
Introduction  Table of Contents  Supplementary Material 
Mathematical Surveys and Monographs 2010; 318 pp; hardcover Volume: 169 ISBN10: 0821851896 ISBN13: 9780821851890 List Price: US$97 Member Price: US$77.60 Order Code: SURV/169 See also: The Connective KTheory of Finite Groups  R R Bruner and J P C Greenlees Parametrized Homotopy Theory  J P May and J Sigurdsson  This book is about equivariant real and complex topological \(K\)theory for finite groups. Its main focus is on the study of real connective \(K\)theory including \(ko^*(BG)\) as a ring and \(ko_*(BG)\) as a module over it. In the course of their study the authors define equivariant versions of connective \(KO\)theory and connective \(K\)theory with reality, in the sense of Atiyah, which give wellbehaved, Noetherian, uncompleted versions of the theory. They prove local cohomology and completion theorems for these theories, giving a means of calculation as well as establishing their formal credentials. In passing from the complex to the real theories in the connective case, the authors describe the known failure of descent and explain how the \(\eta\)Bockstein spectral sequence provides an effective substitute. This formal framework allows the authors to give a systematic calculation scheme to quantify the expectation that \(ko^*(BG)\) should be a mixture of representation theory and group cohomology. It is characteristic that this starts with \(ku^*(BG)\) and then uses the local cohomology theorem and the Bockstein spectral sequence to calculate \(ku_*(BG)\), \(ko^*(BG)\), and \(ko_*(BG)\). To give the skeleton of the answer, the authors provide a theory of \(ko\)characteristic classes for representations, with the Pontrjagin classes of quaternionic representations being the most important. Building on the general results, and their previous calculations, the authors spend the bulk of the book giving a large number of detailed calculations for specific groups (cyclic, quaternion, dihedral, \(A_4\), and elementary abelian 2groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification of the GromovLawsonRosenberg conjecture for several new classes of finite groups. Readership Graduate students and research mathematicians interested in connective \(K\)theory. Reviews "The book is very carefully written, including many diagrams and tables, and also a thorough review of the authors' previous work on the complex case."  Donald M. Davis, Mathematical Reviews 


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