Elements of Homology Theory This page intentionally left blank Elements of Homology Theory

V. V. Prasolov

Graduate Studies in Mathematics Volume 81

.^^% muek American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V.Ivanov Steven G. Krantz

B. B. IIpacojiOB

9JIEMEHTbI TEOPHM TOMOJIOrHM

MIIHMO, MocKBa, 2005 This work was originally published in Russian by MIIHMO under the title "9jieMeHTbi TeopHH roMOJiorHH" © 2005. The present translation was created under license for the American Mathematical Society and is published by permission.

Translated from the Russian by Olga Sipacheva

2000 Mathematics Subject Classification. Primary 55-01.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-81

Library of Congress Cataloging-in-Publication Data Prasolov, V. V. (Viktor Vasil'evich) [Elementy teorii gomologii. English] Elements of homology theory / V. V. Prasolov. p. cm. — (Graduate studies in mathematics ; v. 81) Includes bibliographical references and index. ISBN-13: 978-0-8218-3812-9 (alk. paper) ISBN-10: 0-8218-3812-1 (alk. paper) 1. Homology theory. I. Title. QA612.3.P73 2007 514'.23—dc22 2006047074

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionQams.org. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 Contents

Preface vii

Notation ix

Chapter 1. Simplicial Homology 1 §1. Definition and Some Properties 1 §2. Invariance of Homology 6 §3. Relative Homology 12 §4. Cohomology and Universal Coefficient Theorem 21 §5. Calculations 35 §6. The Euler Characteristic and the Lefschetz Theorem 51 Chapter 2. Cohomology Rings 59 §1. Multiplication in Cohomology 59 §2. Homology and Cohomology of Manifolds 69 §3. The Kunneth Theorem 95

Chapter 3. Applications of Simplicial Homology 111 §1. Homology and Homotopy 111 §2. Characteristic Classes 131 §3. Group Actions 173 §4. Steenrod Squares 184

Chapter 4. Singular Homology 195 §1. Basic Definitions and Properties 195 VI Contents

§2. The Poincare and Lefschetz Isomorphisms for Topological Manifolds 227 §3. Characteristic Classes: Continuation 252 Chapter 5. Cech Cohomology and de Rham Cohomology 263 §1. Sheaf Cohomology 263 §2. De Rham Cohomology 275 §3. The de Rham Theorem 289 Chapter 6. Miscellany 301 §1. The Alexander Polynomial 301 §2. The Arf Invariant 317 §3. Embeddings and Immersions 325 §4. Complex Manifolds 339 §5. Lie Groups and //-Spaces 344 Hints and Solutions 365 Bibliography 403 Index 411 Preface

This book is a natural continuation of the author's earlier book Elements of Combinatorial and Differential Topology (American Mathematical Society, Providence, RI, 2006), which we refer to as Part I here. (A corrected Russian version of Part I is available at http://www.mccme.ru/prasolov.) In Chapter 1, we define simplicial homology and cohomology and give many examples of their calculations and applications. At this point, the book diverges from standard modern courses in algebraic topology, which usually begin with introducing singular homology. Simplicial homology has a simpler and more natural definition. Moreover, it is simplicial homology that is usually involved in calculations. For this reason, we introduce sin• gular homology near the end of the book and use it only when it is indeed necessary, mainly in studying topological manifolds. Homology and cohomology groups with arbitrary coefficients are expres• sed in terms of integral homology by means of the functors Tor and Ext. The properties of these functors are very important for homology theory, so we discuss them in detail. We first prove the Poincare duality theorem for simplicial (co)homology. This proof applies only to smooth (to be more precise, triangulable) mani• folds. There is no triangularization theorem for topological manifolds, and the proof of the Poincare duality theorem for them uses, of necessity, singular (co)homology. This proof is given in Chapter 4; it is very cumbersome. Chapter 2 considers an important algebraic structure on cohomology, the cup product of Kolmogorov and Alexander. It is particularly useful in the case of manifolds. Multiplication in cohomology is related to many topo• logical invariants of manifolds, such as the intersection form and signature.

