search for books and compare prices
Tables of Contents for A Concise Course in Algebraic Topology
Chapter/Section Title
Page #
Page Count
Introduction
1
4
The fundamental group and some of its applications
5
8
What is algebraic topology?
5
1
The fundamental group
6
1
Dependence on the basepoint
7
1
Homotopy invariance
7
1
Calculations: π1(R) = 0 and π (S1) = Z
8
2
The Brouwer fixed point theorem
10
1
The fundamental theorem of algebra
10
3
Categorical language and the van Kampen theorem
13
8
Categories
13
1
Functors
13
1
Natural transformations
14
1
Homotopy categories and homotopy equivalences
14
1
The fundamental groupoid
15
1
Limits and colimits
16
1
The van Kampen theorem
17
2
Examples of the van Kampen theorem
19
2
Covering spaces
21
12
The definition of covering spaces
21
1
The unique path lifting property
22
1
Coverings of groupoids
22
1
Group actions and orbit categories
23
2
The classification of coverings of groupoids
25
2
The construction of coverings of groupoids
27
1
The classification of coverings of spaces
28
1
The construction of coverings of spaces
29
4
Graphs
33
4
The definition of graphs
33
1
Edge paths and trees
33
1
The homotopy types of graphs
34
1
Covers of graphs and Euler characteristics
35
1
Applications to groups
35
2
Compactly generated spaces
37
4
The definition of compactly generated spaces
37
1
The category of compactly generated spaces
38
3
Cofibrations
41
6
The definition of cofibrations
41
1
Mapping cylinders and cofbrations
42
1
Replacing maps by cofibrations
43
1
A criterion for a map to be a cofibration
43
1
Cofiber homotopy equivalence
44
3
Fibrations
47
8
The definition of fibrations
47
1
Path lifting functions and fibrations
47
1
Replacing maps by fibrations
48
1
A criterion for a map to be a fibration
49
1
Fiber homotopy equivalence
50
1
Change of fiber
51
4
Based cofiber and fiber sequences
55
8
Based homotopy classes of maps
55
1
Cones, suspensions, paths, loops
55
1
Based cofibrations
56
1
Cofiber sequences
57
2
Based fibrations
59
1
Fiber sequences
59
2
Connections between cofiber and fiber sequences
61
2
Higher homotopy groups
63
8
The definition of homotopy groups
63
1
Long exact sequences associated to pairs
63
1
Long exact sequences associated to fibrations
64
1
A few calculations
64
2
Change of basepoint
66
1
n-Equivalences, weak equivalences, and a technical lemma
67
4
CW complexes
71
10
The definition and some examples of CW complexes
71
1
Some constructions on CW complexes
72
1
Help and the Whitehead theorem
73
1
The cellular approximation theorem
74
1
Approximation of spaces by CW complexes
75
1
Approximation of pairs by CW pairs
76
1
Approximation of excisive triads by CW triads
77
4
The homotopy excision and suspension theorems
81
8
Statement of the homotopy excision theorem
81
2
The Freudenthal suspension theorem
83
1
Proof of the homotopy excision theorem
84
5
A little homological algebra
89
4
Chain complexes
89
1
Maps and homotopies of maps of chain complexes
89
1
Tensor products of chain complexes
90
1
Short and long exact sequences
91
2
Axiomatic and cellular homology theory
93
12
Axioms for homology
93
1
Cellular homology
94
4
Verification of the axioms
98
1
The cellular chains of products
99
2
Some examples: T, K, and RPn
101
4
Derivations of properties from the axioms
105
10
Reduced homology; based versus unbased spaces
105
1
Cofibrations and the homology of pairs
106
1
Suspension and the long exact sequence of pairs
107
1
Axioms for reduced homology
108
2
Mayer-Vietoris sequences
110
2
The homology of colimits
112
3
The Hurewicz and uniqueness theorems
115
6
The Hurewicz theorem
115
2
The uniqueness of the homology of CW complexes
117
4
Singular homology theory
121
8
