  A topological version of the famous Hedetniemi conjecture says: The mapping
index of the Cartesian product of two $\mathbb Z/2$-spaces is equal to the
minimum of their $\mathbb Z/2$-indexes. The main purpose of this article is to
study the topological version of the Hedetniemi conjecture for $G$-spaces.
Indeed, we show that the topological Hedetniemi conjecture cannot be valid for
general pairs of $G$-spaces. More precisely, we show that this conjecture can
possibly survive if the group $G$ is either a cyclic $p$-group or a generalized
quaternion group whose size is a power of 2.
