In this paper the concept of join of two permutation representations is defined and the cohomology of this join is computed and shown to have trivial cup-products. This computation is then used to compute the cohomology groups of the p-Sylow subgroup of a symmetric group of order n acting on the set of n elements, and it is shown that the ring structure on these groups is not finitely generated, although it is transitive.

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.

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.

No abstract