CompactMATH 1931 – 1997 \begin{matrix}c \\ ◯ \end{matrix}FIZ Karlsruhe & Springer-Verlag Berlin-Heidelberg

Zbl. Math. 731.20001

Benson, D.J. Representations and cohomology. II: Cohomology of groups and modules. (English) Cambridge Studies in Advanced Mathematics, 31. New York etc.: Cambridge University Press x, 278 p. £35.00; $ 59.95 (1991). [ISBN 0-521-36135-4/hbk]

Classification:

20-01 Textbooks (group theory)

20C15 Ordinary representations and characters of groups

20J06 Cohomology of finite groups 19D55 K-theory and homology 18G10 Resolutions (homological algebra) 18G15 Ext and Tor, generalizations 18G30 Simplicial objects in a category 55Q05 General theory of homotopy groups 18G40 Spectral sequences (homological algebra) 55U15 Chain complexes 55U25 Homology of a product, Kuenneth formula 55U20 Universal coefficient theorems, Bockstein operator 19A22 Frobenius induction, etc. 20-02 Research monographs (group theory)

Keywords:

cohomology of groups; singular homology groups; CW-complexes; cellular homology; fibrations; fibre bundles; simplicial homology; Eilenberg-MacLane spaces; K-theory; representation ring; stable cohomotopy theory; classifying space; Burnside ring; Hochschild homology; cyclic homology; spectral sequences; Evens norm map; Steenrod operations; products of Bocksteins; cohomology rings; varieties for modules; actions of finite groups; simplicial complexes; chain complex; subgroup complexes; simple modules; Chevalley groups

The second volume of the author’s book on representations and cohomology deals mainly with cohomology of groups and modules [for a review of Vol. I cf. Zbl. 718.20001]. An interesting feature is that the author approaches the subject both from the algebraic and the topological point of view. However, the algebraic aspects are usually treated in more detail, the topological arguments are more sketchy and informal. Several of the topics discussed appear in book form for the first time. The book will certainly have a considerable influence on the further development of the subject. The book is divided into seven chapters, the first of which presents background material from algebraic topology, including homotopy and singular homology groups, CW-complexes and cellular homology, fibrations and fibre bundles, paracompact spaces and simplicial homology. The second chapter gives various approaches to group cohomology, the algebraic definition, the definition in terms of Eilenberg-MacLane spaces, and the definition via principal G-bundles. Then the K-theory of spaces is developed, and Atiyah’s theorem relating the K-theory of the classifying space of a finite group to its representation ring is stated. Next Chern and Stiefel-Whitney classes of representations are introduced. The transfer for generalized cohomology theories is defined and applied to stable cohomotopy theory; in particular, Segal’s conjecture (proved by G. Carlsson) relating the stable cohomotopy of the classifying space of a finite group to its Burnside ring is stated. Then Quillen’s computation of the cohomology of finite general linear groups is outlined. This leads to the definition of algebraic K-groups for a ring. These are then related to Hochschild homology and cyclic homology. The third chapter introduces the formalism and main examples of spectral sequences, including those of Lyndon-Hochschild-Serre, Kuenneth and Eilenberg-Moore. An application is Venkov’s theorem on the finite generation of cohomology. Chapter 4 defines the Evens norm map and presents the algebraic proof of Venkov’s theorem. Moreover, the Steenrod operations are introduced algebraically, and Serre’s theorem on products of Bocksteins is proved.

Chapter 5 constitutes the main part of the book. The complexity of a module is defined, and the Alperin-Evens theorem is proved. Applications are Chouinard’s theorem on projective modules and Quillen’s theorem on the Krull dimension of cohomology rings. The cohomology of extraspecial 2-groups is computed. After a short introduction into the necessary commutative algebra, Quillen’s stratification of the cohomology ring is proved. Then varieties for modules are introduced, and the Avrunin-Scott theorem is proved. Next rank varieties for modules over elementary abelian groups are defined and shown to coincide with the cohomological varieties. The theory is applied to periodic modules and to modules for the dihedral 2-groups. Carlson’s indecomposability theorem is proved. The Benson-Carlson results on resolutions via tensor products of periodic complexes are presented. Applications are theorems on gaps in cohomology and on isomorphic cohomology rings. The chapter finishes with a discussion of Poincare duality and Cohen-Macaulay cohomology rings. Chapter 6 deals with actions of finite groups on posets and simplicial complexes, in particular the poset of nontrivial p-subgroups. The corresponding chain complex, the Euler characteristic and the Lefschetz virtual module are introduced. The equivalence of various subgroup complexes is discussed. The Steinberg virtual module is defined and shown to be projective. An application is K. Brown’s theorem on the Euler characteristic of a finite group. A short introduction to finite Chevalley groups shows the relation with the Tits building and the classical Steinberg module. As another application of the techniques in this chapter, Robinson’s reformulation of Alperin’s conjecture is presented. The final chapter develops to some extent the Ronan-Smith theory of sheaves on simplicial complexes with group actions. Main examples are the classical groups and fixed point sheaves. Moreover, Smith’s theorem on simple modules for Chevalley groups is stated.

B.Kuelshammer (Augsburg) Citations: Zbl.718.20001 Published: 1991 Document Type: B