r-
M(DCLU(O)(D**'(DcLu(,)au(,+,)(2)
...
0 CLa(k)
ESk+i
1=0
F-k(MC
0 b 0 a2 0
ak)
0
1: sgn(a)m (9 aa(k) (9 acr(O) 0
(_I)k+l
(2) CLCT(k-1)
-rESk+l
a_1(O)
F-k(Mb
0
(9
(9 a2 0
C
ek(M 0 bc 0
a2
ak) + ek(mc (9) b (D ak)
a2 0
...
0
ak) (6.3.3)
that
proof
For the
[e] k
is well-defined it suffices to show that for all f 1,
Derc (A, M) the continuous mapping Hochschild cocycle. We calculate
Af
:=
pkEk(fl
ek (f 1 0
*
*
0 f k)
&
*
0
fk) A6,, :
by using
-
-
-
1
aA
fk E
M is
the fact that all f 1
a
are
derivations:
pkEk(fl
fk)(al
(9
(2)
ak+l)
+
(-l)'al+,Af (a,
(9
0
0
ak+l) k
a,Af ((12
0
(2)
(-l)'a,Af (a,
(2)
...
(2) a,-, (2) a1+1 (9
0
ak+l)
k
+
a
0 a, 0 a1+2 0
o
ak+,)
+ ak+1 Af (a, (9
...
(9
ak)
0.
=
By
...
similar argument the
morphism 174k
is well-defined
(cf. [112,
Lem.
1.3.14]).
finitely generated as A-module. M and finitely generated a projective topologically a topological algebra Then M is the topological direct summand in a topological A-module of the form For the
A ftl,
proof of the last part of the claim
suppose that A is
hence it suffices to prove the claim for the
6.3.6 Lemma For every
gebra
we
case
that M
A. Now
we
have
topologically finitely generated complete locally convex finitely generated as
A the space Der, (A, A) of continuous derivations is
A-module.
=
alan
Homology
194
PROOF
of Algebras of Smooth Fbnctions
By assumption on A there exists a finite system (el, subspace generated by the polynomials in the el derivation 5 E Der, (A, A) with 6 (el) :A 0, then set
OF THE LEMMA:
of elements of A such that the in A. If there exists
a
81 that
Suppose
one can
=
6 (el)
construct
and otherwi se
5,
51,
51
=
,
-
-
,
en)
0.
51 such that 6k (ei)
-
-
is dense
=
0 for i < k < 1 and
such that for every continuous derivation 5 there exist coefficients a,, such that , al 5 Sr a151 -. alb, vanishes on all ei, i < 1. If there exist 5 and a,, a, with 6r(ei) 0 for i < 1 but 6,(el+,) =A 0, then we set -
=
-
-
-
-
.
.
=
61+1 In every
case
-
-
-
,
br(el+,)
and otherwise
5,
easily that the system 51, polynomials in ek are dense in A,
51+1
checks
one
hypothesis. As the 5 1,
=
=
0.
61+1 fulfills the induction recursively defined family
the
5,, generates the A-module Der, (A, A).
PROOF
6.3.5,
OF
k 7t
H
:
k
By ek
CONTINUED:
[Ef(J) (D
Ba
(A' jq)
q'
k
,
0 the map
=
...
(9
f(k)]
-4
E f (1) A
A f (k)
image of 7t' lies
it remains to show that the
continuous, hence
is well-defined and
P"
o
in
-k
A Der, (A, M). By
6.3.3 the
--4
a
mapping
E E f (,(,)) (a) 0 f (,(2)) (a2) (9
(9 f (a(k))
(ak)
aESk
is
a
continuous derivation for every Hochschild
cocycle
f(l)
(9 f (k) and all
(9
ak E A. Next we need a denumeration of the set I
consisting of all k-tuples i2 give I the structure of ik) (ii, a well-ordered set. One obtains such a well-ordering by defining (il, ik) as smaller < if but for holds k. Let cy.(j), 1 than (ii, < some it-, it j1_1 j, ik), ii ii, < the element of and for be e (D cy.= i-the 1, e, i e,, (oc 04) G I(") 04 k We now define recursively a2,
of the form ii
<
<
...
ik. For that it suffices to
<
-
-
-
,
-
-
=
-
-
-
=
-
,
)
:=
A,
=
Af (e, (9
A.(j+,)
=
A.(j)
where Af
f, A
=
...
A
+
...
(2)
eA;)51
(Af (e,,(j+,))
fk. By
an
A -
-
-
*
...
A
5k)
Ax(j) (e,,(j+,))) 5,.,(j+,)
induction argument
one
A
...
A
5,,,(j+,),
concludes that
A, (.)
=
Af.
k
k
As A
,.(.)
lies
by
construction in
A Der, (A, A),
the
proposition
now
is proven.
El
k
_k
Corollary Under the assumptions of the proposition MaAfl /k is in a natural way a topologically direct summand of Hk (A, X). If A isfinitely generated as a topological algebra and M a topologically projective finitely generated complete locally 6.3.7
k
convex
H k(A,
A-module, '
M).
then
A Der, (A, M)
is
a
canonical
topological
direct summand of
6.4 Hochschild
homology
Hochschild
6.4
195
of algebras of smooth functions
of
homology
of smooth
algebras
functions First let
6.4.1
functions k-forms
Now let
M
topology on the algebra C'(M) of smooth the natural Fr6chet topology on the space a' (M) of
recall the natural Fr6chet
manifold M and
on a
over
us
(see Appendix C.1).
suppose that there exists
us
global
a
nowhere
vector field V
vanishing
on M. For example this is the case, if M is an open set of some Euclidean space or if M is compact with Euler characteristic 0. Under these assumptions on M ALAIN CONNEs has constructed in [451 a topologically projective resolution of C'(M), which
computation of the continuous Hochschild (co)homology of C'(M). In the following we will give the construction of the resolution according to CONNEs [45] and which essentially relies on the Koszul complex. To this end let us choose on M a torsionfree affine connection and denote by exp, the corresponding exponential function. Let X: Mx M -4 [0, 11 be a smooth function such that exp '(x) is defined for all (x, -U) E supp X and such that X is identical to 1 on a neighborhood of the diagonal. bundle over M x M, which can Moreover, let Ek pr2 (AkT(C*M) be the complex vector be obtained by pullback of the k-th exterior product of the complexified cotangent allows the
=
bundle
E*1
TZM
is the
projection pr2: M x M -4 M onto the second coordinate. Then bundle of the complexified tangent bundle TCM via pr2, hence
via the
pullback
Z(x, ij) defines Now
smooth section Z of
we can
6.4.2
sion
a
n
-
comprises
a
:
0
E
CI(E*1)
It does not vanishes outside the
diagonal
of M
x
M.
