The Postnikov Tower and the Steenrod problem

Abstract

The Steenrod problem asks: given a G-module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a ZG-module is ZG-realizable if and only if it is ZHrealizable for all p-Sylow subgroups H, for all primes pl GI. Let G be a finite group. Let M be a finitely generated ZG-module. We say that M is a Steenrod representation if there exists a Moore space X with G-action such that the homology of X is isomorphic to M as a ZG-module. Recall that a Moore space is a topological space whose reduced homology vanishes in all dimensions except one. Without loss of generality, M will be assumed 2-free [1] from now on. In this paper, the following statement will be proved: M is a Steenrod representation as a ZG-module if and only if M is a Steenrod representation as a 7H-module for all p-Sylow subgroups H of G, for all primes pII GIC. This statement is inspired by papers by J. Arnold [1, 2] and P. Vogel [18]. Arnold [2] showed that if G is cyclic, then every finitely generated ZG-module is a Steenrod representation. And Vogel showed that if for any ZG-module M, we can always find a G-Moore space X realizes M, then G has only cyclic Sylow subgroups. Let us first consider a Moore space X whose homology is isomorphic to M as a 2-module. Let G(X) denote the space of self homotopy equivalences of X and let BG(X) [9] denote the classifying space of the H-space G(X) for a certain fibration defined by Dold and Lashof [10]; see also Stasheff [15]. Since a homotopy equivalence of X induces an automorphism on M, there is a map G(X) -* Aut(M). And the map G(X) -* Aut(M) induces another map a : BG(X) -' BAut(M). On the other hand, since M is a ZG-module, there is a map G -* Aut(M) (= GL(n, 2) if M = Zn), which induces .p : BG -) BAut(M). Following Cooke's theory [9], as pointed out in [12], M is a Steenrod representation as a ZG-module if and only if the lifting f exists in