$\overline{Spec\mathbb Z}$ and the Gromov norm
Abstract
We define the homology of a simplicial set with coefficients in a Segal's $\Gamma$set ($\mathbf S$module). We show the relevance of this new homology with values in $\mathbf S$modules by proving that taking as coefficients the $\mathbf S$modules at the archimedean place over the structure sheaf on $\overline{Spec\mathbb Z}$ introduced in our previous work, 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 DOI:
 10.48550/arXiv.1905.03310
 arXiv:
 arXiv:1905.03310
 Bibcode:
 2019arXiv190503310C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Number Theory;
 16Y60;
 20N20;
 18G55;
 18G30;
 18G35;
 18G55;
 18G60;
 14G40
 EPrint:
 21 pages, 6 figures