vn Vlll Preface

One possible approach to constructing multiplication in cohomology is based on a theorem of Kiinneth, which expresses the (co)homology ofXx7 in terms of those of X and Y and is of independent interest. Chapter 3 is devoted to various applications of (simplicial) homology and cohomology. Many of them are related to obstruction theory. One of such applications is the construction of the characteristic classes of vector bundles. Other approaches to constructing characteristic classes (namely, the universal bundle and axiomatic approaches) are also discussed. Then, we consider the (co)homological properties of spaces with actions of groups; we construct transfers and Smith's exact sequences. We conclude the chap• ter with constructing Steenrod squares, which generalize multiplication in cohomology. In Chapter 4, we define singular (co)homology and describe some of its applications; in particular, we obtain certain properties of characteristic classes. (Technically, it is more convenient to prove them by using singular cohomology, although the assertions themselves can be stated for simplicial cohomology.) Chapter 5 considers yet another approach to constructing cohomology theory, namely, Cech cohomology and de Rham cohomology, which are closely related to each other. For the de Rham cohomology, we prove the Poincare duality theorem. Then, we carry over the construction of de Rham, which was originally introduced for smooth manifolds, to arbitrary simplicial complexes. The final Chapter 6 is devoted to various applications of homology the• ory, largely to the topology of manifolds. We begin with a detailed account of the Alexander polynomials, which we construct by using the homology of cyclic coverings; the Arf invariant is also considered. Then, we prove the strong Whitney embedding theorem. We also give a formula for calculating the Chern classes of complete intersections and discuss some homological properties of Lie groups and i^-spaces. The book contains many problems (with solutions) and exercises. The problems are based on the materials of topology seminars for second-year students held by the author at the Independent University of Moscow in 2003. The basic notation, as well as theorems and other assertions, of Part I are mostly used without explanations; in some cases, we give references to the corresponding places in Part I. This work was financially supported by the Russian Foundation for Basic Research (project nos. 05-01-01012a, 05-01-02805-NTsNIL_a, and 05- 01-02806-NTsNIL_a). Notation

Hk(X] G), the fc-dimensional homology group of X with coefficients in G; Hk(X', G), the k-dimensional cohomology group of X with coefficients in G; Hom(A, B), the group of homomorphisms A —> £; A ® B, the tensor product of the Abelian groups A and B; Tor(A,B), seep. 29; Ext (A, 5), seep. 29; Coker a, the cokernel of the homomorphism a (see p. 15); [Mn], the fundamental class of the manifold Mn; %(X), the Euler characteristic of X; A(/), the Lefschetz number of the map /; o-(M4n), the signature of the manifold M4n; ek, the fc-dimensional trivial vector bundle; u>fc(0> the fcth Stiefel-Whitney class of the bundle £; Cfe(^), the feth Chern class of the bundle £; Pjfe(0» the fcth Pontryagin class of the bundle £; Sq\ the Steenrod square.

IX This page intentionally left blank Bibliography

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G-complex anticommutativity of cup product, 63 regular, 174 Arf simplicial, 173 invariant G-space, 173 of a knot, 319 H-space, 358 of a link, 320 K{ir,n) space, 122 of a quadratic form, 317 5-equivalent matrices, 314 theorem, 318 r-transnormal embedding, 56 associated sheaf, 265 attaching a handle, 312 augmentation, 6, 17 action axiom effective, 177 dimension, 203, 204 free, 183 exactness, 203, 204 simplicial, 173 acyclic excision, 200, 203, 204 functor, 103 noncommutative, 224 model, 104 homotopy, 203, 204 theorem, 104 axiomatic approach to Stiefel-Whitney simplicial complex, 6 classes, 159 support theorem, 7 admissible set, 231 Alexander basis duality, 81 of a free Abelian group, 4 ideal, 308 of a module, 305 polynomial, 308 symplectic, 317 in Conway's normalization, 315 Betti numbers, 3 theorem, 205 bi-invariant form, 348 Alexander-Pontryagin duality, 271 bilinear map Alexander-Whitney diagonal nondegenerate, 109 approximation, 105, 214 of Abelian groups, 27 algebra Bockstein homomorphism, 14, 92, 142 Hopf, 361 Borromean rings, 85 connected, 361 multidimensional, 87 Lie, 344 Borsuk-Ulam theorem, 76 cohomology, 349 boundary, 2 algebraically trivial map, 120 homomorphism, 2 almost parallelizable manifold, 141 of a simplex, 1