JZ4
C' (M
x
M, En)
-!IK4
C- (M
x
M, Ei) -% ff (M
topologically projective Hereby iz
embedding of M
to CBfrom C" M M
as
rIB-
_eoo(m )
smooth manifold of dimenM
--
TM. Let the bundle
be defined like above. Then the sequence
--->
C1 (M) 6,C' (M).
the natural
(6.4.1)
x(x, ii)) V(ii)
Proposition (CONNES [45], p. 127ff) Let M be a with a nowhere vanishing smooth vector field V :
C" M
M)
El*.
exp l (x) + i (1
formulate CONNES' result.
EA, and the section Z
-
X(x, -y)
=
the
over
...
x
M) -A-*-
C' (M)
resolution of the module Cl (M)
means
over
the canonical insertion of Z in
diagonal
of M
x
0
---
a
M. A continuous chain
the identical map is
given by Fk
:
C' (M
x
form and A
homotopy
CCKk M,
__4
CBar
M,k)
where
FkCV(7-)1J)XW- )Xk):-:::: (Z(Xl)IJ) for x, -y, xi (E M and
w E
A- A
Z(Xk)IJ))(-V(X)IJ))
(6.4.2)
CIK
Kk
Coo (M X give the proof like in [45]. Every one of the modules Mk in hence particular topologically projective. M, Ek) is finitely generated projective, is a topologically projective resolution 0, hence for the proof that CIK Obviously Vz M _4 CCK.+J Of 1Z. CCK" we only have to construct a continuously linear "section" S M, M,
PROOF:
We
now
=
196
Homology
To this end
choose
we
the support of X,
X'
=
a
cut-off function X, X'
1
on a
Z(x, -y) Then there exists
a
For
M, Ek)
C1 (M
w E
that
wx(ij)
=
x
Hereby
0 is
define
and
w(x,u) for all 'y
an
exp,-,'(x)
=
=
open
E
According
neighborhood
t
dt H*X
But then
lution of Next for
d(izxil, )
Sk-l!ZxWx
C'(M)
on
X'. =
diagonal
1
expX
of M
x
(1g)),
(x
-LJ
E
0
M with supp X, C 0. Now
w'X A wx,
+(I-X
DH-,(ij,t).-! at
by using
t
+
X
(6 k Fk W) (X) 1J) X1)
fo
izx
+
-
W
E
Z(x,V)
=
C(Ek)
(6.4.3)
obtain for all
we
tZxSkWx
=
dt
H*X
d7l,,
Km,l
t
w, Hence
C". M,
H*X 71-, + H* X
is
Km,l i1x
-q,.
=
topologically projective
a
reso-
indeed.
must show that
we
Coo (M
W E
1
=
with suppil C 0
1
fo,
X'
M and such that
x
one
of the
H* (d(Xxwx))
to Lemma 5.2.1 and
C'(E,,)
such that
of M
E supp
Ht(x,ij)=Hx,t(-y)=exp,
by
0
11 E
(X, -U)
[0, 1]
dt
f
X
--+
diagonal
x M, Ei) with (w', Z supp (I-x) 1. denotes by wx the form E nk (M, (C) such M. We now define a homotopy
M
E
X
1
Sk (W)x
for all
M
x
smooth section W' E C`0 (M
H:Ox[0,11-- M
we can
neighborhood
M
:
of the
of Algebras of Smooth Functions
by F. there
is
given
a
chain map. To this end
compute
we
M, Ek): *
*
*
)
Xk-1)
Fw (x, -ij, x, x,
=
...
)
Xk)
k-1 -
E'(-l)' Fw (x, ij, xi,
-
-
-
)
Xi) Xi)
*
*
*
)
Xk-1)
+
(_l)k Fw (x,
U, x1,
-
-
-
)
Xk-1)
1)
i=1 =
(Z(x,ij)AZ(xl,lj)A'**AZ(Xk-l,IJ),W(X,IJ))
=
(F iZW) (X) 1J) Xb
hence F is
a
*
*
*
)
Xk-1)
chain map.
According
El
to the work
[165] of TELEMAN and [182] Of WASSERMAN it is possible algebras of smooth functions the "global" Hochschild homology from the "local" homology. In the following we will explain this in more detail according to [165]. To this end let X be an (A)-stratified space which is embeddable in a geodesically complete Riemannian manifold (M, R) and assume X inherits the smooth structure from M. Let d,, be the geodesic distance on M and x: R :' --) [0, 11 a smooth function with support in [0, 11 and X 1 over [0, 1/2]. Then the function to calculate for
=
6k
:
Xk+1
__4
R :O, (X0,... Xk)
is smooth and
)
C124N, X0 + d2tL(X1) X2) +
the distance to the
diagonal,11k+1 T-neighborhood of the diagonal I (xo, x1,
measures
U, the so-called
--i
According to TELEMAN following properties:
5k,r
:=:
XT
'
2
2
dR(Xk-1) Xk) + d L(Xk) XO)
of Xk+1
For
T >
0 denote
by
Xk) 5k (XO) Xk) < T1. 5k with X,(t) X(r/t) has the *
the function
+
*
*
*
)
'
*
)
=
6.4 Hochschild
of algebras of smooth functions
homology
commutes with the Hochschild
(1) Multiplication by 5.,, comprises
a
(2)
The support Of
(3)
The map Now
c
one
in the
cyclic homomorphisms
5k,-r
decompose
can
EX6C00(X)CBar X,k
boundary operator,
of CONNES
as
6.,,
[45].
lies in U,.
is identical to 1
5k,r
sense
197
over
the
neighborhoodUT/2.
every Hochschild
in the form
5k,rC
C
(1
+
-
cycle (or analogously
every
cocycle)
Then the second summand in
6k,,)c.
decomposition turns out to be acyclic, hence does not contribute to homology. precise we define for a complete locally convex C'(M)-module NE the following complexes: this
To make this
C X"
X
The inductive limit with
-r --)
Theorem 3.2 of TELEMAN
Localization theorem
6.4.3 are
[165]
NC CX, 'C,*:=(1-5.,,)-Home.(x) (CB-) X
and
r
Now, by by Co,m,. resp. Cx,m,*. X 0 immediately the following
0 will be denoted we
obtain
(cf. [165,
See.
