Download A User's Guide to Algebraic Topology

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
\
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181
189
204
.
209
210
219
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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
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283
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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