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