3])
The
complexes
CO,M,.
and
X
Cx,M',* 0
acyclic. The localization theorem entails that for every
locally
finite
covering
(Uj)jErq
of
X and every Hochschild cycle c E X OC"(X) C"k there exist Hochschild cycles cj X, such that c and E cj are homologous. An analogous arguwith support in
Uj ll
ment.obviously holds for Hochschild cocycles. (co)homology of algebras of smooth functions functions on locally closed stratified subspaces Next
we
consider the
6.4.4 Lemma Let M be
Then every Hochschild metric
Thus the can
computation of Hochschild
be reduced to the
one
of smooth
of Rn.
antisymmetrization operators of Proposition 6.3.4. a
manifold and JVC
(co)homology
a
complete locally convex eOO (M) -module. can be represented by an antisym-
class in JVC
(co) cycle.
PROOF:
By
the localization theorem it suffices to prove the claim for the
case
that
according to Proposition 6.4.2 we have for e'(M) the Let [c] and the chain map F: Co'. -- C` topologically projective resolution CII. M'. M, M, Then there be a Hochschild homology class in M, represented by the cycle c E Cpar M k. exists an element W E C!00(M x M, Ek), such that c and Fw are homologous. But by definition of F Fw is antisymmetric and represents [c] by 6.4.2. Hence the claim holds D in the homology case; analogously one shows the claim in the cohomology case. M is
an
open set in R. Then
.
a manifold and NC a complete locally convex Is'-module. fl* (M) (&.(m) M. Moreover, isomorphy H. (CI (M), M) if M is finitely generated projective, then H* (e" (M), M) el (M, A*TM) ae. (m) M. fl* (M) and H* (e,00 (M), C!" (M)) In particular H. (e' (M), 12' (M))
6.4.5 Theorem Let M be
Then
one
has the canonical
=
=
=
eOO (M,
A*TM).
of Algebras of Smooth F znctions
Homology
198
The claim follows
PROOF:
6.3.5, if fields
immediately from
recalls that the derivations
one
on
the
preceding lemma and Proposition are given by the smooth vector
&O(M)
M.
on
Cl
goal of the following considerations to transfer the results about the homology of smooth functions on manifolds to certain cone spaces of class C1. To this end we consider a compact (A)-stratified subspace L C S` with the smooth structure inherited from S` and a smooth manifold S, on which there exists a nowhere vanishing vector field V. Additionally we assume that the cone CL C Rn It is the
Hochschild
is
Let X be the stratified space S
convex.
structure. Now
want to transfer the
we
defined above to the stratified
CCK X,k Each
one
Next
we
and
S
7T :
x
Rn
element
an
6.4.6
w
exponential
a
Cc' and obtains Let
CCK X
:
on
projection. One can insert the element izcu E Cc' X,k-I
L, S and X be
morphism Fk
:
Cc'
X,k
Xk)
FkW(_X)1J)'Xb
:_=
a product product on Rn,
vector field Z into
as
above. Then the sequence
(CXn
A continuous chain map the
by
cK
0
over
X
e -(X)
X)
resolution of the module
topologically projective
a
X)-module.
x,y E X.
S and the Euclidean scalar
__ 4 C ]'J __ 4 e00(X comprises
x
which is defined like above
an
X,k
Proposition
el(M)
function of the Levi-Civita connection of
Riemannian metric
S is the canonical
--
G
finitely generated e"(X
Rn)
=X(x,ij)exp 1(x) +i(1-X(X,1J))V(7t(1J)),
exp is the
by
resolution of
ff(X)aff(X, AkTC*W).
of the spaces Cc' then is a projective X,k a vector field Z : X x X -- T (S x
metric formed
R1 with the induced smooth
x
topologically projective
need
Z(x,ij) Hereby
CL C S
We define:
case.
:=
x
the
identity
_4
0
121(X)
over
CI(X
r-lBar
from C" to CBar X X
-C-(X)
is
x
X).
given by
CB,k with
__4
X,
(Z(-x1),x) A
A
Z(Xk) X)) W(X)
X) 1J) Xi E
X)
W
EC XCIK,k*
(6.4.5) only the
proof can be performed almost exactly like for Proposition 6.4.2. One regard that the there defined homotopy H is compatible with X, which by
The
PROOF:
has to
convexity assumption
Analogously
on
like for manifolds
6.4.7 Theorem Let X be is
an
(A)-stratified space
a
we now
obtain the
El
following
stratified space of the form S
x
result.
CL,
where L C Sn-1 c Rn
with the induced smooth structure and such that CL C Rn is
complete locally convex topological C' (X) -module one has the Q* (X) ae. isomorphy H. (C' (X), M) (x) M. Moreover, if M is finitely genA* Der (C,' (X), M) holds. In particular one projective, then H* (C' (X), M)
convex.
Then for every
canonical erated
CL C R' is satisfied indeed.
has H. (Cl (X), C' (X))
=
=
=
Q* (X) and H* (E!"O (X), C,00 (X))
=
A*
Der(C- (X), C- (X)).
6.4 Hochschild
homology
of algebras of smooth functions
199
closely connected to the theorem of the for that TELEMAN piecewise differentiable functions algebra [165] by proven on a simplicial complex the Hochschild homology is isomorphic to the piecewise differentiable forms. By this and the result above we conjecture that a HOCHSCHILDKoSTANT-RoSENBERG theorem holds for arbitrary cone spaces of class C'. More6.4.8 Remark The last result of this section is
over, it remains to
clarify
the connection to the work B RASSELET-LEG RAND
[23].
Appendix
A
from linear
Supplements
and functional
algebra
analysis
The vector space distance
A.1
give a natural distance function on the GraflMannian Grk (Rn) of subspaces of Rn. To keep notation reasonable we mention at this will always denote in this section vector subspaces of Vk)' V, W, V1,
In this section
we
will
k-dimensional linear
point that Rn.
The vector space distance of V and W is defined
dGr(V)W)
by
infJJJv-wJJJWEWJ-
SUP vEV, JIV11=1
general dGr is not symmetric in V and W, hence does not comprise a metric on subspaces of R. But we will see that the restriction of dGr onto any GraBmannian comprises a metric. Denote by Py the orthogonal projection onto V with respect to the Euclidean scalar product on Rn, PW the orthogonal projection onto W and QW idRn PW the one onto the orthogonal space W'. Then one can write the vector space distance in
In
the set of all
=
the
following form using the operator
norm:
dGr(V)W) A.1.1
Proposition
(1) dGr (2)
takes
The relation
only
(3)
If dim V
(4)
For all
=
IIQWPVII-
(A. 1. 2) following properties:
from 0 to 1.
dGr(V) W)
if V n W-L
=
The vector space distance has the
only values
=
0 holds if and
only if V
C
W, and dGr(V) W)
=A {01.
