The cohomological index of free $\Z/p$-actions is not additive with respect to join

TL;DR

This paper demonstrates through examples that the cohomological index of joins of free $ ext{Z}/p$-actions can vary non-additively, showing the sharpness of previously established inequalities.

Contribution

It provides explicit examples illustrating the non-additivity of the cohomological index under join operations for free $ ext{Z}/p$-actions, confirming the sharpness of known bounds.

Findings

Examples where join index is $k+l+1$

Examples where join index is $k+l-1$

Sharpness of inequalities for cohomological index behavior

Abstract

Let $p$ denote an odd prime. We show by example that the inequalities obtained in arXiv:1704.05827 for the behaviour of the cohomological index of a join of free $\Z/p$-actions are sharp. Namely, for all odd integers $k,l$ at least one of which is greater than one, we give examples of finite free $\Z/p$-CW complexes of cohomological indices $k$ and $l$ whose join has index $k+l+1$ and also examples where the join has index $k+l-1$.