A guide to mackey functors

Abstract

Publisher Summary This chapter provides an overview of Mackey functors. A Mackey functor is an algebraic structure possessing operations which behave in a similar manner as the induction, restriction, and conjugation mappings in group representation theory. Such operations appear in a variety of diverse contexts, such as group cohomology, the algebraic K-theory of group rings, and algebraic number theory. The method of stable elements formulated for all Mackey functors are described in the chapter. Induction theorems are the most important methods of computation. They have a very well-developed and well-known theory, in the context of group representations. The technical machinery is developed to present the applications, which they necessitate, and is carried in an order, which to a large extent reflects chronology. The theory of Mackey functors became more elaborate as it is apparent that they are algebraic structures in their own right, with a theory that fits in the framework of representations of algebras. A new notion of Mackey functor—namely, that of a globally-defined Mackey functor—are structures which have a definition on all finite groups. The uses for these functors discussed are: a method of computing group cohomology, an approach to the stable decomposition of classifying spaces BG, and a framework in which Dade's group of endopermutation modules plays a fundamental role.