dim W, then
V1) V2) V3
the
-
dGr (V) W)
=
dGr (W) V)
triangle inequality
holds:
dGr (V1) V3) :5 dGr (V1) V2) + dGr (V2) V3)
M.J. Pflaum: LNM 1768, pp.201- 203, 2001 © Springer-Verlag Berlin Heidelberg 2001
=
1 if and
202
Supplements
(5)
If V1,
Vk
are
from linear
algebra
and functional
analysis
pairwise orthogonal, then
dGr (V1
ED
*
*
Vk) IV)
ED
*
<
dGr (V1) W) +
dG, (Vk) W)
+
PROOF:
The properties (1), (2) and (5) follow immediately from (A. 1.2). Under the dim W the relation dGr (V) W) 0 (resp. dGr (V W) assumption that dim V 1) is 0 (resp. dGr (W) V) by (2) fulfilled, if and only if dGr (W) V) 1). That (3) holds for 0 < dGr (V) W) < I as well, is a consequence of the following consideration. If =
=
-=
Y
=:
dimV
=
dimW
dGr (V) W)
then choose
=
the relation PWv
the
:A
=
spanv and W
dGr (W) V) fOllOWS- If
=
unit vector
a
(w, v) 1. By
hence V
1, I (v, w) I
=
V
E
V with dG, (V,
same
jjvjj
!
11W
-
W)
=
I 1v
(PVW, W) 11pVW11
PVWII
I (PVPWV' V)
j(w,v)j
11PWV11
jjwjj
=
the other hand dim V -
Pwv 11. By 0
=
>
11pVW11
=
<
=
=
1, then
dim W >
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
<
adjoint operator of g*g is a selfadjoint,
A.3
Topological
tensor
products
203
positive definite operator on V. We now define smooth functions UN : UN -4 GL(V) : UN -4 GL(V) with the desired properties. To this end let us first determine the Taylor-coefficients of the analytic function hN : BN(0) -) R O z 1-4 Vz_+N around and SN
the
origin; they
are
given by
k
hkN
i= 1
Q
i)) N
-
2
1/2-k ,
k E N.
By definition of
UN SN,g
=
hN (g* g
-
N
hkN (g* g
idv)
-
N
idv)
k
k N is well-defined for all g E
As hN
values in the set of
only
assumes
UN and depends analytically, in particular smoothly
positive
(SN,g )2
definite. By definition of hN the relation
operator UN,g
:=
9 S-1 N,g depends analytically
U*N,g.UN,g that
means
If we
=
(SN *'grl
9
9
real
=
g.
numbers, positive g* g holds as well. Moreover, the
=
g and satisfies
on
SN,g
on
SN,g must be
(SN,g )2 S-Ig N,
SN,g
=
idv,
UN,g is unitary. Thus we obtain the polar decomposition 9 UN,9 SN,gyet show the uniqueness of the polar decomposition for every g E GL(V), =
can
(resp. SN)
then the functions uN
have to coincide
thus define the desired functions
u
and
on
the intersections of their
So let u, u' be
s.
unitary and
domains,
s, s' be
positive
definite such that g u' s. Then g * us S U:-1 us s ic-1, hence S2 (us)* S12 has definite As definite one a only positive g* g positive operator square root, u'. This proves the claim. s' follows, hence u s =
=
=
=
=
=
=
.
=
=
A.3 One
Topological
tensor
consider many different
products the tensor
product VOW R or k locally convex (with C) such that these topologies are induced by the ones of V and W. The most natural one is the 7t-tOP010gy that means the finest locally convex topology on VOW such that the natural mapping 0 : V x W --i V (D W is continuous. V & W together with this topology will be denoted by V&7r W, its completion by Va,W. The 7r-topology has the following compatibility properties: can
of two
locally convex topologies
k-vector spaces V and W
(TP 1)
0
(TP2)
For every
:
V
x
W
--i
k
=
on
=
V 0 W is continuous.
pair (e, f)
E
V'
x
W' of continuous linear forms the
mapping
vow -4e(v)f(w)
e0f:V0W--4k, is continuous.
The
here,
(see
7r-topology
then is the strongest and the
is the weakest among the
GROTHENDIECK
[70]
or
topologies
TRhVES
At the end of this section let convex
topological
category
in the
us
sense
of DELIGNE
for
details).
note that the
&, [50].
vector spaces with
on
[171]
e-topology, which will not be explained compatible with'o in this sense
V0W
as
tensor
category of all complete locally
product functor comprises
a
tensor
Appendix
B
Ka**hler differentials
The space of Kiffiler differentials
B.1
Let JZ be
B.I.1
5
:
A
--)
a
commutative
A derivation
module.
M, such
A
on
that 5 (ab)
over
ring, A
an
3Z-Algebra (with unit),
T with values in M then is
an
and M
T-linear
an
A-
mapping
ab (b) + 6 (a) b for all a, b E A. The space of all such
=
by DerjZ(A, M). By the space of Kdhler differentials of A over JZ one understands an A-module QA1jz together with a derivation d: A -- f2A/,p called Kdhler derivative such that the following universal property is satisfied: derivations will be denoted
,
(KA)
For every A-module M and every derivation 5
3Z-Iinear
mapping i5
:
f2A/-%
:
M such that the
-4
A
-4
M there exists
a
unique
diagram
5
A
-_
M
dj Z f2A/JZ commutes.
The pair
(f2A/jz, d)
given by
the
is determined uniquely by following proposition. Thus
Deriz (A, M) B.1.2
Proposition
Let A be
=
this universal property; its existence is
HomA (nAlp,, M).
(B.1.2)
9z-algebra. Then the space f2A/JZ of represented by either of the following
commutative
a
K,ihler differentials of A exists.
It
be
can
spaces:
(1)
Let Q be the free A-module submodule
generated by
d(Aa + gb)
d(ab) Then
f2l A/3Z
WO
over
the
symbols
da with
a
E
A,
the relations
-
-
Ada
adb
and d: A
--
M.J. Pflaum: LNM 1768, pp. 205 - 208, 2001 © Springer-Verlag Berlin Heidelberg 2001
-
-
JZ,
Ldb
=
0,
;k,
bda
=
0,
a, b E A.
QAIR,
a
-4
da +
L E
a, b E
A,
and 9 the A-
206
(2)
Kiffiler differentials Let B be the let
B
z :
-4
ring A o9z A. Give B the structure of an A-algebra by a -4 A be the homomorphism (a, b) -4 ab. If 9 denotes the ideal
then the A-module f1lA universal derivation is
associated to
E
(3)
bj
aj 0
Let 9Z
=
A
:
PROOF:
A
A
field,
a
local
a
Then
splits.
m/m2,
-4 a
a
Q,1,j/,,Z.
