In 1979 Cohen, Moore and Neisendorfer determined the decomposition
into indecomposable pieces, up to homotopy, of the loop space on the
mod Moore
space for primes
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
are still open. In this paper we reduce to algebra the determination
of the base indecomposable factor in the decomposition of the
mod
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
Moore space or to an improvement in the known bounds for the exponent of the
–torsion
in the homotopy groups of spheres.
Keywords
mod $2$ Moore spaces, homotopy decomposition, modular
representation theory of the symmetric groups