Integration in \v{C}ech theories and a bound on entropy

TL;DR

This paper develops a new integration framework linking Alexander-Spanier cohomology and Čech homology, and applies it to establish a lower bound on topological entropy for continuous maps on compact spaces.

Contribution

It introduces an explicit integration pairing between Alexander-Spanier cohomology and Čech homology, and generalizes Manning's entropy bound to all compact spaces.

Findings

Established a non-degenerate pairing between cohomology and homology.

Generalized Manning's entropy bound to arbitrary compact spaces.

Provided a new perspective on cohomology-homology integration at finite scales.

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 \v{C}ech homology classes. The integral defines an explicit and non-degenerate pairing between the Alexander-Spanier cohomology and the \v{C}ech 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 $f \colon X \to X$ is bounded from below by the logarithm of the spectral radius of the map induced in the first \v{C}ech cohomology group.