This expertly written volume presents a useful, coherent account of the theory of the cohomology ring of a finite group. The book employs a modern approach from the point of view of homological algebra, and covers themes such as finite generation theorems, the cohomology of wreath products, the norm map, and variety theory. Prerequisites comprise a familiarity with modern algebra comparable to that offered in introductory graduate courses, although otherwise the book is self-contained. As a result, it will be useful for those already engaged or commencing research in this area of mathematics by providing an up-to-date survey of important techniques and their applications to finite group theory.
Author(s): Leonard Evens
Series: Oxford Mathematical Monographs
Publisher: Oxford University Press
Year: 1991
Language: English
Pages: 169
Tags: Математика;Топология;
The Cohomology of Groups......Page 1
Preface......Page 3
Contents......Page 9
1.1 Definitions......Page 11
1.2 Note on sign conventions......Page 14
2.1 Cyclic groups......Page 15
2.2 Free groups......Page 16
2.3 The bar resolution......Page 17
2.4 Minimal resolutions......Page 21
2.5 Building new resolutions from old resolutions......Page 26
3.1 Definitions......Page 31
3.2 Computations......Page 35
3.3 Connecting homomorphisms and Bocksteins......Page 36
3.4 The Universal Coefficient Theorem......Page 39
3.5 Cohomology rings of direct products and abelian groups......Page 42
4.1 Restriction and the Eckmann-Shapiro Lemma......Page 45
4.2 Transfer or corestriction......Page 48
5.1 Tensor induced modules......Page 55
5.2 Wreath products and the monomial representation......Page 56
5.3 Cohomology of wreath products......Page 59
5.4 Odd degree and other variations on the theme......Page 64
6.1 Definition of the norm map......Page 67
6.2 Proofs of the properties of the norm......Page 69
6.3 The norm map for elementary abelian p-groups......Page 72
6.4 Serre's theorem......Page 74
7.1 The spectral sequence of a double complex......Page 79
7.2 The LHS spectral sequence of a group extension......Page 82
7.3 Multiplicative structure in the spectral sequence......Page 90
7.4 Finiteness theorems......Page 97
8.1 The variety of a module......Page 103
8.2 Subgroups......Page 107
8.3 Relations with elementary abelian p-subgroups......Page 110
8.4 Complexity......Page 113
9.1 The Quillen stratification of X_G......Page 119
9.2 Quillen's homeomorphism......Page 127
9.3 Avrunin-Scott stratification......Page 132
9.4 The rank variety......Page 134
Appendix 1: The Bockstein......Page 138
Appendix 2: The Yoneda product......Page 139
10.1 The tensor product theorem and applications......Page 141
10.2 Varieties and corestriction......Page 148
10.3 Depth......Page 151
References......Page 157
Table of notation......Page 163
Index......Page 165
