# Measure homology and singular homology are isometrically isomorphic

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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 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 Gromov's proportionality principle of simplicial volume.

## Keywords

Measure homology singular homology simplicial volume## Mathematics Subject Classification (2000)

Primary: 55N35 Secondary: 55N10 57N65## References

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