We prove that all groups can be realized as fundamental groups of compact spaces if and only if no measurable cardinals exist. If the cardinality of a group G is nonmeasurable then the compact space K such that G = π₁K may be chosen so that it is path connected.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm192-1-6,
author = {Adam Prze\'zdziecki},
title = {Measurable cardinals and fundamental groups of compact spaces},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {87-92},
zbl = {1115.03076},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm192-1-6}
}
Adam Przeździecki. Measurable cardinals and fundamental groups of compact spaces. Fundamenta Mathematicae, Tome 189 (2006) pp. 87-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm192-1-6/