# On ${\rm mod}~p$ $A_p$-spaces

## TL;DR

This paper establishes a necessary condition for mod p A_p-structures on spaces, proves finiteness of such spaces of given rank, and classifies rank 3 mod 3 homotopy associative H-spaces.

## Contribution

It introduces a new necessary condition for the existence of mod p A_p-structures and provides a classification of rank 3 mod 3 homotopy associative H-spaces.

## Key findings

- Necessary condition for mod p A_p-structures
- Finiteness of mod p A_p-spaces of fixed rank
- Classification of rank 3 mod 3 homotopy associative H-spaces

## Abstract

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