-4
differentials, where the morphism i6 : QAII z -4 M
The
M in the universal property has the form
--i
With maximal ideal m, and
k-algebra
f'A/k
:
A
--i
k
a
=
M/M2
is
k a
0
-4
module of Kihler differentials for A
+ M2 its universal derivation.
j (a)
-
A
-4 m -4
B.1.3 Remark It follows
by Proposition 13.1.2 that every element with finitely many aj, bj E A. aj
E,
of
fIA13Z
can
be
dbj
First fundamental exact sequence Let B
commutative
j
[125]
MATSUMURA
written in the form
13.1.4
ker F-,
space of Kghler
a
such that the sequence
is exact and -4
:
1 and
=
E aj 6 (bj).
F-4
0
and A
forms d
given by
derivation 6
a
k be
morphism
J/J2
=
9Z
a0
9
Then there exists
9Z-algebras.
an
A be
--
homomorphism
a
of
exact sequence of A-modules of the
form
where
oc(db 0 a)
PROOF:
=
adb and
WEIBEL
fIA/JZ 04 f2A/B
0'
OB A
f2B/JZ
0(da)
=
-4
0)
da.
[183,9.2.6]
Second fundamental exact sequence Let 3 C A be 9Z-algebra A. Then the sequence
B.1.5
ideal of the
an
com-
mutative
j/j2-54 f2A/ z (DA AIJ a) fl(A/J)/9Z is exact, where 5 PROOF:
13.1.6 where
:
J/J2
WEIBEL
Starting
--)
an
from
f1A/9z
fl /Oz
-
d
da 0 1.
a t-4
one can
build the k-th exterior
flAl_%.
:
A
---)
we
flAl_%
call
as
has
a
The direct
product
Q'/
sum
usual the exterior
unique extension
'j a =
=AkflA/_T'
(D k cri Qk Al j Z
algebra of A. Moreover, to an 3Z-Iinear mapping
such that
d(ocA and d
is the A-module map
A and
-4
=
AIJ
El
A-algebra which
the Kdhler derivative d
d:
OA
[183,9.2.7]
obviously
then becomes
QA1_%
0
--
daA
0. Thus
( fl
d
The reader will find tioned literature.
we
0
+
(_j)k aA dO
oc
obtain the so-called deRham
A
do
proofs
d'
1
jaA
T
2
E
flk4/jz,
complex
d2
fjA
and details about the exterior
0
E
f2,1,q/a
of A: dk
k
n!
Z
algebra
in the above
men-
B.2
Topological
B.2
version
207
version
Topological
R or C, and case that the k-algebra A is defined over k complete locally convex topology in the sense of Section 6.1. Then there exists for every complete locally convex A-module M the space Derc (A, M) of continuous derivations from A to M. The functor M -4 Derc (A, M) is representable as well that means there exists a uniquely determined complete locally convex A-module f2A/k together with a continuous mapping d: A - f2A/k such that We
now
consider the
that it carries
=
a
HomA (?!Alk, M)
--)
Der, (A, NQ,
1
F-4
i
-
d
isomorphism. Note hereby that HomA (-, -) means the space of continuous morphisms. Applying Proposition B.1.2, point (2) we obtain a simple form, in which ?!Alk can be represented. First set f2A/k J/J2' where 9 and J2 are the J2 in the ideals 9 and the of tensor product A6,A. closures completed topological Then the Kdhler derivative d: A 4 f2A/k " ?!Alk becomes a continuous mapping. Now assign to every continuous derivation 5 : A -- M the continuous mapping is
an
A-linear
-
=
i5 : A6,A
M,
--
E aj 0 bj
aj
5(bj),
jEN
i6 d holds by the universal property of the 7r tensor product. Then 5 this the determined other words defined In thus by property. uniquely f2A/k represents the functor Der, (A, -) indeed. We call fIA/k the space of topological Kdhler differentials. The topological Kdhler differentials of order k are given by the elements which exists
=
-
and i5 is
-k-'
-k
of the space
fl /k
=
A
flA/k,
where for any
space V the symbol XkV C Va, completed k-th 7r-tensor product. -
B.3 B.3.1
Application Next
we
consider
a
to
-
-
aj
complete locally convex topological vector
means
the
locally ringed
commutative
closure of AIV in the
topological
locally ringed
spaces
space
(X, 0)
over
k
=
R, C.
build for every point x E X the module Q., := flO /k Mx/M!, of Khhler differentials of the stalk 0,,, where m means the maximal ideal of 0.,. Give
Then
we
=
t(OX,k)
U
DO./k
XEX
the finest
topology
such that all
T: V are
mappings of the form -4
t(f2X,k),
x
-4
d[flx
Hereby V C X runs through all open sets in X, and f through all algebra O(V). Then't(f2X,k) becomes the espace 6tal6 of a sheaf which is called the sheaf of Kdhler differentials on (X, 0). (concerning
continuous.
elements of the
2X/k
on
X
details about the espace 6tal.6
of the exterior
product
see
GODEMENT
[60,
and the exterior derivative
Sec.
are
11.1.2]).
As the construction
functorial in A resp.
QA/a,
we
208
Kihler differentials
obtain in
a
precisely
one
natural way the sheaf constructs first
inkX/k(U)
:=
flxk/k
of k-forms
presheaves f1kX/k
A k (QX/k(U)) and
and
X and the sheaf
on
fl /k
Q /k(U)
X
on
:=
jQ /k'
by defining for Qk
(
More
all U C X
(U)
X/k
kEN
The associated sheaves then derivative Q*
X/k
-->
the sectional spaces d
on
fl*
give the desired sheaves f2kX/k and
X/k
.
Altogether
Q /k
:
fl /k provide
-4
thus obtain for every
we
a
f4/k
sheaf
locally ringed
The Kdhler
morphism d a complex of
space
sheaves:
(f4/k d) B.3.2 rian
or
Let
us
carry
a
do
(9
:
-
d'
f1i
)
X/k
suppose that all stalks
complete locally
dk-1
n2
__
___
X/k
X/k
0, of the locally ringed
topology.
convex
Then
dk
f2k
we
space
denote
(X, 0) by T"X
are
Noethe-
the Zariski
tangent space of X over x that means the set of all linear resp. continuously linear mappings X : f2,, -4 k. Hence, by the universal property (Kk) the Zariski tangent space represents the set of all (continuous) derivations from 0-, to k. Let F
=
that in the
(f, (D)
(X, Ox)
:
topological
case
continuous. Then F induces
--
(Y, Oy)
a
so-called
T,Z,X If G then
=
(g, T) : (Y, Oy)
obviously TZG
-
--
TZF
(Z, Oz) =
be
morphism of locally ringed spaces such homomorphism (D,, : (9yf(.,,) -- (9x,-, is TzY by tangent map TzF: TzX a
every canonical
D A
is
a
TZ(G F) -
-4
A
-
(D,,
E
Tz
Y
further such morphism of holds.
locally ringed
spaces
C
Appendix
Jets, Whitney functions and
few
a
C'-mappings C.1
Frechet
topologies
The
algebra CI(M)
in
canonical way the structure of
a
of smooth functions
seminorms, defining the M
by
for ff-functions
chart domains
topology
Uj
c
on
a
manifold M of dimension
on a
Fr6chet
CI(M).
algebra.
