A topological version of Hedetniemi's conjecture for equivariant spaces

TL;DR

This paper investigates a topological analogue of Hedetniemi's conjecture for G-spaces, revealing it generally fails but may hold for specific groups like cyclic p-groups or generalized quaternion groups of order a power of two.

Contribution

The paper extends the topological Hedetniemi conjecture to G-spaces and identifies conditions under which it might still be valid.

Findings

The conjecture does not hold for general G-spaces.

It may be valid for cyclic p-groups.

It may also hold for generalized quaternion groups of order a power of two.

Abstract

A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two $\mathbb Z/2$-spaces is equal to the minimum of their $\mathbb Z/2$-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for $G$-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of $G$-spaces. More precisely, we show that this conjecture can possibly survive if the group $G$ is either a cyclic $p$-group or a generalized quaternion group whose size is a power of 2.