Milnor-Thurston homology of some wild topological spaces

Janusz Przewocki
Thursday, July 25, 2013 - from 17:30 to 17:55
Milnor-Thurston homology theory is a construction similar to singular homology. The first mention of this theory can be found in Thurston's lecture notes "Geometry and topology of 3-manifolds", where it was used, for example, to prove that the volume of a hyperbolic manifold is a topological invariant. Zastrow and Hansen independently proved that this homology theory coincides with singular homology for CW-complexes. However, its behaviour for spaces with a more complicated local structure is still unknown. In this talk, we will provide some results in this direction. Milnor-Thurston homology for the Warsaw Circle shall be calculated. Moreover, we will show that its zeroth homology is non-Hausdorff in the weak topology considered by Berlanga. Further, the above results lead us to some natural questions concerning Milnor-Thurston homology. Namely, is the natural homomorphism between singular homology and Milnor-Thyrston homology always an injection? Are there any conditions under which zeroth Milnor-Thurston homology group coincides with singular homology group?