The Classifying Space of a Permutation Representation

Abstract

Abstract. In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. An example of this classifying space is given by a generalization of the infinite join construction that defines the standard example of a classifying space of a group. In a previous paper of the author, the join of two permutation representations was defined, and it was shown that the cohomology ring of the join was trivial. In this paper the classifying space of the join of two permutation representations is shown to be the topological join of the two classifying spaces and from this the triviality of the cup-product is derived topologically.I. Introduction. In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. It will be shown that the cohomology groups of this classifying space (with coefficients in Z) are equal to the corresponding cohomology groups of the permutation representation. (The cohomology theory of permutation representations can be found in Snapper [9] and Harris [6].) The standard example of a classifying space of a group will be generalized to a classifying space of a permutation representation.In a previous paper of the author [1], it was proved that the cohomology of the join of two permutation representations (to be defined later) has trivial cup-products. That paper noted that the algebraic, computational proof of this result was lengthy. In this paper it will be shown that a classifying space for the join of two permutation representations is given by the topological join of their classifying spaces, which will lead to a brief topological proof of this result.

On the sectional category of subgroup inclusions and Adamson cohomology theory

“…In an analogous way to the non-relative case, Hochschild defined the relative extension functor as Note that the Adamson relative cohomology defined like this can be seen as a particular case of the cohomology of a permutation representation, with G/H as the base G-set, see Blowers [4].…”