Products and Independence in Concurrence Topology

Suppose we have two groups of variables each displaying negative association, manifested in nontrivial concurrence homology, in dimensions $p$ and $q$, say, when each group is considered individually. Suppose, however, that the two groups are statistically independent of each other. Then when combined the two groups should, with sufficiently large sample, produce non-trivial concurrence homology in dimension $p+q+1$. Generalization to more than two groups is straight forward. We propose turning this necessary condition into the definition of a form of "independent-like" behavior. This version is just an "extended abstract" of work in progress.