Z2-indices and Hedetniemi's conjecture

The ${\mathbb{Z}}_2$-index $\mathrm{i}\mathrm{n}\mathrm{d}(X)$ 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 ${\rm ind}(X \times Y) = \min { {\rm ind}(X) , {\rm ind}(Y)}$foreverypair$X$and$Y$offinite$$\mathbb{Z}$_2$-complexes.