On the First Homology of Peano Continua

TL;DR

This paper investigates the properties of the first homology group in locally connected compact metric spaces, revealing it is either uncountable or finitely generated, with specific examples and clarifications of prior claims.

Contribution

It establishes a new dichotomy for the first homology group of such spaces and clarifies previous misconceptions with new proofs and examples.

Findings

First homology group is either uncountable or finitely generated

Existence of spaces with trivial first homology but uncountable fundamental group

Clarification and correction of prior claims by Pawlikowski

Abstract

We show that the first homology group of a locally connected compact metric space is either uncountable or is finitely generated. This is related to Shelah's well-known result which shows that the fundamental group of such a space satisfies a similar criterion. We give an example of such a space whose fundamental group is uncountable but whose first homology is trivial, showing that our result doesn't follow from Shelah's. We clarify a claim made by Pawlikowski and offer a proof of the clarification.