Weighted cohomology
type: Item
instance of: scholarly article
title: Weighted cohomology
zbMATH Open document ID: 0849.11047
DOI: 10.1007/BF01231560
author: M. Goresky
author: Günter Harder
author: Robert MacPherson
published in: Inventiones Mathematicae
publication date: 1996-11-12T00:00:00Z
full work available at URL: https://eudml.org/doc/144187
review text: Let \(G\) be a semisimple algebraic group defined over \(\mathbb{Q}\). Let \(K\) be a maximal compact subgroup of the group \(G(\mathbb{R})\) of real points of \(G\), and denote by \(X= G(\mathbb{R})/ K\) the associated symmetric space. Let \((\tau, E)\) be a finite-dimensional (complex) rational representation of \(G (\mathbb{R})\). Given an arithmetic torsion free subgroup \(\Gamma\) of \(G (\mathbb{Q})\) there are various cohomology groups attached to the locally symmetric space \(\Gamma \backslash X\); these are defined in analytical, topological or algebraic terms. In this paper the weighted cohomology groups of \(\Gamma\) are introduced. The weighted cohomology groups \(W^p H^i (\Gamma, E)\) depend on an auxiliary parameter \(p\) called a weight profile. The group \(W^p H^i (\Gamma, E)\) is a cohomology theory on a compactification \(\widetilde {\Gamma \backslash X}\) of \(\Gamma \backslash X\) which is called the reductive Borel-Serre compactification; it is the cohomology of a complex of sheaves \(W^p C^\bullet (E)\) on \(\widetilde {\Gamma \backslash X}\). This complex may be thought of as the sheaf \(E\) on \(\Gamma \backslash X< \widetilde {\Gamma \backslash X}\) extended to \(\widetilde {\Gamma \backslash X}\) with certain ``support conditions'' at \((\widetilde {\Gamma \backslash X}) \backslash (\Gamma \backslash X)\), depending on the weight profile. The compactification \(\widetilde {\Gamma \backslash X}\) is ``almost nonsingular'': it has singularities that are particularly amenable to computations. The collection of weighted cohomology groups of an arithmetic group should be seen as analogous to the collection of intersection cohomology groups of a variety. In fact, if \(\Gamma \backslash X\) is Hermitian, for a weight profile \(\mu\) called the middle weight profile there is a canonical isomorphism, \(W^\mu H^i (\Gamma, E)\cong IH^i (\widehat {\Gamma \backslash X}, E)\) where \(IH^i (\widehat {\Gamma \backslash X}, E)\) is the intersection cohomolo…
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 55N33
Mathematics Subject Classification ID: 14F43
zbMATH DE Number: 554016
zbMATH Keywords: weighted cohomology groups
zbMATH Keywords: weight profile
zbMATH Keywords: reductive Borel-Serre compactification
zbMATH Keywords: arithmetic group
zbMATH Keywords: intersection cohomology groups
zbMATH Keywords: topological trace formula
zbMATH Keywords: fixed point formula
zbMATH Keywords: Lefschetz number
zbMATH Keywords: Hecke correspondence
Wikidata QID: Q56679323
reviewed by: Joachim Schwermer
MaRDI profile type: Publication
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OpenAlex ID: W4238737044
description: scientific article; zbMATH DE number 554016