A solution of a problem of Steenrod for cyclic groups of prime order

Abstract:

Given a $Z[G]$ module A, we will say a simply connected CW complex X is of type (A, n) if X admits a cellular G action, and ${\tilde H_i}(X) = 0,i \ne n,{H_n}(X) \simeq A$ as $Z[G]$ modules. In [5], R. Swan considers the problem posed by Steenrod of whether or not there are finite complexes of type (A, n) for all finitely generated A and finite G. Using an invariant defined in terms of ${G_0}(Z[G])$, solutions were obtained for $A = {Z_p}$ (p-prime) and $G \subseteq \operatorname {Aut}\;({Z_p})$. The question of infinite complexes of type (A, n) was left open. In this paper we obtain the following complete solution for $Z[{Z_p}]$ modules: There are complexes of type $(A,n)\;(n \geqslant 3)$, and there are finite complexes of type (A, n) if and only if the invariant which corresponds to Swan’s invariant for these modules vanishes.