Table Of ContentMonografie Matematyczne
lnstytut Matematyczny Polskiej Akademii Nauk (lMPAN)
Volume 63
(New Series)
Founded in 1932 by
S. Banach, B. Knaster, K. Kuratowski,
S. Mazurkiewicz, W. Sierpinski, H. Steinhaus
Managing Editor:
Przemyslaw Wojtaszczyk, IMPAN and Warsaw University
Editorial Board:
Jean Bourgain (lAS, Princeton, USA)
Tadeusz Iwaniec (Syracuse University, USA)
Tom Korner (Cambridge, UK)
Krystyna Kuperberg (Auburn University, USA)
Tomasz Luczak (Poznan University, Poland)
Ludomir Newelski (Wrodaw University, Poland)
Gilles Pisier (Universite Paris 6, France)
Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences)
Grzegorz Swi1!tek (Pennsylvania State University, USA)
Jerzy Zabczyk (Institute of Mathematics, Polish Academy of Sciences)
Volumes 31-62 of the series
Monografie Matematyczne were published by
PWN - Polish Scientific Publishers, Warsaw
J6rg Schiirmann
Topology of Singular
Spaces and
Constructible Sheaves
Springer Basel AG
Author's address:
J6rg Schiirmann
Westfălische Wilhelms-Universităt
SFB 478
Hittorfstr. 27
48149 Mtinster
Germany
e-mail: [email protected]
2000 Mathematics Subject Classification 14-02, 14F17, 14F25, 14F45, 14Pxx, 32-02, 32C18, 32F10,
32F27, 32B20, 32Sxx, 03C64, 55N25, 55N30, 55N33, 55N91
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-0348-9424-1 ISBN 978-3-0348-8061-9 (eBook)
DOI 10.1007/978-3-0348-8061-9
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© 2003 Springer Basel AG
Originally published by Birkhăuser Verlag, Basel, Switzerland in 2001
Cover design: Micha Lotrovsky, CH-4106 Therwil, Switzerland
Printed on acid-free paper produced of chlorine-free pulp. TCF
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987654321 www.birkhauser-science.com
For my Family: Friederike,
Philipp, David and Hendrik
Preface
Sheaf theory and homological algebra first arose in classical algebraic topology
in terms of homology and cohomology of topological spaces, and was used as a
language for describing properties of geometrical objects. This language was devel
oped in different periods for applications to algebraic topology, complex analysis
and algebraic geometry. For a beautiful account of the history of this subject we
refer to the article "A Short History: Les debuts de la tMorie des faisceaux" of C.
Houzel (contained in the book "Sheaves on Manifolds" of M. Kashiwara and P.
Schapira).
The last period deals with sheaf theory in the framework of derived and tri
angulated categories developed by A. Grothendieck and J.L. Verdier, especially for
applications to algebraic geometry. But over the years, this very abstract language
has found more and more applications in quite different fields of mathematics. Es
pecially, the new "intersection (co ) homology theory" and the theory of "perverse
sheaves" have found spectacular applications to the topology of singular spaces.
There exists now a number of excellent books devoted to the introduction to
the theory of derived and triangulated categories, especially also to sheaf theory in
this framework. But applications of this theory to the topology of singular spaces
are only worked out in a few places for some special examples. Moreover, a detailed
translation of this abstract language into the concrete underlying geometrical ideas
is often missing.
Assuming that the reader is familiar with some basic facts from sheaf the
ory, we give in this book a self-contained exposition of the theory of constructible
sheaves related to many different kinds of singular spaces: cell-complexes, trian
gulated spaces, semialgebraic and subanalytic sets, complex algebraic or analytic
sets, stratified spaces, quotient spaces, . .. .
We work out in detail its relation to the underlying geometrical ideas, and
explain many applications of this abstract theory to the topology of such spaces.
But as the reader will easily recognize by this list and the length of the book, this
theory and its applications are related to many different fields of mathematics.
For this reason we have written up this book in different parts, whose state
ments and results (but not all of their proofs) themselves are (essentially) indepen
dent from one part to another. All chapters have their own detailed introduction
viii Preface
so that the reader can switch to the part of the book, which is of interest for him
or her.
But first we give a global introduction to the whole book, where we explain a
little bit the history of this subject and the main ideas and results of the different
chapters. Moreover, we try to explain our point of view about the underlying
philosophy that holds all the different aspects together. We also give references to
similar results in algebraic geometry, which are related to this viewpoint.
Acknowledgments
It is a pleasure to thank J.P. Brasselet, L. Brocker, A. Dimca, H.A. Hamm,
L. Kaup, Le D.T., D. Massey, P. Schapira and S. Yokura for valuable conver
sations about this subjet and possible applications.
This book is an improved and expanded version of the author's Habilitations
schrift with the same title, which was submitted to the Fachbereich Mathematik
der Universitiit Hamburg in the end of 2000. Here I express my gratitude to
O. Riemenschneider for his encouragement during the completion of this habil
itation.
This book was finished at the Sonderforschungsbereich 478 "Geometrische
Strukturen in der Mathematik" in Munster, supported by the Deutsche Forschungs
gemeinschaft.
Finally I want to thank P. Pragacz, the Institute of Mathematics of the Pol
ish Academy of Sciences and the Birkhauser Verlag for getting the book published
in this form.
