Intersection cohomology

General Introduction
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Online References
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Paper References
Goresky and MacPherson 74?
Borel et al: Intersection cohomology
Borel: Intersection cohomology. Modern Birkhäuser classics, reprinted 2008
Book by Kirwan and Woolf, online?
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1205.7057 Rational Homotopy and Intersection Cohomology from arXiv Front: math.AT by David Chataur, Martinxo SaralegiAranguren, Daniel Tanré In this text, we extend Sullivan's presentation of rational homotopy type to Goresky and MacPherson's intersection cohomology, providing also a homotopical framework where intersection cohomology is a representable functor.
We choose the context of face sets, also called simplicial sets without degeneracies, introduced by Rourke and Sanderson. We define filtered face sets, perverse local systems over them and intersection cohomology with coefficients in a perverse local system. In particular, we get a perverse local system of cochains quasiisomorphic to the intersection cochains of Goresky and MacPherson, over Z. We show also that these two complexes are quasiisomorphic to a filtered version of Sullivan's differential forms over the field Q.
We construct a pair of contravariant adjoint functors between the category of filtered face sets and a category of perverse commutative differential graded algebras (cdga's) due to Hovey. This correspondence gives bijections between homotopy classes, with the usual restriction to fibrant and cofibrant objects. We establish also the existence and unicity of a positively graded, minimal model of some perverse cdga's, including the perverse forms over a filtered face set and their intersection cohomology. This brings a definition of formality in the intersection setting. Finally, we prove the topological invariance of the minimal model of a locally conelike space and this theory creates new topological invariants.
http://front.math.ucdavis.edu/1104.0241 Motivic intersection complex from arXiv Front: math.AG by Jörg Wildeshaus In this article, we give an unconditional definition of the motivic analogue of the intersection complex, establish its basic properties, and prove its existence in certain cases.
arXiv:1101.4883 Deformation of Singularities and the Homology of Intersection Spaces from arXiv Front: math.AT by Markus Banagl, Laurentiu Maxim While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface.
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