  We consider a natural graph operation $\Omega_k$ that is a certain inverse
(formally: the right adjoint) to taking the k-th power of a graph. We show that
it preserves the topology (the $\mathbb{Z}_2$-homotopy type) of the box
complex, a basic tool in topological combinatorics. Moreover, we prove that the
box complex of a graph G admits a $\mathbb{Z}_2$-map (an equivariant,
continuous map) to the box complex of a graph H if and only if the graph
$\Omega_k(G)$ admits a homomorphism to H, for high enough k.
  This allows to show that if Hedetniemi's conjecture on the chromatic number
of graph products were true for n-colorings, then the following analogous
conjecture in topology would also also true: If X,Y are $\mathbb{Z}_2$-spaces
(finite $\mathbb{Z}_2$-simplicial complexes) such that X x Y admits a
$\mathbb{Z}_2$-map to the (n-2)-dimensional sphere, then X or Y itself admits
such a map. We discuss this and other implications, arguing the importance of
the topological conjecture.
