A Basic Course in Algebraic Topology

Front Cover
Springer Science & Business Media, 1991 - Mathematics - 428 pages
The main purpose of this book is to give a systematic treatment of singular homology and cohomology theory. It is in some sense a sequel to the author's previous book in this Springer-Verlag series entitled Algebraic Topology: An Introduction. This earlier book is definitely not a logical prerequisite for the present volume. However, it would certainly be advantageous for a prospective reader to have an acquaintance with some of the topics treated in that earlier volume, such as 2-dimensional manifolds and the funda mental group. Singular homology and cohomology theory has been the subject of a number of textbooks in the last couple of decades, so the basic outline of the theory is fairly well established. Therefore, from the point of view of the mathematics involved, there can be little that is new or original in a book such as this. On the other hand, there is still room for a great deal of variety and originality in the details of the exposition. In this volume the author has tried to give a straightforward treatment of the subject matter, stripped of all unnecessary definitions, terminology, and technical machinery. He has also tried, wherever feasible, to emphasize the geometric motivation behind the various concepts.
 

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Contents

4 Examples of Compact Connected 2Manifolds
5
6 Triangulations of Compact Surfaces
14
8 The Euler Characteristic of a Surface
26
The Fundamental Group
35
4 The Effect of a Continuous Mapping on the Fundamental Group
42
The Brouwer FixedPoint Theorem in Dimension 2
51
7 The Fundamental Group of a Product Space
52
8 Homotopy Type and Homotopy Equivalence of Spaces
54
CHAPTER VIII
186
3 Homology of Finite Graphs
192
4 Homology of Compact Surfaces
201
5 The MayerVietoris Exact Sequence
207
6 The JordanBrouwer Separation Theorem and Invariance of Domain
211
7 The Relation between the Fundamental Group and the First Homology Group
217
References
224
Homology of CWCompIexes
225

References
59
CHAPTER HI Free Groups and Free Products of Groups
60
3 Free Abelian Groups
63
4 Free Products of Groups
71
5 Free Groups
75
6 The Presentation of Groups by Generators and Relations
78
7 Universal Mapping Problems
81
References
85
Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces Applications
86
2 Statement and Proof of the Theorem of Seifert and Van Kampen
87
3 First Application of Theorem 2 1
91
4 Second Application of Theorem 2 1
95
5 Structure of the Fundamental Group of a Compact Surface
96
6 Application to Knot Theory
103
7 Proof of Lemma 2 4
108
References
116
Covering Spaces
117
3 Lifting of Paths to a Covering Space
123
4 The Fundamental Group of a Covering Space
126
5 Lifting of Arbitrary Maps to a Covering Space
127
6 Homomorphisms and Automorphisms of Covering Spaces
130
7 The Action of the Group nXx on the Set pl x
133
8 Regular Covering Spaces and Quotient Spaces
135
The BorsukUlam Theorem for the 2Sphere
138
10 The Existence Theorem for Covering Spaces
140
References
146
CHAPTER VI
147
3 Some Examples of Problems which Motivated the Development of Homology Theory in the Nineteenth Century
149
References
157
CHAPTER VII
158
3 The Homomorphism Induced by a Continuous Map
163
4 The Homotopy Property of the Induced Homomorphisms
166
5 The Exact Homology Sequence of a Pair
169
6 The Main Properties of Relative Homology Groups
173
7 The Subdivision of Singular Cubes and the Proof of Theorem 6 4
178
3 CWComplexes
228
4 The Homology Groups of a CWComplex
232
5 Incidence Numbers and Orientations of Cells
238
6 Regular CWComplexes
243
7 Determination of Incidence Numbers for a Regular Cell Complex
244
8 Homology Groups of a Pseudomanifold
249
References
253
CHAPTER x
254
3 Definition and Basic Properties of Homology with Arbitrary Coefficients
262
4 Intuitive Geometric Picture of a Cycle with Coefficients in G
266
5 Coefficient Homomorphisms and Coefficient Exact Sequences
267
6 The Universal Coefficient Theorem
269
7 Further Properties of Homology with Arbitrary Coefficients
274
References
278
CHAPTER XI
279
2 The Product of CWComplexes and the Tensor Product of Chain Complexes
280
3 The Singular Chain Complex of a Product Space
282
4 The Homology of the Tensor Product of Chain Complexes The Kiinneth Theorem
284
5 Proof of the EilenbergZilber Theorem
286
6 Formulas for the Homology Groups of Product Spaces
300
References
303
CHAPTER XII
305
5 Geometric Interpretation of Cochains Cocycles etc
316
CHAPTER XIII
323
10 Remarks on the Coefficients for the Various Products
343
Duality Theorems for the Homology of Manifolds
350
3 Cohomology with Compact Supports
358
5 Applications of the Poincare Duality Theorem
365
7 Duality Theorems for Manifolds with Boundary
375
CHAPTER XV
394
4 The Hopf Invariant
402
2 Differentiable Singular Chains
408
APPENDIX
419
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About the author (1991)

William S. Massey (1920-2017) was an American mathematician known for his work in algebraic topology. The Blakers-Massey theorem and the Massey product were both named for him. His textbooks Singular Homology Theory and Algebraic Topology: An Introduction are also in the Graduate Texts in Mathematics series.

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