On the coincidence of zeroth Milnor-Thurston homology with singular homology

TL;DR

This paper proves that for Peano Continua, the zeroth Milnor-Thurston homology matches singular homology, but the canonical map between them may not always be injective unless the space has Borel path-components.

Contribution

It establishes the equivalence of zeroth Milnor-Thurston and singular homology for Peano Continua and identifies conditions for the injectivity of the canonical homomorphism.

Findings

Zeroth Milnor-Thurston homology coincides with singular homology for Peano Continua.

The canonical homomorphism may not be injective in general.

Injectivity holds when the space has Borel path-components.

Abstract

In this paper we prove that the zeroth Milnor-Thurston homology group coincides with singular homology for Peano Continua. More- over, we show that the canonical homomorphism between these ho- mology theories may not be injective. However, it is proved that it is injective when a space has Borel path-components.