Measure homology and singular homology are isometrically isomorphic |
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Authors: | Clara Löh |
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Institution: | 1. Graduiertenkolleg Analytische Topologie und Metageometrie, Universit?t Münster, Einsteinstr. 62, 48149, Münster, Germany
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Abstract: | 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 ℓ1-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 Gromov's proportionality principle of simplicial volume. |
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Keywords: | Measure homology singular homology simplicial volume |
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