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Spec(Z) and the Gromov Norm

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Alain Connes and Caterina Consani

We define the homology of a simplicial set with coefficients in a Segal's Gamma-set (s-module). We show the relevance of this new homology with values in s-modules by proving that taking as coefficients the s-modules at the archimedean place over the structure sheaf on Spec(Z), one obtains on the singular homology with real coefficients of a topological space X, a norm equivalent to the Gromov norm. Moreover, we prove that the two norms agree when X is an oriented compact Riemann surface.

Keywords:
Gamma spaces, Gamma rings, Site, Gromov norm, Arakelov geometry, Homology theory

2010 MSC:
16Y60; 20N20; 18G55; 18G30; 18G35; 18G55; 18G60; 14G40

*Theory and Applications of Categories,*
Vol. 35, 2020,
No. 6, pp 155-178.

Published 2020-02-18.

http://www.tac.mta.ca/tac/volumes/35/6/35-06.pdf

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