The decomposition of the loop space of the mod $2$ Moore space

Abstract

In 1979 Cohen, Moore and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod $p$ Moore space for primes $p>2$ and used the results to find the best possible exponent for the homotopy groups of spheres and for Moore spaces at such primes. The corresponding problems for $p=2$ are still open. In this paper we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod $2$ Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod $2$ Moore space or to an improvement in the known bounds for the exponent of the $2$–torsion in the homotopy groups of spheres.