On the cohomology of \(\operatorname{SL}_n(\mathbb{Z})\)
type: Item
instance of: scholarly article
title: On the cohomology of \(\operatorname{SL}_n(\mathbb{Z})\)
author: Avner Ash
published in: Advances in Mathematics
publication date: 2024-08-24T00:00:00Z
review text: This paper is concerned with the cohomology of the arithmetic group \(\mathrm{SL}_n(\mathbb Z)\), which is of interest to algebraists and number theorists. This group has a classifying space for proper actions of dimension \(\nu_n = \tfrac {n(n-1)}2\) (a retract of its associated locally symmetric space) and in fact its virtual cohomological dimension equals \(\nu_n\). However, it is expected that the groups \(H^{\nu_n - i}(\mathrm{SL}_n(\mathbb Z), \mathbb Q)\) themselves vanish for \(i < (n-1)\) (see [\textit{T. Church} et al., Contemp. Math. 620, 55--70 (2014; Zbl 1377.11065)]). In this paper the author proves that this would be sharp for integers \(n\) congruent to 1 or 0 mod 3 by constructing explicit \(\mathbb Q\)-cocycles in degrees \(\nu_n-i\) for \(i=n-1, n\) which do not vanish in cohomology. The construction is inductive starting from \(n=3, 1, 4\), using a composition of cocycles in the so-called ``sharbly complex'' (introduced a while ago by the authors and collaborators to study this kind of problem) which computes the cohomology via Borel-Serre duality.
reviewed by: Jean Raimbault
Mathematics Subject Classification ID: 11F75
zbMATH DE Number: 7900772
zbMATH Keywords: cohomology
zbMATH Keywords: arithmetic group
MaRDI profile type: Publication
cites work: Resolutions of the Steinberg module for \(\mathrm{GL}(n)\)
cites work: Unstable cohomology of \(SL(n,O)\)
cites work: A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
cites work: Perfect forms, \(K\)-theory and the cohomology of modular groups
cites work: Characteristic classes of bundles of \(K3\) manifolds and the Nielsen realization problem
cites work: On unstable cohomology classes of SL n ( Z )
cites work: On the torsion in \(K_4(\mathbb{Z})\) and \(K_5(\mathbb{Z})\)
cites work: The cohomology of \(SL_3(Z)\)
DOI: 10.1016/J.AIM.2024.109868
description: scientific article; zbMATH DE number 7900772