Intersection homology

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 "L"2 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
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 formula_3-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 formula_3-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 formula_5—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 formula_3-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
of "X" by closed subspaces such that:
If "X" is a topological pseudomanifold, the "i"-dimensional stratum of "X" is the space formula_9.
Examples:
Perversities.
Intersection homology groups formula_17 depend on a choice of perversity formula_18, which measures how far cycles are allowed to deviate from transversality. (The origin of the name "perversity" was explained by .) A perversity formula_18 is a function
from integers formula_21 to the integers such that
The second condition is used to show invariance of intersection homology groups under change of stratification.
The complementary perversity formula_24 of formula_18 is the one with
Intersection homology groups of complementary dimension and complementary perversity are dually paired.
Singular intersection homology.
Fix a topological pseudomanifold "X" of dimension "n" with some stratification, and a perversity "p".
A map σ from the standard "i"-simplex formula_32 to "X" (a singular simplex) is called allowable if
is contained in the formula_34 skeleton of formula_32.
The complex formula_36 is a subcomplex of the complex of singular chains on "X" that consists of all singular chains such that both the chain and its boundary are linear combinations of allowable singular simplexes. The singular intersection homology groups (with perversity "p")
are the homology groups of this complex.
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".
If "X" is a topological manifold, then the intersection homology groups (for any perversity) are the same as the usual homology groups.
Small resolutions.
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 2"r". Roughly speaking, this means that most fibers are small. In this case the morphism induces an isomorphism from the (intersection) homology of "X" to the intersection homology of "Y" (with the middle perversity).
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.
Sheaf theory.
Deligne's formula for intersection cohomology states that
where formula_40 is a certain complex of constructible sheaves on "X" (considered as an element of the derived category, so the cohomology on the right means the hypercohomology of the complex). The complex formula_40 is given by starting with the constant sheaf on the open set formula_42 and repeatedly extending it to larger open sets formula_43 and then truncating it in the derived category; more precisely it is given by Deligne's formula
where formula_45 is a truncation functor in the derived category, formula_46 is the inclusion of formula_43 into formula_48, and formula_49 is the constant sheaf on formula_42.
By replacing the constant sheaf on formula_42 with a local system, one can use Deligne's formula to define intersection cohomology with coefficients in a local system.
Examples.
Given a smooth elliptic curve formula_52 defined by a cubic homogeneous polynomial formula_53pg 281-282, such as formula_54, the affine coneformula_55has an isolated singularity at the origin since formula_56 and all partial derivatives formula_57 vanish. This is because it is homogeneous of degree formula_58, and the derivatives are homogeneous of degree 2. Setting formula_59 and formula_60 the inclusion map, the intersection complex formula_61 is given asformula_62This can be computed explicitly by looking at the stalks of the cohomology. At formula_63 where formula_64 the derived pushforward is the identity map on a smooth point, hence the only possible cohomology is concentrated in degree formula_65. For formula_66 the cohomology is more interesting sinceformula_67for formula_68 where the closure of formula_69 contains the origin. Since any formula_68 can be refined by considering the intersection of an open disk in formula_71 with formula_72, we can just compute the cohomology of formula_73. This can be done by observing formula_72 is a formula_75 bundle over the elliptic curve formula_76, the hyperplane bundle, and the Wang sequence gives the cohomology groupsformula_77hence the cohomology sheaves at the stalk formula_78 areformula_79truncating this gives the nontrivial cohomology sheaves formula_80, hence the intersection cohomology sheaf isformula_81The last decomposition follows from the Decomposition theorem.
Properties of the complex IC("X").
The complex IC"p"("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 IC"p" to IC"q" shifted by "n" = dim("X") in the derived category.