Global questions in the topology of singular spaces. (English) Zbl 0612.57012
Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 213-235 (1984).
[For the entire collection see Zbl 0553.00001.]
This is a survey article on intersection homology, which was developed by M. Goresky and the author to extend Poincaré duality to singular spaces [see M. Goresky and the author, Topology 19, 135-165 (1980; Zbl 0448.55004); Invent. Math. 72, 77-129 (1983; Zbl 0529.55007); T. Springer, Astérisque 92/93, 249-273 (1982; Zbl 0526.22014)].
Also other concepts like Morse theory, fixed point indices are extended to singular spaces. The description of intersection homology via sheaf theory is given, and applications to analysis (L\({}^ 2\)-cohomology), algebraic geometry (decomposition of the push forward of an intersection homology sheaf, specialization), group theory (Weyl group representations, Hecke algebras, representations of Lie algebras) are explained.
This is a survey article on intersection homology, which was developed by M. Goresky and the author to extend Poincaré duality to singular spaces [see M. Goresky and the author, Topology 19, 135-165 (1980; Zbl 0448.55004); Invent. Math. 72, 77-129 (1983; Zbl 0529.55007); T. Springer, Astérisque 92/93, 249-273 (1982; Zbl 0526.22014)].
Also other concepts like Morse theory, fixed point indices are extended to singular spaces. The description of intersection homology via sheaf theory is given, and applications to analysis (L\({}^ 2\)-cohomology), algebraic geometry (decomposition of the push forward of an intersection homology sheaf, specialization), group theory (Weyl group representations, Hecke algebras, representations of Lie algebras) are explained.
Reviewer: E.Vogt
MSC:
| 57N35 | Embeddings and immersions in topological manifolds |
| 55M05 | Duality in algebraic topology |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 58A99 | General theory of differentiable manifolds |
| 11F27 | Theta series; Weil representation; theta correspondences |
| 22E46 | Semisimple Lie groups and their representations |