411 412 Index

braid Stiefel-Whitney, 141 colored, 122 of a manifold, 150 group, 123 Thorn, 237, 255 bundle Wu, 259 associated with a divisor, 339 closed form, 277 conjugate, 170 coboundary, 22 dual, 171 formula, 186 induced, 133 homomorphism, 267 nonorientable, 143 cochain, 21, 266 orientable, 143 difference, 117 vector, 131 relative, 23 equivalent, 132 with compact supports, 48 isomorphic, 132 cocycle, 22 smooth, 131 noncommutative, 222 stably equivalent, 148 cohomologous, 222 with structure group, 272 coefficient group, 203, 204 cofinal set, 266 canonical cohomology map, 9 Cech, 267, 268 vector bundle, 153 cross product, 106 cap product, 70 de Rham, 277 Cart an formula, 189 fundamental class, 125 Cartesian product of Abelian groups, 21 group, 22 category, 103 Lie algebra, 349 with models, 103 noncommutative, 222, 274 Cech cohomology, 267, 268 operation, 127 cellular homology, 37, 210 reduced, 23 centralizer, 351 relative, 23 chain, 2 singular, 202 complex, 4 with compact supports, 48 free, 4 with local coefficients, 136 nonnegative, 4 cokernel, 15 ordered, 60 collar theorem, 78, 247 total, 60 colored braid group, 122 homotopy, 5, 196 commutator, 346 map, 4 relative, 12 of vector fields, 276 with closed supports, 48 subgroup, 112 characteristic complete intersection, 341 class complex Chern, 163 chain, 4 Chern of a complex manifold, 171 Stiefel manifold, 162 Euler, 144 vector bundle, 162 Pontryagin, 172 complexification, 172 Stiefel-Whitney, 141 conjugate bundle, 170 Euler, 51, 90 connected of a pair, 55, 182 Hopf algebra, 361 Chern characteristic class, 163 sum of manifolds, 336 of a complex manifold, 171 connecting homomorphism, 12, 14 class consistent family, 264 Chern, 163 constant presheaf, 264 of a complex manifold, 171 contravariant functor, 103 Euler, 144 Conway polynomial, 316 fundamental, 36 coproduct, 360 cohomology, 125 covariant functor, 103 Pontryagin, 172 cross product, 106 primitive homology, 49 cup product, 59, 62 Index 413

cycle, 2 class, 144 homologous, 3 exact form, 277 degree of a map, 36 sequence Dehn twist, 50 Mayer-Vietoris, 18 de Rham of a pair, 12 cohomology, 277 Smith, 181 theorem, 289 split, 24 diagonal approximation, 105, 184, 214 exactness axiom, 203, 204 Alexander-Whitney, 105, 214 excision difference cochain, 117 axiom, 200, 203, 204 differential form noncommutative, 224 closed, 277 isomorphism, 12 exact, 277 theorem, 198 polynomial extraordinary (co) homology theory, 204 on a complex, 296 on a simplex, 296 five lemma, 15 smooth, 296 form dimension axiom, 203, 204 bi-invariant, 348 direct closed, 277 limit, 265 exact, 277 product of vector bundles, 134 intersection, 88, 260 sum left-invariant, 348 of Abelian groups, 21 polynomial of bundles, 134 on a complex, 296 directed set, 264 on a simplex, 296 of Abelian groups, 265 quadratic over Z2, 317 divisible group, 32 right-invariant, 348 domain invariance theorem, 205 Seifert, 304 double point, 325 smooth, 296 dual formula bundle, 171 Cartan, 189 Stiefel-Whitney class, 149 coboundary, 186 duality of universal coefficients, 33 Alexander, 81 Thorn, 257 Alexander-Pontryagin, 271 Whitney, 147 Poincare, 44 Wu, 193 dunce hat, 115 free effective action, 177 action, 183 Eilenberg theorem, 117 chain complex, 4 Eilenberg-MacLane space, 122 functor, 103 Eilenberg-Zilber theorem, 213 module, 305 element resolution of an Abelian group, 28 regular, 353 functor singular, 353 acyclic, 103 elementary ideal, 307 contravariant, 103 embedding covariant, 103 r-transnormal, 56 free, 103 transnormal, 56 fundamental class, 36 equivalent cohomology, 125 microbundles, 254 of a topological manifold, 232 vector bundles, 132 with boundary, 249 equivariant map, 173 Euler generalized (co)homology theory, 204 characteristic, 51, 90 germ, 265 of a pair, 55, 182 Gromov norm, 221 414 Index

