Z2-indices and Hedetniemi's conjecture

Abstract

The ${\mathbb{Z}}_2$-index $\mathrm{ind}(X)$ of a ${\mathbb{Z}}_2$-CW-complex $X$ is the smallest number $n$ such that there is a ${\mathbb{Z}}_2$-map from $X$ to $S^n$. Here we consider $S^n$ as a ${\mathbb{Z}}_2$-space by the antipodal map. Hedetniemi's conjecture is a long standing conjecture in graph theory concerning the graph coloring problem of tensor products of finite graphs. We show that if Hedetniemi's conjecture is true, then $\mathrm{ind}(X\times Y)=\min \{\mathrm{ind}(X),\mathrm{ind}(Y)\}$ for every pair $X$ and $Y$ of finite ${\mathbb{Z}}_2$-complexes.

arXiv comment11 pages, final version, to appear in Discrete & Computational Geometry