An uncountable homology group, where each element is an infinite product of commutators

Abstract

In 1986 Umed Karimov constructed a space as a one-point compactification of a CW-complex, where each homotopy class of loops can be represented as an infinite product of commutators. We prove Karimov's conjecture that the first singular homology group of this space is uncountable. The proof uses methods of combinatorial group theory for one-relator groups and methods of geometric topology for smoothing singularities in van-Kampen diagrams corresponding to infinite words.