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Concept
Intersection homology
Summary
In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.
Intersection cohomology was used to prove the Kazhdan–Lusztig conjectures and the Riemann–Hilbert correspondence. It is closely related to L2 cohomology.
The homology groups of a compact, oriented, connected, n-dimensional manifold X have a fundamental property called Poincaré duality: there is a perfect pairing
Classically—going back, for instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of
is represented by a j-dimensional cycle. If an i-dimensional and an -dimensional cycle are in general position, then their intersection is a finite collection of points. Using the orientation of X one may assign to each of these points a sign; in other words intersection yields a 0-dimensional cycle. One may prove that the homology class of this cycle depends only on the homology classes of the original i- and -dimensional cycles; one may furthermore prove that this pairing is perfect.
When X has singularities—that is, when the space has places that do not look like —these ideas break down. For example, it is no longer possible to make sense of the notion of "general position" for cycles. Goresky and MacPherson introduced a class of "allowable" cycles for which general position does make sense. They introduced an equivalence relation for allowable cycles (where only "allowable boundaries" are equivalent to zero), and called the group
of i-dimensional allowable cycles modulo this equivalence relation "intersection homology". They furthermore showed that the intersection of an i- and an -dimensional allowable cycle gives an (ordinary) zero-cycle whose homology class is well-defined.
Intersection homology was originally defined on suitable spaces with a stratification, though the groups often turn out to be independent of the choice of stratification.
Official source
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