  This paper is devoted to studying a topological version of the famous
Hedetniemi conjecture which says: The $\mathbb Z/2$-index of the Cartesian
product of two $\mathbb Z/2$-spaces is equal to the minimum of their $\mathbb
Z/2$-indexes. We fully confirm the version of this conjecture for the
homological index via establishing a stronger formula for the homological index
of the join of $\mathbb Z/2$-spaces. Moreover, we confirm the original
conjecture for the case when one of the factors is an $n$-sphere. Analogous
results for $\mathbb Z/p$-spaces are presented as well. In addition, we answer
a question about computing the index of some non-trivial products, raised by
Marcin Wrochna. Finally, some new topological lower bounds for the chromatic
number of the Categorical product of (hyper-)graphs are presented.
