Measure homology and singular homology are isometrically isomorphic

Details Measure homology and singular homology are isometrically isomorphic Measure homology and singular homology are isometrically isomorphic

2005-04-06

arxiv.org/abs/math/0504103v2

Mathematics

Algebraic Topology

20 pages, typos corrected, see also http://www.math.uni-muenster.de/u/clara.loeh/preprints.html, accepted by Mathematische Z

Scientific paper

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the l^1-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology -- as already claimed by Thurston. For example, measure homology can be used to prove the proportionality principle of simplicial volume.

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