Let
us
To this end let
indicate
(UAErq
be
M such that there exist compact subsets
further
(Xj)jEq
be
a
If lm:
E 11aXjf11Kj) 0,
f E
a
Kj
family of interior sets Kj' C Uj is a covering of M family of differentiable charts xj : Uj -- Rn. Then
the property that the
n
possesses
sequence of
a
covering c
as
Uj
of
with
well. Let
COO(M)
ijocl
defines for every natural the compact set
m a
seminorm
and
the
on
C- (M), where
Kj
denotes the seminorm
derivatives in the coordinates
higher partial given Kj, axj by xj. With some patience and a few computations one now checks that the seminorms I I turn the algebra C' (M) into a Fr6chet algebra, and that the Fr6chet topology is independent of the special choice of the initial data. It is a well-known result from functional analysis that for two manifolds M and N the completed 7r-tensor product C'(M)CD,C'(N) is canonically isomorphic to the Fr6chet algebra CI(M x N). A proof of this fact can be found for instance in TRhVES [171]. Besides spaces of smooth functions one can also supply the space fl(M) of smooth over
-
m
1-forms
on
canonical
M
in other words the space of Kdhler differentials of
or
topology
C'(M)-module, topology
becomes
on
a
fl(M): n
n
E 11axjWj,111Kj) OC
W
i'lOCKM
w G
Q(M), wluj
wj,ldxjl.
1=1
Similarly the higher alternating products Q(M) Fr6chet
with
Fr6chet
a
hence in
detail: via the above Fr6chet
C'(M)
finitely generated projective fl(M) particular is topologically projective (see Section 6.3). In inducing the covering by charts (Xj)jEN fix seminorms
such that
C'(M)-modules.
M.J. Pflaum: LNM 1768, pp. 209 - 214, 2001 © Springer-Verlag Berlin Heidelberg 2001
become
finitely generated projective
210
Jets, Whitney functions The Kdhler derivative d
C' (M)
:
fl(M)
--
is
and
a
few
C'-mappings
continuous map with respect to
a
the
topologies defined above, hence induces by Section B.2 a uniquely determined continuous C'(M)-linear mapping T: lie-(M)/R -4 fl(M). Note hereby that _f1COO(M)/R has to be a Fr6chet space, as C'(M) is already one. The mapping T is surjective, as its image contains the generating system consisting of the forms df with f E C'(M), and injective, because by the universal property (Kk) Q(M) lies in lie-(myR. By the open mapping theorem fl(M) and lle.(m)/p, then have to be topologically isomorphic.
C.2
Jets
C.2.1
In this
monograph we will occasionally use Landau's notation, in particular jets and Whitney functions. Therefore we will briefly explain this f, g be two real or complex valued functions on the topological space
in connection with
notation. Let
A,
and
E A. Then
z
one
writes
f (X) if there exists
bounded
by
a
C
neighborhood
>
0. If
0
=
U of
(g(x)),
Let A be
a
locally closed A family F
that A C 0 closed. A is called
m-jets
on
order
jet of
A possesses in
is compact, then
f (X)
X)
seminorm
the
one
a
m on
subset of
R1,
A
locally
natural way the structure of
obtains
For all
Jm
:
x
E A and F E
one
Cm(O)
defined
Jm(A)
denotes for every
(F()) 10,1: ,, -4
F-4
1:
differential operator
D", locl
g
a
on
functions
A. The space
F()
J'(A)
of
real vector space. If K C A
P
-->
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,
< m on
D'
IF()(x)l
sup
compact exhaustion of A, i.e.
a
(F(O-'+O))
Jm(A) by
and 0 C R' open such
by
=
topology
{oo}
E N' of continuous
briefl7y only m-jet
or
Jm(A). As there exists A Kj C A with Uj Kj
convex
NU
m c cx
on
sets
Moreover, Jm-101 (A),
and
X --) Z.
,
=
10C,<M
a
\ fzJ
0,
=
g
(g(x))
0
=
IFIK,m=
compact
E U
writes
one
f (X)
on
x
additionally lim
C.2.2
Z,
f(X) I g(X) Iis defined for all
such that
z
x.I EA\fz
holds,
X -4
D'.
with the well-known
C.3
Whitney functions
211
The kernel of J1 has its class C1 and is denoted
Given a
own
by 31(A; 0)
m-jet
F E
J'(A)
polynomial T,';F:
Rn
-4
an
name; it is called the space of
with
and
m < oo
R of order
Tz'F (x)
ker J1 resp.
:=
=
a
3(A; 0)
point
z
A
on
:=
flat functions of
01 (A; 0)
E A one
:=
ker J110.
to F and
assigns
z
m:
E
F'(z).
oc!
10cl
Furthermore
one
sets
R'F z and
interprets R'F
as
=
F
J'(T,F),
-
the "rest term" of F in the
"Taylor expansion" TzmF
up to order
M.
C.3
Whitney functions
C.3.1 Definition a
jet F
a
Whitney function
(RmF) () (-y)
=
X
A
jet F
E
on
A of class
Cm,
m E
N,
one
understands
of order 7n, such that for every compact K C A
J111(A)
E
By
JI(A)
is called
a
o
(Jx
-
ij
lm-l'-I),
IX
Whitney function
-
1JI
of class
--4
0,
X, -9 E K.
'C', if the projection of
F to
J'(A) is for every M E N a Whitney function. The space of all Whitney functions A of class On, M E N U foo}, is linear and will be denoted by 8 '(A). Then
have the
we
C.3.2
following famous
Extension theorem of
result:
Whitney
For every
every open set 0 C Rn with A C 0 closed the
0
In
81(A)
PROOF: In
0'(A; 0)
this sequence
case m < oo
section
---4
--
---)
C!'(0)
locally closed following sequence is ---
F-'(A)
--
set A c Rn and exact:
0.
splits topologically that means there exists
a
continuous
C'(0).
See for instance MALGRANGE
[118,
Thm.
4.11.