Munster, February 2003. Jorg Schurmann
Contents
Introduction 1
1 Thom-Sebastiani Theorem for constructible sheaves 17
Introduction. . . . . . . . . . . . . . . . . . . . . . . 17
1.1 Milnor fibration. . . . . . . . . . . . . . . . . . 23
1.1.1 Cohomological version of a Milnor fibration 24
1.1.2 Examples................... 32
1.2 Thom-Sebastiani Theorem. . . . . . . . . . . . . . 38
1.2.1 Preliminaries and Thom-Sebastiani for additive functions 38
1.2.2 Thom-Sebastiani Theorem for sheaves . . . . . . . . 50
1.3 The Thom-Sebastiani Isomorphism in the derived category 62
1.4 Appendix: Kiinneth formula. . . . . . . . . . . . . . . . . . 74
2 Constructible sheaves in geometric categories 81
Introduction. . . . . . . . . . 81
2.0.1 The basic results 81
2.0.2 Definable spaces 89
2.1 Geometric categories . . 93
2.2 Constructible sheaves 98
2.3 Constructible functions 122
3 Localization results for equivariant constructible sheaves 141
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.1 Equivariant sheaves. . . . . . . . . . . . . . . . . . . 149
3.1.1 Equivariant sheaves and monodromic complexes 149
3.1.2 Equivariant derived categories 155
3.1.3 Examples and stalk formulae . . . . . . . . . . . 159
3.2 Localization results for additive functions . . . . . . . . 170
3.3 Localization results for Grothendieck groups and trace formulae. 176
3.3.1 Grothendieck groups 176
3.3.2 Trace formulae . 192
3.4 Equivariant cohomology . . 200
ix
x Contents
4 Stratification theory and constructible sheaves 207
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
4.1 Stratification theory . . . . . . . . . . . . . . . . . . . . . 217
4.1.1 A cohomological version of the first isotopy lemma 217
4.1.2 Comparison of different regularity conditions ... 223
4.1.3 Micro-local characterization of constructible sheaves 229
4.2 Constructible sheaves on stratified spaces .. . . 232
4.2.1 Cohomologically cone-like stratifications . 232
4.2.2 Stability results for constructible sheaves 241
4.3 Base change properties . . . . . . . . . . . . . 252
4.3.1 Some constructions for stratifications. 253
4.3.2 Base change isomorphisms. . . . . . . 259
5 Morse theory for constructible sheaves 269
Introduction ............... . 269
5.0.1 Real stratified Morse theory ... 270
5.0.2 Complex stratified Morse theory 281
5.0.3 Introduction to characteristic cycles 283
5.1 Stratified Morse theory, part I . 296
5.1.1 Local Morse data .......... . 296
5.1.2 Normal Morse data ......... . 307
5.1.3 Morse theory for a stratified space with corners 316
5.2 Characteristic cycles and index formulae ... 320
5.2.1 Index formulae and Euler obstruction 321
5.2.2 A specialization argument ...... . 328
5.3 Stratified Morse theory, part II . . . . . . . . 334
5.3.1 Normal Morse data are independent of choices 334
5.3.2 Splitting of the local Morse data ..... . 345
5.3.3 Normal Morse data and micro-localization. 355
5.4 Vanishing cycles .................. . 361
6 Vanishing theorems for constructible sheaves 375
Introduction: Results and examples ............. . 375
6.0.1 (Co)stalk properties .............. . 375
6.0.2 Intersection (co)homology and perverse sheaves 381
6.0.3 Vanishing results in the complex context. 385
6.0.4 Nearby and vanishing cycles ...... . 402
6.0.5 Artin-Grothendieck type theorems . . . 408
6.0.6 Applications to constructible functions . 412
6.1 Proof of the results ............... . 414
Bibliography 433
Index 447
Introduction
Sheaf theory has its origin in classical algebraic topology. One of the highlights
of algebraic topology is the Poincare duality isomorphism between the cohomol
ogy and homology of a compact oriented manifold. A powerful generalization ap
peared in the 1960s through the work of Verdier [SHSl. This Poincare-Verdier
duality can only be formulated in the abstract framework of derived and triangu
lated categories, and is related to the functorial formalism of sheaf theory, what
Grothendieck calls the "six operations on sheaves", i.e. the six functors
Ri*, 1*, Ril, 1', ®L, Rhom.
This is also related to the Borel-Moore homology theory [BMl. This Borel
Moore homology is the right homology theory for non-compact spaces, as shown
by Borel-Haefiiger [BHl. They showed that a complex (or real) analytic set has
a fundamental class in the top-dimensional Borel-Moore homology group with
coefficients in Z (or Z2), one of the first important results about the topology of
singular spaces. This was later on extended by Schwartz [Schwa], Sullivan [Sul
and MacPherson [MPl to a theory of Stiefel-Whitney and Chern homology classes
of real and complex analytic spaces. The first approach of Schwartz uses "radial
vector fields" in relation to Poincare-Hopf theorems for singular complex spaces
(cf. [Schwa2, Schwa3, BSchwa]). The other approach is based upon the observation
of Sullivan [Su], that the link of a point in a complex (or real) analytic space has
vanishing Euler characteristic (modulo 2). Note that these characteristic classes
of a singular space lie in homology, not in cohomology, unlike the corresponding
characteristic classes of a vector bundle.
This shows that questions about the topology of singular spaces are related
to cohomology and homology. This viewpoint motivated the introduction of the
new formalism of bivariant theories by Fulton-MacPherson [FM], which gives the
categorical framework for the study of theorems of Riemann-Roch type in many
different categories. In the context of constructible sheaves, this formalism is re
lated to the "six operation on sheaves" (as above). Moreover, the relation to the
characteristic homology classes is through the theory of constructible functions,
which are closely related to constructible sheaves. Especially, the theory of Chern
classes of complex analytic (or algebraic) spaces depends on the Euler obstruction,
a constructible function introduced in [MPl.
J. Schürmann, Topology of Singular Spaces and Constructible Sheaves
© Birkhäuser Verlag 2003
