This self-contained text is suitable for advanced undergraduate and graduate students and may be used either after or concurrently with courses in general topology and algebra. It surveys several algebraic invariants: the fundamental group, singular and Cech homology groups, and a variety of cohomology groups.
Proceeding from the view of topology as a form of geometry, Wallace emphasizes geometrical motivations and interpretations. Once beyond the singular homology groups, however, the author advances an understanding of the subject's algebraic patterns, leaving geometry aside in order to study these patterns as pure algebra. Numerous exercises appear throughout the text. In addition to developing students' thinking in terms of algebraic topology, the exercises also unify the text, since many of them feature results that appear in later expositions. Extensive appendixes offer helpful reviews of background material.
Reprint of the W. A. Benjamin, Inc., New York, 1970 edition.
Author(s): Andrew H. Wallace
Series: Dover Books on Mathematics (Reprint 1970)
Publisher: Dover Publications
Year: 2007
Language: English
Pages: C+X+272+B
Cover
S Title
Algebraic Topology Homology and Cohomology
Copyright
© 1970 by W. A. Benjamin, Inc
ISBN 0-8053-9482-6
LCCN 79-108005
Copyright Dover
1970, 1993 by Andrew H. Wallac
ISBN-I 3: 978-0-486-46239-4
ISBN-10: 0-486-46239-0
QA612.W33 2007 514'.2--dc22
LCCN 2007010621
Preface
Contents
1 Singular Homology Theory
1-1. EUCLIDEAN SIMPLEXES
1-2. LINEAR MAPS
1-3. SINGULAR SIMPLEXES AND CHAINS
1-4. THE BOUNDARY OPERATOR
1-5. CYCLES AND HOMOLOGY
1-6. INDUCED HOMOMORPHISMS
1-7. THE MAIN THEOREMS
1-8. THE DIMENSION THEOREM
1-9. THE EXACTNESS THEOREM
1-10. THE HOMOTOPY THEOREM
1-11. THE EXCISION THEOREM
2 Singular and Simplicial Homology
2-1. THE AXIOMATIC APPROAC
2-2. SIMPLICIAL COMPLEXES
2-3. THE DIRECT SUM THEOREM
2-4. THE DIRECT SUM THEOREM FOR COMPLEXES
2-5. HOMOLOGY GROUPS OF CELLS AND SPHERES
2-6. ORIENTATION
2-7. HOMOLOGY GROUPS OF A SIMPLICIAL PAIR
2-8. FORMAL DESCRIPTION OF SIMPLICIAL HOMOLOGY
2-9. CELL COMPLEXES
2-10. CANONICAL BASES
3 Chain Complexes-Homologyand Cohomology
3-1. A PAUSE FOR MOTIVATION
3-2. CHAIN COMPLEXES
3-3. CHAIN HOMORPHISMS
3-4. INDUCED HOMOMORPHISMS ON HOMOLOGY AND COHOMOLOGY GROUPS
3-5. CHAIN HOMOTOPY
3-6. THE ALGEBRAIC HOMOTOPY THEOREM
3-7. SOME APPLICATIONS OF ALGEBRAIC HOMOTOPY
3-8. SUBCOMPLEXES AND QUOTIENT COMPLEXES
3-9. COMPUTATION OF COHOMOLOGY GROUPS
3-10. ATTACHING CONES AND CELLS
4 The Cohomology Ring
4-1. MOTIVATION
4-2. U PRODUCT FOR SINGULAR COHOMOLOGY
4-3. DEFINITION OF THE SINGULAR COHOMOLOGY RING
4-4. SOME EXAMPLES OF COMPUTATIONS
4-5. TENSOR PRODUCTS AND CUP PRODUCTS
4-6. TENSOR PRODUCTS OF CHAIN AND COCHAIN COMPLEXES
4-7. COHOMOLOGY OF A TENSOR PRODUCT
4-8. TENSOR PRODUCTS AND TOPOLOGICAL PRODUCTS
4-9. SINGULAR CHAINS ON A PRODUCT
4-10. SOME ACYCLICITY ARGUMENTS
4-11. THE DIAGONAL MAP
4-12. SUMMING UP
4-13. THE x PRODUCT
4-14. ANTICOMMUTATIVITY OF THE U PRODUCT
4-15. THE COHOMOLOGY RING OF A PRODUCT SPACE
4-16. SOME COMPUTATIONS
5 Cech Homology Theory -- The Construction
5-1. INTRODUCTION
5-2. THE NERVE OF A COVERING
5-3. HOMOLOGY GROUPS OF A COVERING
5-4. INVERSE LIMITS
5-5. LIMITS OF HOMOMORPHISMS
5-6. CONSTRUCTION OF THE CECH HOMOLOGY GROUPS
5-7. CECH HOMOLOGY FOR SIMPLICIAL PAIRS
6 Further Properties of Cech Homology
6-1. INTRODUCTION
6-2. THE HOMOTOPY THEOREM FOR CECH HOMOLOGY
6-3. THE EXCISION THEOREM IN CECH HOMOLOGY
6-4. THE PARTIAL EXACTNESS THEOREM
6-5. THE CONTINUITY THEOREM
6-6. COMPARISON OF SINGULAR AND CECH HOMOLOGY THEORIES
6-7. RELATIVE HOMEOMORPHISMS
6-8. THE UNIQUENESS THEOREM FOR CECH HOMOLOGY
7 Cech Cohomology Theory
7-1. INTRODUCTION
7-2. DIRECT LIMITS
7-3. THE DEFINITION OF CECH COHOMOLOGY GRO UPS
7-4. INDUCED HOMOMORPHISMS
7-5. THE EXCISION AND HOMOTOPY THEOREMS
7-6. THE EXACTNESS THEOREM
7-7. THE CONTINUITY THEOREM
Appendix A The Fundamental Group
A-1. PATHS, THEIR COMPOSITIONS AND HOMOTOPIES
A-2. GROUP SPACES
Appendix B General Topology
B-1. SET THEORY
B-2. GEOMETRIC NOTIONS
B-3. TOPOLOGICAL SPACES
B-4. CONTINUOUS MAPS
B-5. PRODUCT AND QUOTIENT SPACES
B-6. COMPACT SPACES
B-7. CONNECTED SPACES
Bibliography
Index
Back Cover
