Equivariant Moore spaces and the Dade group

Let G be a finite p-group and k be a field of characteristic p. A topological space X is called an n-Moore space if its reduced homology is nonzero only in dimension n. We call a G-CW-complex X an n_-Moore G-space over k if for every subgroup H of G, the fixed point set XH is an n_(H)-Moore space with coefficients in k, where n_(H) is a function of H. We show that if X is a finite n_-Moore G-space, then the reduced homology module of X is an endo-permutation kG-module generated by relative syzygies. A kG-module M is an endo-permutation module if Endk(M)=M⊗kM⁎ is a permutation kG-module. We consider the Grothendieck group of finite Moore G-spaces M(G), with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies.