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

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 ${\bar{H}}_{\ast }(X;\text{zmodp})$ of $X$, we use a decomposition of $\Omega\Sigma X$ due to the second author and computations in modular representation theory to show there are arbitrarily large integers $i$ such that $\mathrm{\Omega }{\mathrm{\Sigma }}^{i}X$ is a homotopy retract of $\mathrm{\Omega }\mathrm{\Sigma }X$. This implies the stable homotopy groups of $\mathrm{\Sigma }X$ are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for $\mathrm{\Sigma }X$. Under additional assumptions on ${\bar{H}}_{\ast }(X;\text{zmodp})$, we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of $\mathrm{\Omega }\mathrm{\Sigma }X$ that has infinitely many finite $H$-spaces as factors.