Download A User's Guide to Algebraic Topology
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A User's Guide to Algebraic Topology by C. T. J. Dodson University of Toronto, Ontario, Canada and Phillip E. Parker Wichita State University, Kansas, USA. KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON Contents Preface xi Introduction and Overview 1 1 Basics of Extension and Lifting Problems 5 1.1 1.2 1.3 1.4 1.5 2 Up 2.1 2.2 2.3 2.4 2.5 2.6 Existence problems . . . . ; . . Retractions Separation Transcription of problems by functors The shape of things to come to Homotopy is Good Enough Introducing homotopy Fibrations and cofibrations Commuting up to homotopy Homotopy groups The fundamental group First applications 3 Homotopy Group Theory 3.1 Introduction 3.1.1 Exact sequences 3.2 Relative homotopy 3.3 Relative and exact properties 3.4 Fiberings 3.4.1 Applications 3.5 CW-complexes 3.5.1 Attaching cells and homotopy properties 3.5.2 Simplicial complexes 3.5.3 Computing fundamental groups 5 8 8 9 10 \. . . 17 17 27 37 44 45 47 51 51 53 56 58 62 67 69 76 80 86 vi CONTENTS 3.6 3.7 3.8 3.9 Simplicial and cellular approximation Weak homotopy equivalence is good enough in CW Exploiting n-connectedness Extracting homotopy groups from known bundles 89 93 96 99 4 Homology and Cohomology Theories 4.1 Introduction 4.2 Homology and cohomology theories 4.3 Deductions from the axioms 4.3.1 Reduction and unreduction 4.3.2 Deductions from homology 4.3.3 The Lefschetz theorem 4.4 Homology of chain complexes . 4.4.1 Universal coefficient theorems 4.5 Homology and cohomology of CW-complexes 105 105 106 108 116 118 122 125 133 135 5 Examples in Homology and Cohomology 5.1 Cubical singular homology 5.2 Simplicial singular homology 5.3 Cup product 5.4 Geometric simplicial homology 5.5 Computing simplicial homology groups 5.6 Relative simplicial homology 5.7 Geometric simplicial singular homology 5.8 Bordism homology 5.9 de Rham cohomology 5.10 Geometric simplicial cohomology 5.11 More on products 5.12 Cech cohomology theories 143 144 147 150 151 155 157 164 165 165 167 168 172 6 Sheaf and Spectral Theories 6.1 Some sheaf theory 6.2 Generalization to spectral theories 6.3 Spectral sequences 6.3.1 Review and moral tale 7 Bundle Theory 7.1 Elemental theory 7.1.1 Pullbacks 7.1.2 The Milnor construction \ 175 175 181 189 204 . 209 210 219 222 CONTENTS vii 7.2 Stabilization 7.2.1 Linear stabilization 7.2.2 Nonlinear stabilization 7.2.3 Linear ^-theory 7.3 Homology and cohomology 7.3.1 The Gysin sequence 7.3.2 The Wang sequence 7.3.3 Transgression and the Serre sequence 7.3.4 The Leray-Hirsch theorem 7.3.5 Thom isomorphism theorem 7.3.6 Zeeman comparison theorem 7.4 Characteristic classes 7.5 Nonabelian cohomology 8 Obstruction Theory 8.1 Preliminary ideas 8.2 Eilenberg-Maclane spaces K(TT, n) 8.3 Moore-Postnikov decomposition of a 8.4 Homotopy cofunctors 8.5 Postnikov invariants 9 Applications 9.1 Those already done 9.2 Two classical results 9.3 Theorems of Geroch and Stiefel 9.4 The power 9.4.1 Piecewise linear structures 9.4.2 Smoothing PL structures 9.4.3 Almost-complex structures 9.5 Marcus's theorem 9.6 Meta structures 9.7 Other signatures A Algebra A.I Sets and maps A.2 Categories and functors A.2.1 A triangular view A.2.2 Limits of diagrams A.3 Groups and actions A.3.1 Groups A.3.2 Group actions 224 227 230 233 234 243 244 244 245 246 248 248 250 fibration 257 257 260 267 271 277 283 283 285 286 288 \ . . 288 288 289 289 291 292 295 295 297 297 298 300 301 305 viii CONTENTS B Topology B.I B.2 B.3 B.4 B.5 B.6 B.7 309 Topological spaces Separation properties Compactness Paracompactness Connectedness Peano's space-filling curve Collected examples on general topology 310 315 316 318 319 321 322 C Manifolds and Bundles 331 C.I Manifolds C.2 Tangent spaces C.3 Calculus on manifolds C.3.1 Summary of formulae C.4 Bundles C.5 Metrics and connections C.5.1 Principal bundles C.5.2 Linear connections C.5.3 Levi-Civita connection C.6 Fibered manifolds C.7 Systems of connections and universal connections 331 332 335 335 348 349 351 352 354 362 362 D Tables of Homotopy Groups D.I D.2 D.3 D.4 D.5 Spheres Three special unitary and symplectic groups Symplectic groups Two spin and two exceptional groups, and CP 2 Real Stiefel manifolds 365 - 366 368 368 369 370 E Computational Algebraic Topology 381 Bibliography 385 Index 393