  The $\mathbb{Z}_2$-index ${\rm ind}(X)$ 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 ${\rm ind}(X \times Y) =
\min \{ {\rm ind}(X) , {\rm ind}(Y)\}$ for every pair $X$ and $Y$ of finite
$\mathbb{Z}_2$-complexes.
