Homology and some homotopy decompositions for the James filtration on spheres

The filtrations on the James construction on spheres, $J_{k}\left (S^{2n}\right )$, have played a major role in the study of the double suspension $S^{2n-1}\to \Omega ^2 S^{2n+1}$ and have been used to get information about the homotopy groups of spheres and Moore spaces and to construct product decompositions of related spaces. In this paper we calculate $H_*\left ( \Omega J_{k}\left (S^{2n}\right ); {\mathbb {Z}}/p{\mathbb {Z}}\right )$ for odd primes $p$. When $k$ has the form $p^t-1$, the result is well known, but these are exceptional cases in which the homology has polynomial growth. We find that in general the homology has exponential growth and in some cases also has higher $p$-torsion. The calculations are applied to construct a $p$-local product decomposition of $\Omega J_{k}\left (S^{2n}\right )$ for $k<p^2-p$ which demonstrates a mod $p$ homotopy exponent in these cases.