Zeroth Milnor-Thurston Homology Groups or How to connect different path-components with a measure chain

Andreas Zastrow
Thursday, July 25, 2013 - from 17:00 to 17:25
Computing the classical algebaric invariants for wild (e.g.\ non-triangulable) spaces has yield one or the other psychologically surprising result (e.g.\ Milnor's \& Barratt's example of a wild two-dimensional complex with a non-trivial three-dimensinal singular homologiy group, the fact that a one-point union of simply connected spaces may be non-simply connected, or the fact that the Hawaiian Earring has a non-free fundamental group and an astonishingly complicated 3-summands formula for its first singular homology group, the latter shown by Eda in 2000). In the the very recent years by Diestel, Spr\"ussel and Georgakopoulos papers have been published which may be interpreted as a kind of search for a homology theory which behaves more sensible on wild spaces than does classical singular homology theory. Now the analysis of some of these phenomena has left the impression that the origin for all the strange behaviour of the classical algebraic invariants is that they are based on classical finite arithmetics, while wild spaces often have a topological structure which can only suitably be mirrored by infinite algebraic operations. Since Milnor-Thurston homology theory is based on considering measures and thus allows to some extent infinite summation, although it was created by a complete different motivation, the idea arose to examine how these homolopy groups behave on wild topological spaces. Milnor-Thurston homology theory fulfils the Eilenberg-Steenrod-Axioms and thus coincides with singular homology theory on triangulable spaces, but it is also known that it does not automatically coincide for arbitrary spaces. During the years 2011--12 Przewocki found a way to compute the Milnor-Thurston homology groups for the Warsaw Circle, with the result that the first and all higher homology group are trivial and thus coincide with singular homology, but the zeroth homology group is uncountably dimensional. This result motivated the current research on the the general behaviour of zeroth Milnor-Thurston homology groups. While the Warsaw Circle already shows that the dimension of the zeroth Milnor-Thurston group is not automatically the number of path-components and in particular can be much bigger, it was meanwhile shown by Przewocki that in case of Peano-Continua the zerorth Milnor-Thurston group is always one-dimensional and thus coincides with singular homology theory. It remains the question whether it can be smaller than the singular homology group, which would make it necesary to construct measures on one-simplices that connect points in different path components. While by a result of Przewocki this can be ruled out for spaces with measurable path-components, that talk is about to present an example showing that this can well happen for spaces with non-measurable path components. The talk shall briefly explain the basic concept and original motivation of the construction of Milnor-Thurston homology theory, then sum up Przewocki's results on zeroth Milnor-Thurston homology groups, and then focus on the last above-mentioned result.