13
general F-I(A) need not be a closed subspace of J'(A). In other words EI(A) is always complete with respect to the seminorms I JK,, (but see Corollary 1.6.13
not
-
for criteria which entail that space of
Whitney
F-I(A)
runs
IX
Therefore
one
defines
on
the
*
(RTF) () (-y) SUP -,IIEY,, -Ov
Now, if K
JI(A)). 11 IIK,,, by
is closed in
functions the seminorms
JJFJ1K,m::--JF1K,m+
-
1JJ--'0"
F E
Vn(A).
through a compact exhaustion of A and m through all natural numbers, 11, IIK,,, provide a locally convex topology which turns Fm(A) into
then the seminorms a
on
Fr6chet space.
212
Jets, Whitney functions
C.3.3 Lemma Under the
0)2
000 (A;
12'-mappings
prerequisites of Theorem C.3.2 the equality a' (A; 0)
See TOUGERON
[170,
Lem.
CA
Smoothing
The
of this section is to construct
of the
C.4.1 Lemma There exists the
few
a
holds.
PROOF:
goal
and
2.4].
El
angle
a
function which intuitively smoothes the
smooth function
a
[0, 11
:
p
[0, 11
x
[0, 11
---4
angle.
[0, 11
x
with
following properties:
(1)
p is
homeomorphism
a
(2)
The restriction of p to
(3)
D y (s,
t)
(1, 0)
(4)
p
(5)
p (s,
is
=
Such
2
1
3
function
a
.
a
([0, 11
[0, 11) \ {(1, 0)) 2
x
[0, 11
for all S, t E
2
2s)
x
fO})
101
C
for 0 <
s <
-
-
with
(s) t) :A (1, 2 0).
[0, 11
x
1 and
p (s,
6
is smooth.
t)
=
chosen to be either smooth
can
1] ([1, 2
and y
(s
on
for
-
2
fO})
x
3
6
its domain
Let x
R
:
s <
--
[0, 11
1 and
smooth function
p
be
x(s) :
smooth function such that
a
=
1 for
[0, 11 x10, 11
s >
1. Besides that let
[0, 11
--
[0, 11 by O
x
X(s) R
=
R(6s+2t-3) t
(Lt3 (s
3
(Lt3
+
X(
6s+2t-3
8t
t
s-
(Lt3
+
3
COS(7r(6s+2t-3))) R(6s-t-3) 8t
2
(1-2s) 3 (6s+2t-3) t
(2t3
-
(Lt3
+
-
1 3
sin( 7r(6s+2t-3) 8t
6s+2t-3
X
X(6s-t-3)
+
(2t3
-
!sin (7r(6s+2t-3))) 3
8t
R(6s-t-3
3
easy
:=
=
QL2
0 is
[0,
1
+
11. 1, 3
-
2
computation
J2t 1], 3
j 2
1
31 2
First check that
one
proves that
6
J3t
2
6
2
+
14t 1], 6
s
<
X. Then
-
for
s E
I,t)
for
s E
J2t)
for
S
E
J3t)
for
s
E
J4t)
for
s E
J5t)
for
s
E
It)
for
s
E
12t)
for
s E
for
s
E
J4t)
for
s
E
J5t)
1, 6
1 +
2
0, we
with
t
t
and J5t
0 for
=
1
t
7r(6s+2t-3) !,sin( 3 8t t
where I't
such that
t
(1-2s)
t
101.
-
t
CoS(7r(6s+2t-3)))
'(6s-t-3) j)X 2
-
Cos( 7t(6s+2t-3) 8t
3
1
CP2(S) t)
-
( 01) CP2)
3
CP 1 (s, t)
x
< s <
or
t
t
[0) 11
c
{(1, 0)}. 2
for 0 <
> 0
image.
([0, 11
y
-
p
relative
PROOF:
define
(0, 0),
t)
tempered
x'(s)
bijective
onto its
+
J3t
2
3
Cp satisfies (5) by definition. By a lengthy, but pj (resp. CP2) is monotone increasing in t for fixed
CA
213
angle
increasing (resp. decreasing) in s for fixed t. We show this for CP 1 and s by differentiation; the other proofs of monotony are similar. Together
and
s
of the
Smoothing
in detail
,9(t, s)
6s+2t-3 =
we
have
S))
+
E
J2t
with
t
Cpi(t, S) 3 t
7t
(2
3
(8
cos
-
7r
7t
S))+
71 (t,
(8
sin
71 (t,
3
-
6s
S)8.t
X
0I (t, s))
7T
+
(8 (t, S) (8 (t, s) )
-Cos
3 1
7r
7r
+
-
sin
-11
24
3
(X (71 (t, sm
71
(71 (t, s))
x
0,
>
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 <
thus obtain the claim.
we
N such that M
M'
[0, 11,
2
smooth function y
Then there exists
and aN.
s <
fo r
lengthy image. is bijective for all (s, t) :A
Proposition
C.4.2
a(M
a
X
for
that the function
see
[0, 1]
2s)
" 0)
-
Another
onto its
derivative D y (s,
--
x
relative
Furthermore it is easy to
-
=
on
a
N'.
Before
PROOF:
we
will
come
0
which smoothes the
:
to
[0, 11
our
matter of
[0, 11
X
angle according
-4
to the
concern
[0, 1]
X
let
first get
us
the M
preceding
Then
lemma.
aM
and a M
x
x
only
aM x N U M x
remains to
provide
a
supply the product of the
we
manifold structure for
loss of generality that aN, N) As N aN and both of the x sets 1[ 1[. [0, [0, aMx]O, 1[xaN x [0, 1[ open a N 1 1 differentiable in x canonical structures x a [0, [ 10, [ carry way, one only
boundary =
a (M
function
[0, 1].
interior M'xN' of MxN witlithe natural differentiable structure of the two manifolds M' and N'. Hence it
a
x
=
=
so we can assume
214
Jets, Whitney functions and
a
few
C'-mappings
has to find in the
neighborhood of each of the points (x, 0, -Y, 0) with x E a M, 'Y E aN compatible with all other charts. More precisely we look for a homeomorphism y : U -4 Rn x V!' of a neighborhood U of (x, 0, U, 0) into Euclidean (half-) space such that y is a diffeomorphism around each of the points (x, s,'Y, t) with (s, t) =A (0, 0) and such that y (a (M x N)) C Rn-' x f0j. Now the homeomorphism O comes into the game; we denote the inverse function of (p by Then one defines a chart y on a sufficiently small U by a
differentiable chart
Y(XI)S)IJI)t)=(YM(Xi))YN(IJI))ql(S)t))V2(S)t))) where ym and YN characteristic
are
differentiable charts of aM resp. aN around
properties of
(p the map y is
that the thus defined differentiable structure critical
(XI,S,-Y'j)C-U,
points (x, 0, -U, 0)
on
a on
x
resp. -Lj.
