$\overline{Spec\mathbb Z}$ and the Gromov norm

Alain Connes; Caterina Consani

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arXiv · 1905.03310

$\overline{Spec\mathbb Z}$ and the Gromov norm

Alain Connes·Caterina Consani

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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.

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BibTeX RIS

Alain Connes, Caterina Consani. 2019-05-08. $\overline{Spec\mathbb Z}$ and the Gromov norm. https://arxiv.org/abs/1905.03310

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