\(L_{(2)}\)-cohomology of orbit spaces
type: Item
instance of: scholarly article
title: \(L_{(2)}\)-cohomology of orbit spaces
zbMATH Open document ID: 0756.57021
DOI: 10.1016/0166-8641(92)90057-7
author: Reyer Sjamaar
published in: Topology and its Applications
publication date: 1992-09-27T00:00:00Z
review text: Let \(M\) be a smooth compact Riemannian manifold and \(G\) be a compact Lie group acting on \(M\) by isometries such that the orbit space \(M/G\) is connected. Let \((K)\) be a maximal orbit type, i.e., for any isotropy group \(H\) of the \(G\)-action \(K\) is conjugate to a subgroup of \(H\). Let \(M_{(K)}\) be the \(G\)-subset of \(M\) consisting of those points whose isotropy subgroup is conjugate to \(K\). Then the stratum \(M_{(K)}/G\) is connected, open and dense in \(M/G\). The open and dense stratum of \(M\) is called the principal stratum and denoted by \(M^{princ}/G\). The Riemannian metric on \(M\) induces a Riemannian metric on \(M^{princ}/G\). Let \(H^*_{(2)}(M^{princ}/G)\) be the \(L_{(2)}\)-cohomology of \(M^{princ}/G\) with respect to the Riemannian metric described above. Denote by \(IH_{\bar n}(M/G)\) the intersection cohomology with real coefficients of \(M/G\) with respect to the stratification given by the strata \(M_{(H)}/G\) and the middle perversity \(\bar n\) assigning to a stratum of codimension \(k\) the integer \([k-1/2]\). The main theorem of the article is: Theorem. There is a natural isomorphism \[ H^*_{(2)}(M^{princ}/G)\to IH_{\bar n}(M/G). \] The proof uses a sheaf-theoretic characterization of intersection cohomology and consists of verifying these for the \(L_{(2)}\)-cohomology of the principal stratum \(M^{princ}/G\).
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 55N33
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 64609
zbMATH Keywords: smooth compact Riemannian manifold
zbMATH Keywords: compact Lie group
zbMATH Keywords: isometries
zbMATH Keywords: orbit space
zbMATH Keywords: maximal orbit type
zbMATH Keywords: principal stratum
zbMATH Keywords: intersection cohomology
zbMATH Keywords: middle perversity
zbMATH Keywords: \(L_{(2)}\)-cohomology
reviewed by: Wolfgang Lück
MaRDI profile type: Publication
full work available at URL: 0166-8641(92)90057-7
OpenAlex ID: W2149219351
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description: scientific article; zbMATH DE number 64609