On $\mathrm{mod}p$ $A_p$-spaces

Abstract

We prove a necessary condition for the existence of the $A_p$-structure on $\mathrm{mod}p$ spaces, and also derive a simple proof for the finiteness of the number of $\mathrm{mod}p$ $A_p$-spaces of given rank. As a direct application, we compute a list of possible types of rank $3$ $\mathrm{mod}3$ homotopy associative $H$-spaces.