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In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined initially by Goresky and MacPherson, this is a version of homology which agrees with ordinary homology on manifolds, but also retains crucial properties like Poincare Duality and Hodge Theory on singular (non-)manifolds. The original definition was combinatorial, but it was later re-interpreted in sheaf-theoretic terms (perverse sheaves?).

Back then it certainly looked like an exciting new development. So, I'm curious - where does the field stand today? Is it still thriving, or has it been merged with something else, or just faded away?

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4 Answers 4

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Intersection homology and cohomology are still around, but as a topic they have just substantially been renamed. They are part of the theory of perverse sheaves, which are widely used in the Langlands program, in algebraic geometry approaches to categorification, and elsewhere in algebraic geometry.

To the extent that intersection homology was intended for topology, it has stoked relatively less interest than in algebraic geometry. On the one hand, there has been a trend away from homological algebra in geometric topology. On the other hand, singularities are part of the structure of intersection homology. Singularies are more germane to geometric topology than to algebraic topology in the sense of homotopy theory. Both singularities and homological algebra are major aspects of algebraic geometry.

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Intersection homology is alive and well in a large number of guises. It's true that a lot of the work trended to algebraic geometry, representation theory, and categorical constructions, such as perverse sheaves, through the 90s, but there also continues to be work in the more topological settings by people such as me, Cappell, Shaneson, Markus Banagl, Laurentiu Maxim, and many others. At least some of this work is dedicated to extending classical manifold invariants, such as characteristic classes, in a meaningful way to stratified spaces, such as algebraic varieties, and there is a lot of recent interest (though slow progress) in figuring out how intersection homology might tie into various algebraic topology constructions. There are also analytic formulations such as L^2 cohomology (initiated by Cheeger), and much more.

Here are some good references to get started in the area:

Books: An Introduction to Intersection Homology by Kirwan and Woolf (mostly concerned with telling the reader about the fancy early applications to algebraic geometry and representation theory, but a great overview nonetheless)

Intersection Cohomology by Borel, et.al. This is a great serious technical introduction to the area and, to my mind, the canonical source for the foundations of the subject)

Topological Invariants of Stratified Spaces by Markus Banagl (topological but mostly from the sheaf point of view)

For an overview of state-of-the-art in intersection homology and related fields, I'm co-editing a volume on Topology of Stratified Spaces that will be published in the MSRI series. Unfortunately, it's not out yet, but look for it soon.

Papers: The original papers of Goresky and MacPherson are quite good.

Topological invariance of intersection homology without sheaves by Henry King is a good introduction to the singular version of the theory.

And for a whole pile of recent papers, I'll shamelessly plug my own web site: http://faculty.tcu.edu/gfriedman/ and Markus Banagl's: http://www.mathi.uni-heidelberg.de/~banagl/

And many further references can be found from these locations.

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i was about to comment on Greg Kuperberg's above answer saying something like "hey, what about Greg Friedman's work, that's topological." anyway, the extent of my knowledge is literally your midwest topology seminar talk... –  Sean Tilson Jun 9 '10 at 15:55
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There do seem to be people thinking about it. I heard Sylvain Cappell lecture a few weeks ago; he spoke about using intersection (co)homology as part of a generalization of characteristic classes to singular varieties.

Here's a recent paper by Cappell-Maxim-Shaneson that he referred to: "Intersection cohomology invariants of complex algebraic varieties".

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Intersection homology quickly found applications in representation theory, starting with the Kazhdan-Lusztig conjectures. Today, the theory of perverse sheaves is an important tool in geometric representation theory.

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