chart of the desired kind. the
product M
the choice of the function T.
x
N
By the
But note
depends
near
the El
Index
G-manifold, 151 G-space, 151 Hamiltonian, 85 E-decomposition, 16 E-manifold, 16 C'-structure, 27 L,,,,-algebra, 89 L.-morphism, 89
complete, 185 topological, 185 compact exhaustion, 33 compatible atlases, 27 homotopy, 118 singular charts, 27 tubular neighborhood, condition of frontier, 15
action, 151 effective, 151 faithful, 151
free,
cone, 17 cone
cone
152 cone
proper, 153
symplectic,
chart, 148 comb, 20 metric, 149
35, 148 data, 127 curvature moderate,
cone
84
space,
control
transitive, 151
acyclic, 187 admissible, 187 algebra Fr6chet, 184 locally convex, 184
,
proper , 128
control structure , 127
183
critical constant , 51 curvature moderate , 102 , 103 , 106
arc, 44
length,
strongly
,
75 cut
chart
,
15
deformation quantization , 87
depth
,
16 , 24 , 148
deRham cohomolo gy, 170
algebraic 104
transformation, domain, 27
cokernel
point distance, 96
decomposition
basic
canonical
103
curve, 44
b-metric, 71 Bar resolution, 189
cohomology, 174 basic complex, 174 bimodule, 189 boundary, 15 boundary set, 50 bounded away from Z,
datum , 21
covering by charts, 27
strong, 39 arc
127
127
equivalent normal, 127
corner
topological, analytic
93
84
deRham
,
177
com p lex ,
206
derivation, 205 derived functor, 189
differentiability set, 47 differential form, 68
228
INDEX
basic, 153 controlled, 170
Hochschild
invariant, 153
Hochschild
differential operator, 80
stratified, dimension, 16 finite, 16
(co)complex
topological,
(co)cycles
continuous,
82
Hochschild
190
190
(co)homology
continuous, 189
homomorphism. topologically linear, 184 horizontal subspace, 95
distance 73
geodesic, 44,
equivariant, 152 escape time, 96 etoile, 26 Euclidean embeddable, 31
image complete 185
exact, 185
controlled, 127 smooth stratified, 29 stratified, 26 inductively embedding, 33 insertion, 70 interior, 15 intersection forms, 170 intersection homology, 170 isotopy, 118 isotropy group, 152
weakly, 186 exterior algebra,
,
topological,
206
factorization canonical , 185 many connected
finitely
185
immersion
components,
-
102
finitely path connected locally 45 fixed point set, 152
,
45
,
flat , 42 , 49 , 211 of order
c,
jet, 43,
53
210
jet mapping,
flow, 134 stratified, 138 focal point distance, 96 formal completion, 177
210
Kdhler derivative , 205 Kiffiler differentials , 205
topological
Fuchs type operators, 149
,
207
kernel
function
topological
,
185
differentiable, 28
smooth,
28
left
fundamental vector
field,
153
geodesic 75 geometric curve 75 geometrically flat 140 gluing functions 22 ,
,
,
,
GraBmannian , 201 Hilbert
basis, 162 homogeneous, 162 minimal, 162 HKR-quasi isomorphism,
88
action, 151 length, 44 geodesic, 72 length space, 74 Lie algebra differential graded, Lie bracket, 67 link, 35 link chart, 148 local triviality, 35 locally closed, 15 locally trivial, 143
88
229
INDEX
manifold-with-boundary, manifold-with-corners,
17
proper
PSA,
21
31
embedding,
39
map
rank, 29 rectifiable, 45 reduction, 85
controlled, 127 of class C', 29 smooth, 28 maximal atlas, 27
regular,
45
(T, 1), 50,
metric cone, 149
51
1,45
module
(A), 36, (B), 37, (C), 40 (T), 39 (W), 39
Fr6chet, 184 locally convex, 184 topological, 184 moment map, 84
orbifold,
168
orbifold
chart,
orbit, 152 principal,
Ferrarotti-Wilson, 51 Whitney-Tougeron, 45 regularly situated, 52 resolution, 147 topologically projective, Riemannian metric, 71 smooth, 71 right action, 151
168
158
orbit bundle
principal,
158
orbit type, 152
principal, 158 order, 42 75
parametrization, by arc length, 75 path, 44 perverse sheaves, 170 perversity, 170 piece, 15 piecewise C', 47 weakly, 47 piecewise linear, 72 PL, 72 Poisson bivector, 83 Poisson bracket, 84 polyhedron, 19 principal bundle opposite, 157 projection, 61 tubular neighborhood, 93 projection d'ensemble semi-analytique,
section, 70 continuous, 186 segment, 75 sequence
canonical, 185 set germ, 23
forms, 171 simplex, 18 simplicial complex, 19 geometric realization, singular atlas, 27 singular chart, 26 weak,28 skeleton, 16, 26 slice, 156 slice representation, 156 shadow
smooth structure, 27
weak,28 space
T-decomposed,
39
projection valued section, projective, 186, 187 topologically, 186, 187
38 38
40
(A)-stratified, (B)-stratified, controllable,
16
38 38
127
19
188
230
INDEX
controlled, 127 decomposed, 15 glued, 22
tube,
induced, 126 normal, 126 tubular function, 93 tubular neighborhood, 61, isomorphic, 118 maximal, 101 morphism, 117 restriction, 117
inner
metric, 74 length, 74 locally ringed, 28 PL, 72 Poisson stratified, 83 reduced, 85 stratified, 23 symplectic stratified, triangulizable, 19 Whitney, 38 Whitney (A), 38 Riemannian, 71 sphere bundle, 156 spiral fast, 18 slow, 18 splitting, 186
neighborhood,
star, 26
product, 87 stratification, 23 by orbit types, Lipschitz, 148 star
typical fiber,
35
uniformly convergent V-manifold,
,
75
168
vector field
93
conformally radial, 135 continuous, 65 controlled, 135 weakly, 134 of class C', 65 radial, 135 smooth, 65 stratified, 65 vector space distance, 201 vertical subspace, 95
159
Whitney
condition
(A), 36, (B), 36,
stratum, 25 submersion
controlled, 127 smooth stratified, stratified, 26
29
support, 118
symmetry group, 152
38 38
Whitney cusp, 20 Whitney form, 177 Whitney function, 43, 211 Whitney umbrella, 19 Whitney-deRham cohomology, Zariski tangent space, 208
tangent bundle, 63
stratified,
63
tangent map, 208 tempered, 57, 118, 147 top stratum, 26
topologically exact, total space, 93 transition map, 27
triangulation, trivial, 143
19
92
type, 50 83
stabilizer group, 152 standard tubular
125
185
177
