Equivariant Analytic Localization Of Group Representations
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Equivariant Analytic Localization of Group Representations
Author: Laura Ann Smithies
language: en
Publisher: American Mathematical Soc.
Release Date: 2001
This book is intended for graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.
Equivariant Orthogonal Spectra and $S$-Modules
Author: M. A. Mandell
language: en
Publisher: American Mathematical Soc.
Release Date: 2002
The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of $S$-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of $S$-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to $S$-modules. We then develop the equivariant theory.For a compact Lie group $G$, we construct a symmetric monoidal model category of orthogonal $G$-spectra whose homotopy category is equivalent to the classical stable homotopy category of $G$-spectra. We also complete the theory of $S_G$-modules and compare the categories of orthogonal $G$-spectra and $S_G$-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.
From Representation Theory to Homotopy Groups
Author: Donald M. Davis
language: en
Publisher: American Mathematical Soc.
Release Date: 2002
A formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations has been obtained by Bousfield. This work applys this theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989.