Representations and Cohomology: Volume 2, Cohomology of Groups and Modules

כריכה קדמית
Cambridge University Press, 1991 - 292 עמודים
This is the second of two volumes which will provide an introduction to modern developments in the representation theory of finite groups and associative algebras. The subject is viewed from the perspective of homological algebra and the theory of representations of finite dimensional algebras; the author emphasises modular representations and the homological algebra associated with their categories. This volume concentrates on the cohomology of groups, always with representations in view, however. It begins with a background reference chapter, then proceeds to an overview of the algebraic topology and K-theory associated with cohomology of groups, especially the work of Quillen. Later chapters look at algebraic and topological proofs of the finite generation of the cohomology ring of a finite group, and an algebraic approach to the Steenrod operations in group cohomology. The book cumulates in a chapter dealing with the theory of varieties for modules. Much of the material presented here has never appeared before in book form. Consequently students and research workers studying group theory, and indeed algebra in general, will be grateful to Dr Benson for supplying an exposition of a good deal of the essential results of modern representation theory.
 

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תוכן

Background from algebraic topology
1
12 Homotopy groups
2
13 The Hurewicz theorem
9
14 The Whitehead theorem
11
15 CWcomplexes and cellular homology
12
16 Fibrations and fibre bundles
16
17 Paracompact spaces
21
18 Simplicial sets
23
46 Adem relations
144
47 Serres theorem on products of Bocksteins
148
48 Steenrod operations and spectral sequences
150
Varieties for modules and multiple complexes
153
52 Restriction to elementary abelian subgroups
155
53 Poincare series and complexity
157
54 Varieties and commutative algebra
161
extraspecial 2groups
169

19 The Milnor exact sequence
27
Cohomology of groups
29
22 EilenbergMac Lane spaces
32
23 Principal Gbundles
35
24 Classifying spaces
37
25 Ktheory
44
26 Characteristic classes
48
27 Transfer
51
28 Stable cohomotopy and the Segal conjecture
56
29 Cohomology of general linear groups
60
210 The plus construction and algebraic Ktheory
68
211 Hochschild homology
73
212 Free loops on BG
77
213 Cyclic homology
80
214 Cyclic sets
85
215 Extended centralisers
90
Spectral sequences
93
32 The spectral sequence of a filtered chain complex
98
33 The spectral sequence of a fibration
104
34 The spectral sequence of a double complex
106
35 The spectral sequence of a group extension
109
36 The Kiinneth spectral sequence
111
37 The EilenbergMoore spectral sequence
112
38 The Atiyah spectral sequence
114
39 Products in spectral sequences
115
310 Equivariant cohomology and finite generation
117
The Evens norm map and the Steenrod algebra
121
42 Finite generation of cohomology
126
43 The Bockstein homomorphism
132
44 Steenrod operations
136
45 Proof of the properties
138
56 The Quillen stratification
172
57 Varieties for modules
176
58 Rank varieties
180
59 The modules L
186
510 Periodic modules
191
511 Andrews theorem
192
512 The variety of an indecomposable kGmodule is connected
194
dihedral 2groups
195
514 Multiple complexes
199
515 Gaps in group cohomology
205
516 Isomorphisms in group cohomology
208
517 Poincare duality
209
518 CohenMacaulay cohornology rings
211
Group actions and the Steinberg module
215
62 Gposets
217
63 The Lefschetz Invariant
218
64 Equivariant homotopy
220
65 Quillens lemma
222
66 Equivalences of subgroup complexes
224
67 The generalised Steinberg module
226
a crash course
228
69 Steinberg module inversion and Alperins conjecture
233
Local coefficients on subgroup complexes
237
72 Constructions on coefficient systems
239
73 Chain complexes and homology of coefficient systems
242
74 Symplectic and orthogonal groups
243
75 Smiths theorem and universal coefficient systems
246
Bibliography
251
Index
269
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