Milnor–Thurston homology groups of the Warsaw Circle

Abstract

Milnor-Thurston homology theory is a construction of homology theory that is based on measures. It is known to be equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor-Thurston homology groups for the Warsaw Circle are trivial except for the zeroth homology group which is uncountable dimensional. Additionally, we prove that the zeroth homology group is non-Hausdorff for this space with respect a natural topology that was proposed by Berlanga.

On the coincidence of zeroth Milnor–Thurston and singular homology

“…In [15] it has been proved that the Warsaw Circle has uncountable-dimensional zeroth Milnor-Thurston homology group. We may suspect that the fact that this space is not locally connected is the reason behind this phenomenon.…”

“…Some results in this direction were provided by the second author [20,Section 6], where it is proved that the canonical homomorphism (defined below) between singular homology and Milnor-Thurston homology is not necessarily an isomorphism. Additionally, in [15] it is proved that the first Milnor-Thurston homology group for the Warsaw Circle is trivial, and that the zeroth homology group is uncountable-dimensional, which is an unexpected result. Now, we shall briefly present the construction of Milnor-Thurston homology theory.…”

“…So in this case it is obviously an injection. Moreover, in [15] it was observed that it is an injection, when X is the Warsaw Circle.…”

“…Additionally, it happens to be an monomorhpism for many wild spaces (e.g. for the zeroth homology of the Warsaw Circle [15, see proof of Theorem 4] or the second author's example [20,Section 6], the last fact can be proved with the methods of [15]).…”

“…In [15] the Milnor-Thurston homology groups of the Warsaw Circle were computed, with the surprising result that the zeroth Milnor-Thurston homology group is infinite-dimensional. Milnor-Thurston homology theory satisfies in principle the Eilenberg-Steenrod axioms, but the determination of the isomorphism type of these homology groups (and thus the "coincidence" of Milnor-Thurston homology groups with singular homology groups) is only guaranteed for triangulable spaces.…”