\(L^ 2\)-cohomology and intersection homology of stratified spaces
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.
scientific article; zbMATH DE number 3868221
scientific article