group Hopf-Whitney theorem, 119 cohomology, 22 Hurewicz reduced, 23 homomorphism, 112 relative, 23 theorem, 113 singular, 202 hyperbolic manifold, 222 colored braid, 122 divisible, 32 ideal free Abelian, 4 Alexander, 308 homology elementary, 307 of a chain complex, 4 induced bundle, 133 singular, 196 infinite Lie, 344 cyclic covering, 304 of braids, 123 Grassmann manifold, 153 of coefficients, 203, 204 infinite-dimensional lens space, 123 periodic, 31 injective resolution, 32 ring, 180 integral form, 280 simplicial homology, 3 intersection Smith homology, 181 complete, 341 Gysin sequence, 256 form, 88, 260 number, 42 Helly's theorem, 208, 209 invariance homologous cycles, 3 of a boundary, 206 homology of a domain, 205 cellular, 37, 210 invariant disk, 183 Arf group of a knot, 319 of a chain complex, 4 of a link, 320 simplicial, 3 of a quadratic form, 317 Smith, 181 Hopf, 219 primitive class, 49 isomorphic vector bundles, 132 reduced, 17 isomorphism relative, 12 excision, 12 sequence Lefschetz, 81 for a triple, 15 for topological manifolds, 250 of a pair, 12 of bundles with structure group, 273 singular, 196 Poincare, 44 sphere, 45, 81, 183 for de Rham cohomology, 285 with closed supports, 48 for topological manifolds, 243 homomorphism with local coefficients, 138 Bockstein, 14, 92, 142 suspension, 20, 201 boundary, 2 Thorn, 238 connecting, 12, 14 Hurewicz, 112 Kiinneth theorem, 99 of presheaves, 263 for relative homology, 215 restriction, 263 for singular homology, 213 transfer, 179 relative, 215 homotopic Kan-Whitehead theorem, 129 trivializations, 133 Kolmogorov-Alexander multiplication, 59 vector fields, 133 homotopy Lefschetz axiom, 203, 204 fixed point theorem, 56 chain, 5, 196 isomorphism, 81 Hopf for topological manifolds, 250 algebra, 361 number, 56 fibration, 219 left-invariant invariant, 219 form, 348 theorem, 337, 362 vector field, 346 Index 415

lemma natural transformation, 103 on extension, 297 naturality Poincare, 283 of cap product, 71 lens space, 93 of Stiefel-Whitney classes, 146 infinite-dimensional, 123 noncommut at i ve Leray-Hirsh theorem, 168 cocycle, 222 Lie cohomology, 222, 274 algebra, 344 excision axiom, 224 group, 344 Mayer-Vietoris sequence, 224 line bundle associated with a divisor, 339 nondegenerate linking number, 46, 83 bilinear map, 109 local system of groups, 136 quadratic form over Z2, 317 nonnegative chain complex, 4 manifold nonorientable bundle, 143 almost parallelizable, 141 normal degree of an immersion, 335 normalizer, 351 Grassmann infinite, 153 number hyperbolic, 222 parallelizable, 132, 260, 338 Betti, 3 Schubert, 157 intersection, 42 stably parallelizable, 156 Lefschetz, 56 Stiefel, 139 linking, 46, 83 complex, 162 self-intersection, 326, 327 topological with boundary orient able, Stiefel-Whitney, 152 248 map object of a category, 103 algebraically trivial, 120 obstruction, 116 chain, 4 to extending sections, 138 equi variant, 173 ordered chain complex, 60 splitting, 167 orientable Massey triple product, 84 bundle, 143 matrix topological manifold, 232 presentation, 305 with boundary, 248 Seifert, 304 orientation maximal torus, 351 of a topological manifold, 232 Mayer-Vietoris sequence, 18, 200 with boundary, 248 for de Rham cohomology, 278 system of groups, 136 for de Rham cohomology with compact oriented topological manifold, 232 supports, 279 noncommutative, 224 parallelizable manifold, 132, 260, 338 relative, 20, 202 partition of an integer, 158 microbundle, 254 periodic group, 31 equivalent, 254 Poincare tangent, 254 duality, 44 Milnor theorem, 338 isomorphism, 44 Minkowski theorem, 177 for de Rham cohomology, 285 model, 103 for topological manifolds, 243 acyclic, 104 with local coefficients, 138 module lemma, 283 finitely generated, 305 theorem, 112 free, 305 point Moore space, 128 double, 325 morphism, 103 self-intersection, 325 Morse inequality, 210 polynomial multidimensional Borromean rings, 87 Alexander, 308 multiplication, 358 in Conway's normalization, 315 Kolmogorov-Alexander, 59 Conway, 316 416 Index

