Higher homotopy associativity of power maps on finite H-spaces

Abstract

Let $p$ be an odd prime, and let $\lambda \in \mathbb{Z}$. Consider the loop space $Y_t=S_{\left(p\right)}^{2t-1}$ for $t\ge 1$ with $t\left|\right(p-1)$. Then we first determine the condition for the power map ${\Phi }_{\lambda }$ on $Y_t$ to be an $A_p$-map. We next assume that $X$ is a simply connected ${\mathbb{F}}_p$-finite $A_p$-space and that $\lambda$ is a primitive $(p-1)$st root of unity mod $p$. Our results show that if the reduced power operations $\{{\mathcal{P}}^i{\}}_{i\ge 1}$ act trivially on the indecomposable module $Q{H}^{\ast }(X;{\mathbb{F}}_p)$ and the power map ${\Phi }_{\lambda }$ on $X$ is an $A_n$-map with $n>(p-1)/2$, then $X$ is ${\mathbb{F}}_p$-acyclic.