Mod Two Homology and Cohomology

539 Pages·2014·6.95 MB·English

by  Jean-Claude Hausmann (auth.)

Universitext
Jean-Claude Hausmann
Mod Two
Homology
and Cohomology
Universitext
Universitext
Series editors
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Vincenzo Capasso
Università degli Studi di Milano, Milan, Italy
Carles Casacuberta
Universitat de Barcelona, Barcelona, Spain
Angus MacIntyre
Queen Mary University of London, London, UK
Kenneth Ribet
University of California, Berkeley, CA, USA
Claude Sabbah
CNRS, École polytechnique Centre de mathématiques, Palaiseau, France
Endre Süli
University of Oxford, Oxford, UK
Wojbor A. Woyczynski
Case Western Reserve University, Cleveland, OH, USA
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Jean-Claude Hausmann
Mod Two Homology
and Cohomology
123
Jean-Claude Hausmann
Universityof Geneva
Geneva
Switzerland
ISSN 0172-5939 ISSN 2191-6675 (electronic)
ISBN 978-3-319-09353-6 ISBN 978-3-319-09354-3 (eBook)
DOI 10.1007/978-3-319-09354-3
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JELClassificationCode:55-01,55N10,55N91,57R91,57R20,55U10,55U25,55S10,57R19,55R91
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Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Simplicial (Co)homology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Definitions of Simplicial (Co)homology. . . . . . . . . . . . . . . . . 11
2.3 Kronecker Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 First Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Reduction to Components . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 0-Dimensional (Co)homology. . . . . . . . . . . . . . . . . . . 20
2.4.3 Pseudomanifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.4 Poincaré Series and Polynomials . . . . . . . . . . . . . . . . 23
2.4.5 (Co)homology of a Cone. . . . . . . . . . . . . . . . . . . . . . 23
2.4.6 The Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . 25
2.4.7 Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 The Homomorphism Induced by a Simplicial Map . . . . . . . . . 30
2.6 Exact Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7 Relative (Co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8 Mayer-Vietoris Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.9 Appendix A: An Acyclic Carrier Result. . . . . . . . . . . . . . . . . 49
2.10 Appendix B: Ordered Simplicial (Co)homology . . . . . . . . . . . 50
2.11 Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Singular and Cellular (Co)homologies. . . . . . . . . . . . . . . . . . . . . 59
3.1 Singular (Co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.2 Relative Singular (Co)homology. . . . . . . . . . . . . . . . . 66
3.1.3 The Homotopy Property . . . . . . . . . . . . . . . . . . . . . . 73
3.1.4 Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1.5 Well Cofibrant Pairs. . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.6 Mayer-Vietoris Sequences. . . . . . . . . . . . . . . . . . . . . 87
v
vi Contents
3.2 Spheres, Disks, Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 Classical Applications of the mod2 (Co)homology. . . . . . . . . 94
3.4 CW-Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5 Cellular (Co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.6 Isomorphisms Between Simplicial and Singular
(Co)homology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.7 CW-Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.8 Eilenberg-MacLane Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.9 Generalized Cohomology Theories . . . . . . . . . . . . . . . . . . . . 123
3.10 Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4 Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1 The Cup Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1.1 The Cup Product in Simplicial Cohomology . . . . . . . . 127
4.1.2 The Cup Product in Singular Cohomology. . . . . . . . . . 131
4.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.1 Disjoint Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.2 Bouquets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.3 Connected Sum(s) of Closed Topological Manifolds. . . 135
4.2.4 Cohomology Algebras of Surfaces . . . . . . . . . . . . . . . 137
4.3 Two-Fold Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.3.1 H1, Fundamental Group and 2-Fold Coverings. . . . . . . 139
4.3.2 The Characteristic Class . . . . . . . . . . . . . . . . . . . . . . 142
4.3.3 The Transfer Exact Sequence of a 2-Fold Covering . . . 144
4.3.4 The Cohomology Ring of RPn. . . . . . . . . . . . . . . . . . 146
4.4 Nilpotency, Lusternik-Schnirelmann Categories
and Topological Complexity. . . . . . . . . . . . . . . . . . . . . . . . . 147
4.5 The Cap Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.6 The Cross Product and the Künneth Theorem. . . . . . . . . . . . . 156
4.7 Some Applications of the Künneth Theorem . . . . . . . . . . . . . 165
4.7.1 Poincaré Series and Euler Characteristic of a Product. . 165
4.7.2 Slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.7.3 The Cohomology Ring of a Product of Spheres . . . . . . 166
4.7.4 Smash Products and Joins . . . . . . . . . . . . . . . . . . . . . 167
4.7.5 The Theorem of Leray-Hirsch . . . . . . . . . . . . . . . . . . 172
4.7.6 The Thom Isomorphism . . . . . . . . . . . . . . . . . . . . . . 180
4.7.7 Bundles Over Spheres. . . . . . . . . . . . . . . . . . . . . . . . 189
4.7.8 The Face Space of a Simplicial Complex . . . . . . . . . . 194
4.7.9 Continuous Multiplications on KðZ ;mÞ . . . . . . . . . . . 196
2
4.8 Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Contents vii
5 Poincaré Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.1 Algebraic Topology and Manifolds. . . . . . . . . . . . . . . . . . . . 201
5.2 Poincaré Duality in Polyhedral Homology Manifolds . . . . . . . 202
