Modular representations and the homotopy of low rank p-local CW-complexes

Abstract

Fix an odd prime p and let X be the p-localization of a finite suspended CW-complex. Given certain conditions on the reduced mod-p homology H̃*(X; ℤp) of X, we use a decomposition of ΩΣX due to the second author and computations in modular representation theory to show there are arbitrarily large integers i such that ΩΣiX is a homotopy retract of ΩΣX. This implies the stable homotopy groups of ΣX are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for ΣX. Under additional assumptions on H̃*(X; ℤp), we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of ΩΣX that has infinitely many finite H-spaces as factors. © 2012 Springer-Verlag.