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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-3,
author = {Gregory R. Conner and Samuel M. Corson},
title = {On the first homology of Peano continua},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {41-48},
zbl = {1335.55004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-3}
}
Gregory R. Conner; Samuel M. Corson. On the first homology of Peano continua. Fundamenta Mathematicae, Tome 233 (2016) pp. 41-48. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-3/