A Basic Course in Algebraic Topology

Front Cover
Springer Science & Business Media, 1991 - Mathematics - 428 pages
3 Reviews
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
K Triangulations of Compact Surfaces
14
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
CHAPTER
60
4 Free Products of Groups
71
Homology of CWCompIexes
225
4 The Homology Groups of a CWComplex
232
5 Incidence Numbers and Orientations of Cells
238
7 Determination of Incidence Numbers
244
CHAPTER X
254
3 Definition and Basic Properties of Homology with
262
6 The Universal Coefficient Theorem
269
CHAPTER XI
279

6 The Presentation of Groups by Generators and Relations
78
Seifert and Van Kampen Theorem on the Fundamental Group
86
4 Second Application of Theorem 2 1
95
6 Application to Knot Theory
103
Covering Spaces
117
10 The Existence Theorem for Covering Spaces
140
CHAPTER VI
147
CHAPTER VII
158
4 The Homotopy Property of the Induced Homomorphisms
166
The Main Properties of Relative Homology Groups
173
CHAPTER VIII
186
3 Homology of Finite Graphs
192
4 Homology of Compact Surfaces
201
5 The MayerVietoris Exact Sequence
207
7 The Relation between the Fundamental Group
217
Proof of the EilenbergZilber Theorem
286
Formulas for the Homology Groups of Product Spaces
300
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
H The Hopf Invariant
402
2 Differentiable Singular Chains
408
APPENDIX
419
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