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87,40 €Chapter 1. Simplicial Homology 12
1. Definition and Some Properties 12
2. Invariance of Homology 17
3. Relative Homology 23
4. Cohomology and Universal Coefficient Theorem 32
5. Calculations 46
6. The Euler Characteristic and the Lefschetz Theorem 62
Chapter 2. Cohomology Rings 70
1. Multiplication in Cohomology 70
2. Homology and Cohomology of Manifolds 80
3. The Künneth Theorem 106
Chapter 3. Applications of Simplicial Homology 122
1. Homology and Homotopy 122
2. Characteristic Classes 142
3. Group Actions 184
4. Steenrod Squares 195
Chapter 4. Singular Homology 206
1. Basic Definitions and Properties 206
2. The Poincaré and Lefschetz Isomorphisms for Topological Manifolds 238
3. Characteristic Classes: Continuation 263
Chapter 5. Cech Cohomology and de Rham Cohomology 274
1. Sheaf Cohomology 274
2. De Rham Cohomology 286
3. The de Rham Theorem 300
Chapter 6. Miscellany 312
1. The Alexander Polynomial 312
2. The Arf Invariant 328
3. Embeddings and Immersions 336
4. Complex Manifolds 350
5. Lie Groups and H-Spaces 355
Hints and Solutions 376
Bibliography 414
The book is a continuation of the previous book by the author (Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, Volume 74, American Mathematical Society, 2006). It starts with the definition of simplicial homology and cohomology, with many examples and applications. Then the Kolmogorov–Alexander multiplication in cohomology is introduced. A significant part of the book is devoted to applications of simplicial homology and cohomology to obstruction theory, in particular, to characteristic classes of vector bundles. The later chapters are concerned with singular homology and cohomology, and Cech and de Rham cohomology. The book ends with various applications of homology to the topology of manifolds, some of which might be of interest to experts in the area.
The book contains many problems; almost all of them are provided with hints or complete solutions.
Author
V. V. Prasolov: Independent University of Moscow, Moscow, Russia
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