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 $\Omega P^m(p^r)$ 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 talk we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod $2$ Moore space. 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.