  We study the topological structure of the quotient of $SU(3)\times SU(3)$ by
diagonal conjugation. This is the simplest nontrivial example for the classical
reduced configuration space of chromodynamics on a spatial lattice in the
Hamiltonian approach. We construct a cell complex structure of the quotient in
such a way that the closures of strata are subcomplexes and we compute the
homology and cohomology groups of the strata and their closures.
