Integration in Čech theories and a bound on entropy

Abstract

The evaluation of Alexander-Spanier cochains over formal simplices in a topological space leads to a notion of integration of Alexander-Spanier cohomology classes over Čech homology classes. The integral defines an explicit and non-degenerate pairing between the Alexander-Spanier cohomology and the Čech homology. Instead of working on the limits that define both groups, most of the discussion is carried out "at scale $\mathcal{U}$", for an open covering $\mathcal{U}$. As an application, we generalize a result of Manning to arbitrary compact spaces $X$: we prove that the topological entropy of $f:X\to X$ is bounded from below by the logarithm of the spectral radius of the map induced in the first Čech cohomology group.