\(L_{(2)}\)-cohomology of orbit spaces
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 \(M^{princ}/G\).
scientific article; zbMATH DE number 64609
scientific article