5.3 Other Forms of Poincaré Duality. . . . . . . . . . . . . . . . . . . . . . 211
5.3.1 Relative Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.3.2 Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . 215
5.3.3 The Intersection Form. . . . . . . . . . . . . . . . . . . . . . . . 217
5.3.4 Non Degeneracy of the Cup Product. . . . . . . . . . . . . . 219
5.3.5 Alexander Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5.4 Poincaré Duality and Submanifolds. . . . . . . . . . . . . . . . . . . . 221
5.4.1 The Poincaré Dual of a Submanifold . . . . . . . . . . . . . 221
5.4.2 The Gysin Homomorphism . . . . . . . . . . . . . . . . . . . . 225
5.4.3 Intersections of Submanifolds. . . . . . . . . . . . . . . . . . . 227
5.4.4 The Linking Number. . . . . . . . . . . . . . . . . . . . . . . . . 231
5.5 Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
6 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
6.1 The Cohomology Ring of Projective Spaces—Hopf
Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
6.2 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
6.2.1 The Borsuk-Ulam Theorem . . . . . . . . . . . . . . . . . . . . 245
6.2.2 Non-singular and Axial Maps. . . . . . . . . . . . . . . . . . . 246
6.3 The Hopf Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.3.2 The Hopf InvariantandContinuous Multiplications. . . . 250
6.3.3 Dimension Restrictions . . . . . . . . . . . . . . . . . . . . . . . 252
6.3.4 Hopf Invariant and Linking Numbers . . . . . . . . . . . . . 253
6.4 Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
7 Equivariant Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.1 Spaces with Involution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
7.3 Localization Theorems and Smith Theory . . . . . . . . . . . . . . . 283
7.4 Equivariant Cross Products and Künneth Theorems. . . . . . . . . 288
7.5 Equivariant Bundles and Euler Classes . . . . . . . . . . . . . . . . . 298
7.6 Equivariant Morse-Bott Theory. . . . . . . . . . . . . . . . . . . . . . . 309
7.7 Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
8 Steenrod Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
8.1 Cohomology Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
8.2 Properties of Steenrod Squares . . . . . . . . . . . . . . . . . . . . . . . 330
8.3 Construction of Steenrod Squares . . . . . . . . . . . . . . . . . . . . . 333
8.4 Adem Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
viii Contents
8.5 The Steenrod Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
8.6 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
8.7 Exercises for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
9 Stiefel-Whitney Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
9.1 Trivializations and Structures on Vector Bundles . . . . . . . . . . 355
9.2 The Class w —Orientability. . . . . . . . . . . . . . . . . . . . . . . . . 363
1
9.3 The Class w_ —Spin Structures. . . . . . . . . . . . . . . . . . . . . . . 367
2
9.4 Definition and Properties of Stiefel-Whitney Classes. . . . . . . . 372
9.5 Real Flag Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
9.5.1 Definitions and Morse Theory . . . . . . . . . . . . . . . . . . 376
9.5.2 Cohomology Rings. . . . . . . . . . . . . . . . . . . . . . . . . . 381
9.5.3 Schubert Cells and Stiefel-Whitney Classes. . . . . . . . . 389
9.6 Splitting Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
9.7 Complex Flag Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
9.8 The Wu Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
9.8.1 Wu’s Classes and Formula. . . . . . . . . . . . . . . . . . . . . 411
9.8.2 Orientability and Spin Structures . . . . . . . . . . . . . . . . 415
9.8.3 Applications to 3-Manifolds. . . . . . . . . . . . . . . . . . . . 418
9.8.4 The Universal Class for Double Points . . . . . . . . . . . . 420
9.9 Thom’s Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
9.9.1 Representing Homology Classes by Manifolds. . . . . . . 425
9.9.2 CobordismandStiefel-WhitneyNumbers. . . . . . . . . . . 429
9.10 Exercises for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
10 Miscellaneous Applications and Developments. . . . . . . . . . . . . . . 433
10.1 Actions with Scattered or Discrete Fixed Point Sets . . . . . . . . 433
10.2 Conjugation Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
10.3 Chain and Polygon Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 444
10.3.1 Definitions and Basic Properties. . . . . . . . . . . . . . . . . 444
10.3.2 Equivariant Cohomology. . . . . . . . . . . . . . . . . . . . . . 449
10.3.3 Non-equivariant Cohomology. . . . . . . . . . . . . . . . . . . 458
10.3.4 The Inverse Problem. . . . . . . . . . . . . . . . . . . . . . . . . 464
10.3.5 Spatial Polygon Spaces and Conjugation Spaces. . . . . . 468
10.4 Equivariant Characteristic Classes. . . . . . . . . . . . . . . . . . . . . 470
10.5 The Equivariant Cohomology of Certain Homogeneous
Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
10.6 The Kervaire Invariant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
10.7 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
Contents ix
11 Hints and Answers for Some Exercises . . . . . . . . . . . . . . . . . . . . 501
11.1 Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
11.2 Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
11.3 Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
11.4 Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
11.5 Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
11.6 Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
11.7 Exercises for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
11.8 Exercises for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
11.9 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527