An uncountable homology group, where each element is an infinite product of commutators
Obtaining homotopy invariants of wild topological spaces (i.e. spaces, which are not locally contractible) is hard due to the absence of standard tools of algebraic topology. Even in the simplest case, the Hawaiian Earring, the fundamental group and the first homology group are hard to obtain and to describe. Another example of such a space was introduced by \textit{U. Kh. Karimov} [Sov. Math., Dokl. 33, 113--117 (1986); translation from Dokl. Akad. Nauk SSSR 286, 531--534 (1986; Zbl 0606.55003)]. He constructed \(X\) as the one-point compactification of a certain simplicial complex, so that each homotopy class of loops can be represented as an infinite product of commutators. In this paper the authors prove that the first homology group of Karimov's space \(X\) is uncountable. The proof uses infinite words and relates them to concrete cycles in \(X\).
singular homology group of a non-triangulable space
scientific article; zbMATH DE number 6517548
scientific article