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A topological version of the famous Hedetniemi conjecture says: The mappingindex of the Cartesian product of two $\mathbb Z/2$-spaces is equal to theminimum of their $\mathbb Z/2$-indexes. The main purpose of this article is tostudy the topological version of the Hedetniemi conjecture for $G$-spaces.Indeed, we show that the topological Hedetniemi conjecture cannot be valid forgeneral pairs of $G$-spaces. More precisely, we show that this conjecture canpossibly survive if the group $G$ is either a cyclic $p$-group or a generalizedquaternion group whose size is a power of 2.

A topological version of Hedetniemi's conjecture for equivariant spaces