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.
Goresky–MacPherson approach The homology groups of a compact, oriented, connected, n-dimensional manifold X have a fundamental property called Poincaré duality: there is a perfect pairing
H i ( X , Q ) × H n − i ( X , Q ) → H 0 ( X , Q ) ≅ Q . {\displaystyle H_{i}(X,\mathbb {Q} )\times H_{n-i}(X,\mathbb {Q} )\to H_{0}(X,\mathbb {Q} )\cong \mathbb {Q} .}
Classically—going back, for instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of
H j ( X ) {\displaystyle H_{j}(X)}
is represented by a j-dimensional cycle. If an i-dimensional and an ( n − i ) {\displaystyle (n-i)} -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 ( n − i ) {\displaystyle (n-i)} -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 R n {\displaystyle \mathbb {R} ^{n}} —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
I H i ( X ) {\displaystyle IH_{i}(X)}
of i-dimensional allowable cycles modulo this equivalence relation "intersection homology". They furthermore showed that the intersection of an i- and an ( n − i ) {\displaystyle (n-i)} -dimensional allowable cycle gives an (ordinary) zero-cycle whose homology class is well-defined.
Stratifications 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. There are many different definitions of stratified spaces. A convenient one for intersection homology is an n-dimensional topological pseudomanifold. This is a (paracompact, Hausdorff) space X that has a filtration
∅ = X − 1 ⊂ X 0 ⊂ X 1 ⊂ ⋯ ⊂ X n = X {\displaystyle \emptyset =X_{-1}\subset X_{0}\subset X_{1}\subset \cdots \subset X_{n}=X}
of X by closed subspaces such that:
For each i and for each point x of X i ∖ X i − 1 {\displaystyle X_{i}\setminus X_{i-1}} , there exists a neighborhood U ⊂ X {\displaystyle U\subset X} of x in X, a compact ( n − i − 1 ) {\displaystyle (n-i-1)} -dimensional stratified space L, and a filtration-preserving homeomorphism U ≅ R i × C L {\displaystyle U\cong \mathbb {R} ^{i}\times CL} . Here C L {\displaystyle CL} is the open cone on L.
X n − 1 = X n − 2 {\displaystyle X_{n-1}=X_{n-2}} .
X ∖ X n − 1 {\displaystyle X\setminus X_{n-1}} is dense in X. If X is a topological pseudomanifold, the i-dimensional stratum of X is the space X i ∖ X i − 1 {\displaystyle X_{i}\setminus X_{i-1}} . Examples:
If X is an n-dimensional simplicial complex such that every simplex is contained in an n-simplex and n−1 simplex is contained in exactly two n-simplexes, then the underlying space of X is a topological pseudomanifold. If X is any complex quasi-projective variety (possibly with singularities) then its underlying space is a topological pseudomanifold, with all strata of even dimension.
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