On the first homology of Peano continua
Volume 232 / 2016
Fundamenta Mathematicae 232 (2016), 41-48
MSC: Primary 14F35; Secondary 03E15.
DOI: 10.4064/fm232-1-3
Abstract
We show that the first homology group of a locally connected compact metric space is either uncountable or finitely generated. This is related to Shelah's well-known result (1988) which shows that the fundamental group of such a space satisfies a similar condition. We give an example of such a space whose fundamental group is uncountable but whose first homology is trivial, showing that our result does not follow from Shelah's. We clarify a claim made by Pawlikowski (1998) and offer a proof of the clarification.
