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.
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.
-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.
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".
-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.
A convenient one for intersection homology is an n-dimensional topological pseudomanifold.
, which measures how far cycles are allowed to deviate from transversality.
to the integers such that The second condition is used to show invariance of intersection homology groups under change of stratification.
Fix a topological pseudomanifold X of dimension n with some stratification, and a perversity p. A map σ from the standard i-simplex
to X (a singular simplex) is called allowable if is contained in the
If X has a triangulation compatible with the stratification, then simplicial intersection homology groups can be defined in a similar way, and are naturally isomorphic to the singular intersection homology groups.
The intersection homology groups are independent of the choice of stratification of X.
A resolution of singularities of a complex variety Y is called a small resolution if for every r > 0, the space of points of Y where the fiber has dimension r is of codimension greater than 2r.
Roughly speaking, this means that most fibers are small.
There is a variety with two different small resolutions that have different ring structures on their cohomology, showing that there is in general no natural ring structure on intersection (co)homology.
Deligne's formula for intersection cohomology states that where
is given by starting with the constant sheaf on the open set
and repeatedly extending it to larger open sets
and then truncating it in the derived category; more precisely it is given by Deligne's formula where
the derived pushforward is the identity map on a smooth point, hence the only possible cohomology is concentrated in degree
can be refined by considering the intersection of an open disk in
, the hyperplane bundle, and the Wang sequence gives the cohomology groups
The complex ICp(X) has the following properties As usual, q is the complementary perversity to p. Moreover, the complex is uniquely characterized by these conditions, up to isomorphism in the derived category.
The conditions do not depend on the choice of stratification, so this shows that intersection cohomology does not depend on the choice of stratification either.
Verdier duality takes ICp to ICq shifted by n = dim(X) in the derived category.