Scaled Homology and Topological Entropy
Bingzhe Hou, Kiyoshi Igusa, and Zihao Liu

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, -homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newly-built homology. We check that -homology satisfies all Eilenberg-Steenrod axioms except exactness axiom whereas its corresponding -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 -homology group.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Mathematical Dynamics and Fractals · Homotopy and Cohomology in Algebraic Topology
