Kiem, Young-Hoon; Woolf, Jonathan Intersection cohomology of symplectic quotients by circle actions. (English) Zbl 1070.55002 J. Lond. Math. Soc., II. Ser. 71, No. 2, 531-544 (2005). Let \(T= U(1)\) and \(M\) be a Hamiltonian \(T\)-space with proper moment map \(\mu: M\to\mathbb{R}\). When \(0\) is not a regular value of \(\mu\), the symplectic quotient \(X= Z/T\), \(Z= \mu^{-1}(0)\), is a singular stratified space. The authors prove that the middle perversity intersection cohomology \(IH^*(X)\) is isomorphic to a subspace of the equivariant cohomology \(H^*_T(Z)\). More precisely, let \(F+1,\dots, F_r\) be the \(T\)-fixed components in \(Z\). For \(1\leq i\leq r\), the normal space of \(F_i\) decomposes into positive and negative weight spaces \(w^+_i\), \(w^-_i\). Let \(d_i={1\over 2}\min(\dim w^+_i, \dim w^-_i)\). Then \(H^*_T(F_i)\cong H^*(F_i)\otimes H^*_T\) and \[ IH^*(X)\cong\{x\in H^*_T(Z)\mid x|_{F_i}\in H^*(F_i)\otimes H_T^{2d_i-2}\}. \] Reviewer: Yves Félix (Louvain-La-Neuve) Cited in 1 Document MSC: 55N33 Intersection homology and cohomology in algebraic topology 55N91 Equivariant homology and cohomology in algebraic topology 53D20 Momentum maps; symplectic reduction Keywords:symplectic action; Hamiltonian \(T\)-space; intersection cohomology; singular stratified space PDFBibTeX XMLCite \textit{Y.-H. Kiem} and \textit{J. Woolf}, J. Lond. Math. Soc., II. Ser. 71, No. 2, 531--544 (2005; Zbl 1070.55002) Full Text: DOI