differential form matrix, 304 on a complex, 296 surface, 305 on a simplex, 296 self-intersection Pontryagin number, 326, 327 characteristic class, 172 point, 325 theorem, 152 sequence presentation matrix, 305 exact of a pair, 12 presheaf, 263 Gysin, 256 constant, 264 Mayer-Vietoris, 18, 200 primitive homology class, 49 for de Rham cohomology, 278 product for de Rham cohomology with Massey triple, 84 compact supports, 279 of Abelian groups noncommutative, 224 Cartesian, 21 relative, 20, 202 tensor, 27 Smith exact, 181 tensor of chain complexes, 97 set vector bundle, 132 admissible, 231 projective cofinal, 266 resolution, 32 directed, 264 projectivization of a vector bundle, 167 of Abelian groups, 265 pullback, 133 sheaf, 264 associated with a presheaf, 265 quadratic form over Z2, 317 generated by a presheaf, 265 nondegenerate, 317 signature of a manifold, 90 rank of a Lie group, 353 of a product, 108 reduced Thorn theorem, 91 cohomology, 23 simplex homology, 17 boundary, 1 regular singular, 195 G-complex, 174 simplicial element, 353 G-complex, 173 immersion, 325 action, 173 relative complex acyclic, 6 chain, 12 homology group, 3 cochain, 23 volume, 221 cohomology, 23 singular homology, 12 cohomology, 202 Kiinneth theorem, 215 element, 353 Mayer-Vietoris sequence, 20, 202 homology, 196 resolution simplex, 195 injective, 32 skein relation, 316 projective, 32 skew-commutativity of cup product, 63 restriction homomorphism, 263 Smith right-invariant form, 348 exact sequence, 181 ring homology group, 181 Borromean, 85 theorem, 183 group, 180 smooth roots, 355 differential form on a complex, 296 Schubert manifold, 157 on a simplex, 296 section, 265 triangulation, 289 of a bundle, 131 vector bundle, 131 zero, 132 space Seifert K(7r,n), 122 form, 304 Eilenberg-MacLane, 122 knot, 310 lens, 93 Index 417

Moore, 128 for singular homology, 213 split exact sequence, 24 relative, 215 splitting map, 167 Kan-Whitehead, 129 stably Lefschetz equivalent vector bundles, 148 fixed point, 56 parallelizable manifold, 156 isomorphism, 81 Steenrod square, 188 Leray-Hirsh, 168 Steenrod's five lemma, 15 Milnor, 338 Steenrod-Eilenberg axioms, 203, 204 Minkowski, 177 Stiefel on a collar, 247 manifold, 139 on acyclic complex, 162 models, 104 theorem, 260 supports, 7 Stiefel-Hopf theorem, 109 on domain invariance, 205 Stiefel-Whitney Poincare, 112 class Pontryagin, 152 characteristic, 141 Smith, 183 dual, 149 Stiefel, 260 of a manifold, 150 Stiefel-Hopf, 109 total, 149 Stokes, 281 number, 152 Thorn, 257 Stokes theorem, 281 signature, 91 strong Whitney embedding theorem, 325 Whitney sum duality, 149 connected of manifolds, 336 strong embedding, 325 direct Wu, 259 of Abelian groups, 21 Thorn of bundles, 134 class, 237, 255 Whitney, 134 formula, 257 support of a chain, 6 isomorphism, 238 suspension isomorphism, 20, 201 theorem, 257 symplectic basis, 317 signature, 91 topological tangent microbundle, 254 generator, 350 tensor product manifold of Abelian groups, 27 orientable, 232 of chain complexes, 97 oriented, 232 theorem with boundary orientable, 248 acyclic model, 104 torsion subgroup, 44 acyclic support, 7 torus, 345 Alexander, 205 maximal, 351 Arf, 318 total Borsuk-Ulam, 76 chain complex, 60 chain homotopy, 5 Stiefel-Whitney class, 149 collar for smooth manifolds, 78 transfer homomorphism, 179 de Rham, 289 transition function, 272 simplicial, 299 transnormal embedding, 56 domain invariance, 205 transversality, 73 Eilenberg, 117 triple Eilenberg-Zilber, 213 homology sequence, 15 excision, 198 Massey product, 84 Helly's, 208, 209 Hopf, 337, 362 trivial vector bundle, 132 Hopf-Whitney, 119 Hurewicz, 113 universal coefficient Kunneth, 99 formulas, 33 for relative homology, 215 theorem, 33 418 Index

vector Whitney bundle, 131 formula, 147 canonical, 153 sum, 134 complex, 162 theorem equivalent, 132 duality, 149 isomorphic, 132 strong embedding, 325 product, 132 trick, 329 smooth, 131 Wu stably equivalent, 148 class, 259 trivial, 132 formula, 193 field theorem, 259 homotopic, 133 left-invariant, 346 zero section, 132 Titles in This Series

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46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing

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