Scaled homology and topological entropy

Abstract

In this paper, we build up a scaled homology theory, l c 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 l c lc -homology satisfies all Eilenberg-Steenrod axioms except the exactness axiom whereas its corresponding l c lc -cohomology satisfies exactness axiom 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 l c lc -homology group.