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.
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