Realization of Permutation Modules via Alexandroff Spaces

Abstract

We raise the question of the realizability of permutation mod ules in the context of Kahn’s realizability problem for abstract groups and the G-Moore space problem. Specifically, given a finite group G, we con sider a collection {Mi}ni=1 of finitely generated ZG-modules that admit asubmodule decomposition on which G acts by permuting the summands. Then we prove the existence of connected finite spaces X that realize each Mi as its i-th homology, G as its group of self-homotopy equiva lences E(X), and the action of G on each Mi as the action of E(X) onHi(X; Z)