\(L_ 2\)-cohomology and group cohomology
type: Item
instance of: scholarly article
title: \(L_ 2\)-cohomology and group cohomology
zbMATH Open document ID: 0597.57020
DOI: 10.1016/0040-9383(86)90039-X
author: Jeff Cheeger
author: Misha Gromov
published in: Topology
publication date: 1986-01-01T00:00:00Z
review text: Simplicial \(L_ 2\)-cohomology is based on the space of square-summable real-valued cochains on a simplicial complex. The authors show how to extend this \(L_ 2\)-cohomology to group equivariant singular \(L_ 2\)- cohomology on arbitrary topological spaces. Using the notion of von Neumann dimension, they then manage to define the appropriate equivariant notions of Betti number and Euler characteristic and prove the usual formal results therefore. They obtain thereby homotopy invariants of group actions with particularly nice properties. Indeed, the Euler characteristic is sometimes defined even though the orbit space of the action has infinite topological type; and the Betti numbers behave multiplicatively under finite coverings. The authors give in some circumstances a local formula for their Euler characteristic and show that the local invariants are particularly easily calculable in the case of group actions by amenable groups. In the amenable case, the invariants can sometimes be expressed in terms of standard Euler characteristics; this fact allows the authors to recover and extend prior results of Gottlieb, Stallings, and Rosset. The authors also relate their results to discrete actions of groups on Riemannian manifolds having bounded covering geometry.
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 3962780
zbMATH Keywords: Simplicial \(L_ 2\)-cohomology
zbMATH Keywords: square-summable real-valued cochains on a simplicial complex
zbMATH Keywords: equivariant singular \(L_ 2\)-cohomology
zbMATH Keywords: von Neumann dimension
zbMATH Keywords: Betti number
zbMATH Keywords: Euler characteristic
zbMATH Keywords: homotopy invariants of group actions
zbMATH Keywords: finite coverings
zbMATH Keywords: local formula for their Euler characteristic
zbMATH Keywords: amenable groups
zbMATH Keywords: discrete actions of groups on Riemannian manifolds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3962780