Scaled Homology and Topological Entropy

TL;DR

This paper introduces a new scaled homology theory, called $lc$-homology, for metric spaces that relaxes smoothness requirements and satisfies most Eilenberg-Steenrod axioms, aiding in entropy conjecture analysis.

Contribution

The paper develops $lc$-homology, a novel homology theory for metric spaces that broadens applicability beyond smooth manifolds and satisfies key axioms except exactness.

Findings

$lc$-homology satisfies all Eilenberg-Steenrod axioms except exactness.

$lc$-cohomology satisfies all Eilenberg-Steenrod axioms.

Entropy conjecture holds for the first $lc$-homology group.

Abstract

In this paper, we build up a scaled homology theory, $lc$-homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newly-built homology. We check that $lc$-homology satisfies all Eilenberg-Steenrod axioms except exactness axiom whereas its corresponding $lc$-cohomology satisfies all axioms for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first $lc$-homology group.