A partial $A_infty$-structure on the cohomology of $C_ntimes C_m$

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.

💡 Research Summary

The paper investigates the partial A∞‑structure on the cohomology algebra of the direct product of two cyclic 2‑groups, Cₙ × Cₘ, where n = 2ʳ and m = 2ˢ with r, s ≥ 2, over a field k of characteristic 2. The cohomology rings H⁎(Cₙ,k) and H⁎(Cₘ,k) are classical: each is isomorphic to a tensor product of an exterior algebra on a degree‑1 generator u and a polynomial algebra on a degree‑2 generator v, i.e. k