The singular chain complex
121
1
Geometric realization
122
1
Proofs of the theorems
123
1
Simplicial objects in algebraic topology
124
2
Classifying spaces and K(π,n)s
126
3
Some more homological algebra
129
6
Universal coefficients in homology
129
1
The Kunneth theorem
130
1
Hom functors and universal coefficients in cohomology
131
2
Proof of the universal coefficient theorem
133
1
Relations between ⨷ and Hom
133
2
Axiomatic and cellular cohomology theory
135
8
Axioms for cohomology
135
1
Cellular and singular cohomology
136
1
Cup products in cohomology
137
1
An example: RP2 and the Borsuk-Ulam theorem
138
2
Obstruction theory
140
3
Derivations of properties from the axioms
143
6
Reduced cohomology groups and their properties
143
1
Axioms for reduced cohomology
144
1
Mayer-Vietoris sequences in cohomology
145
1
Lim1 and the cohomology of colimits
146
1
The uniqueness of the cohomology of CW complexes
147
2
The Poincare duality theorem
149
14
Statement of the theorem
149
2
The definition of the cap product
151
2
Orientations and fundamental classes
153
2
The proof of the vanishing theorem
155
3
The proof of the Poincare duality theorem
158
3
The orientation cover
161
2
The index of manifolds; manifolds with boundary
163
8
The Euler characteristic of compact manifolds
163
1
The index of compact oriented manifolds
164
2
Manifolds with boundary
166
1
Poincare duality for manifolds with boundary
167
2
The index of manifolds that are boundaries
169
2
Homology, cohomology, and K(π,n)
171
12
K(π,n)s and homology
171
2
K(π,n)s and cohomology
173
2
Cup and cap products
175
3
Postnikov systems
178
2
Cohomology operations
180
3
Characteristic classes of vector bundles
183
16
The classification of vector bundles
183
2
Characteristic classes for vector bundles
185
2
Stiefel-Whitney classes of manifolds
187
2
Characteristic numbers of manifolds
189
1
Thom spaces and the Thom isomorphism theorem
190
2
The construction of the Stiefel-Whitney classes
192
1
Clern, Pontryagin, and Euler classes
193
3
A glimpse at the general theory
196
3
An introduction to K-theory
199
16
The definition of K-theory
199
3
The Bott periodicity theorem
202
2
The splitting principle and the Thom isomorphism
204
3
The Chern character; almost complex structures on spheres
207
2
The Adams operations
209
2
The Hopf invariant one problem and its applications
211
4
An introduction to cobordism
215
16
The cobordism groups of smooth closed manifolds
215
1
Sketch proof that N* is isomorphic to π*(TO)
216
3
Prespectra and the algebra H*(TO;Z2)
219
3
The Steenrod algebra and its coaction on H*(TO)
222
2
The relationship to Stiefel-Whitney numbers
224
2
Spectra and the computation of π*(TO) = π*(MO)
226
2
An introduction to the stable category
228
3
Suggestions for further reading
231
8
1. A classic book and historical references
231
1
2. Textbooks in algebraic topology and homotopy theory
231
1
3. Books on CW complexes
232
1
4. Differential forms and Morse theory
232
1
5. Equivariant algebraic topology
233
1
6. Category theory and homological algebra
233
1
7. Simplicial sets in algebraic topology
233
1
8. The Serre spectral sequence and Serre class theory
233
1
9. The Eilenberg-Moore spectral sequence
233
1
10. Cohomology operations
234
1
11. Vector bundles
234
1
12. Characteristic classes
234
1
13. K-theory
235
1
14. Hopf algebras; the Steenrod algebra, Adams spectral sequence
235
1
15. Cobordism
236
1
16. Generalized homology theory and stable homotopy theory
236
1
17. Quillen model categories
236
1
18. Localization and completion; rational homotopy theory
237
1
19. Infinite loop space theory
237
1
20. Complex cobordism and stable homotopy theory
238
1
21. Follow-ups to this book
238
1
Index
239