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Session 37. Wild algebraic and geometric topology
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Milnor-Thurston homology of some wild topological spaces |
Janusz Przewocki, Institute of Mathematics of the Polish Academy of Sciences, Poland
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The talk is partially based on joint work with Andreas Zastrow
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Milnor-Thurston homology theory is a version of homology theory admitting chains with infinite number of singular simplices. The other well known name for this theory is
``measure homology'', since it exploits techniques from measure theory.
This theory was first defined in the context of hyperbolic geometry, where it was used to provide a new proof of the Mostow Rigidity Theorem [3, Chapter 6]. The behaviour of Milnor-Thurston homology theory is well known in the case of CW-complexes, since it satisfies the Eilenberg-Steenrod Axioms. The case of more general spaces is mostly unexplored. The first results in that direction were obtained by Zastrow [4, Section 6]. And the first concrete computation of Milnor-Thurston homology groups was obtained by Przewocki
[1].
The aim of this talk is to present results of the preprint [2]. We show that there is a coincidence of zeroth homology groups for Peano Continua (this does not follow from the Eilengerg-Steenrod Axioms, since Peano Continua are not triangulable in general). We prove that under some condition the canonical homomorphism from singular homology to Milnor-Thurston homology is a monomorphism, and we present a counterexample showing that this condition cannot be omitted.
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References- J. Przewocki, Milnor-Thurston homology groups of the Warsaw Circle , Topology Appl. 160 (2013), no. 13, 1732 - 1741
- J. Przewocki, A. Zastrow, On the coincidence of zeroth Milnor-Thurston homology with singular homology , preprint available at\\ http://www.impan.pl/ jprzew/preprints/MTHomologyCoincidence.pdf
- W. P. Thurston, Geometry and Topology of Three-manifolds , Lecture notes, available at http://www.msri.org/publications/books/gt3m, Princeton, 1978
- A. Zastrow, On the (non)-coincidence of Milnor-Thurston homology theory with singular homology theory , Pacific Journal of Mathematics Vol. 186(1998), No. 2, 369-396
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