\(L^ 2\)-cohomology and intersection homology of stratified spaces
type: Item
instance of: scholarly article
title: \(L^ 2\)-cohomology and intersection homology of stratified spaces
zbMATH Open document ID: 0545.55004
DOI: 10.1215/S0012-7094-83-05015-9
author: Masayoshi Nagase
published in: Duke Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Let X be a singular space with singular locus \(\Sigma\). The author considers the question if for any perversity \(\bar p\) the dual of the intersection homology group \(IH_*^{\bar p}(X)\) with real coefficients is canonically isomorphic to the \(L^ 2\)-cohomology group \(H^*_{(2)}(X-\Sigma)\). Here X-\(\Sigma\) has a metric, which is associated to \(\bar p\). For example on the interior (0,1)\(\times X\) of a cone over X one can consider the metric \(dr\otimes dr+2^{rc}g,\) where g is a metric on X. In the general case a metric g on \(X-\Sigma\) is associated with \(\bar c=(c_ 2,...,c_ n)\) if the restrictions to local cones near points of \(\Sigma\) are of the above type. Also metrics can be associated to a perversity \(\bar p\). The author gives a positive answer to the question if \(\bar p\leq \bar m\) (middle perversion). If \(\bar p=\bar m\) the statement was already due to J. Cheeger. The proof is rather technical. The paper is written before the publishing of the article of \textit{M. Goresky} and \textit{R. MacPherson} [Invent. Math. 72, 77-129 (1983; Zbl 0529.55007)]. It is clear that part of the proof could be simplified if the sheaf-theoretic set-up of that paper is used.
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 57N80
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 32S60
zbMATH DE Number: 3868221
zbMATH Keywords: metrics associated to a perversity
zbMATH Keywords: singular space
zbMATH Keywords: singular locus
zbMATH Keywords: intersection homology group
zbMATH Keywords: \(L^ 2\)-cohomology group
zbMATH Keywords: local cones
reviewed by: Dirk Siersma
MaRDI profile type: Publication
cites work: Manifold Phenomena in the Theory of Polyhedra
cites work: On the spectral geometry of spaces with cone-like singularities
cites work: Q3911109
cites work: Triangulation of Stratified Objects
cites work: Q4187010
cites work: Intersection homology theory
cites work: Stability of \(C^ \infty\) mappings. V: Transversality
cites work: Q4172661
cites work: De Rham-Hodge theory on a manifold with cone-like singularities
full work available at URL: s0012-7094-83-05015-9
OpenAlex ID: W2075798501
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description: scientific article; zbMATH DE number 3868221