$\overline{Spec\mathbb{Z}}$ 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.

21 pages, 6 figures