The cohomology ring of certain families of periodic virtually cyclic groups

TL;DR

This paper computes the integral cohomology rings of specific virtually cyclic groups, revealing their periodic Farrell cohomology, which advances understanding of their algebraic topology.

Contribution

It provides explicit calculations of the cohomology rings for certain families of virtually cyclic groups, a novel contribution to algebraic topology.

Findings

Computed integral cohomology rings for the groups

Established periodicity in Farrell cohomology for these groups

Enhanced understanding of the algebraic structure of these groups

Abstract

Let G be a virtually cyclic of the form (Z_a x Z_b) x Z or [Z_a x (Z b x Q_{2^i})] x Z. We compute the integral cohomology ring of G, and then obtain the periodicity of the Farell cohomology of these groups.