Joining and Independence in Concurrence Topology

"Concurrence topology" (Ellis and Klein \emph{Homology, Homotopy, and Applications,} \textbf{16}) is a TDA method for binary data. The idea is to construct a filtration consisting of Dowker complexes then compute persistent homology. Persistent classes correspond to a form of negative statistical association among the variables. Suppose we have two groups of binary variables each displaying negative association, manifested in nontrivial concurrence homology in dimensions $p$ and in one group and $q$ in the other \emph{when the groups of variables are considered individually.} Suppose, however, that the two \emph{groups} of variables are statistically independent of each other. Now combine the two groups of variables and suppose the sample size is large. Then representative cycles, one from each group of variables, will combine to produce a cycle in dimension $p+q+1$. This is a chain level phenomenon, but we show it has a signature in homology. Looking for this signature can be used to study the dependence among groups of variables.