A partial $A_\infty$-structure on the cohomology of $C_n\times C_m$

TL;DR

This paper investigates the $A_ abla$-structure on the cohomology of product groups $C_n imes C_m$ over a field of characteristic 2, revealing limits on operations and identifying an infinite family of higher operations.

Contribution

It provides a partial characterization of the $A_ abla$-structure on the cohomology of product cyclic groups, including bounds on operations and discovering new higher operations.

Findings

Limits on non-vanishing low-arity operations.

Existence of an infinite family of non-vanishing higher operations.

Results specific to groups with orders as powers of 2.

Abstract

Suppose k is a field of characteristic 2, and $n,m\geq 4$ powers of 2. Then the $A_\infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $(H^*(C_m,k)$ are well known. We give results characterizing an $A_\infty$-structure on $H^*(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.