Integration in Cech 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 Cech homology classes. The integral defines an explicit and non-degenerate pairing between the Alexander-Spanier cohomology and the Cech homology. Instead of working on the limits that define both groups, most of the discussion is carried out "at scale U", for an open covering U. As an application, we generalize a result of Manning to arbitrary compact spaces X: we prove that the topological entropy of f : X → X is bounded from below by the logarithm of the spectral radius of the map induced in the first Cech cohomology group.