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

TL;DR

This paper explores the homotopy properties of low rank p-local CW complexes, demonstrating how their stable and unstable homotopy groups relate through modular representation theory and providing new decompositions involving finite H-spaces.

Contribution

It introduces a novel approach linking modular representation theory with homotopy decompositions of p-local CW complexes, confirming the Moore conjecture for certain cases.

Findings

Existence of arbitrarily large integers i with specific homotopy retractions.

Stable homotopy groups are retracts of unstable homotopy groups under certain conditions.

Decomposition of ΩΣX includes infinitely many finite H-spaces as factors.

Abstract

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