Intersection cohomology
Encyclopedia
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.
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 (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)-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 Rn—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 (n − i)-dimensional allowable cycle gives an (ordinary) zero-cycle whose homology class is well-defined.
, 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 Xi − Xi−1.
Examples:
A perversity p is a function from integers ≥2 to integers such that
The second condition is used to show invariance of intersection homology groups under change of stratification.
The complementary perversity q of p is the one with
Intersection homology groups of complementary dimension and complementary perversity are dually paired.
Examples:
A map σ from the standard i-simplex Δi to X (a singular simplex) is called allowable if
is contained in the i − k + p(k) skeleton of Δi
The complex Ip(X) 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.
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. 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.
where ICp(X) is a certain complex of 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 ICp(X) is given by starting with the constant sheaf on the open set X−Xn−2 and repeatedly extending it to larger open sets X−Xn−k and then truncating it in the derived category; more precisely it is given by Deligne's formula
where τ≤p is a truncation functor in the derived category, and ik is the inclusion of X−Xn−k into X−Xn−k−1 and CX−Xn−2 is the constant sheaf on X−Xn−2. (Warning: there is more than one convention for the way that the perversity enters Deligne's construction: the numbers p(k)−n are sometimes written as p(k).)
By replacing the constant sheaf on X−Xn−2 with a local system, one can use Deligne's formula to define intersection cohomology with coefficients in a local system.
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.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
, discovered by Mark Goresky
Mark Goresky
Robert Mark Goresky is a Canadian mathematician who invented intersection homology with Robert MacPherson.He received his Ph.D. from Brown University in 1976. His thesis, titled Geometric Cohomology and Homology of Stratified Objects, was written under the direction of MacPherson...
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
Riemann–Hilbert correspondence
In mathematics, the Riemann–Hilbert correspondence is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups...
. It is closely related to L2 cohomology.
Goresky–MacPherson approach
The homology groups of a compact, oriented, n-dimensional manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
X have a fundamental property called Poincaré duality
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
: there is a perfect pairing
Classically—going back, for instance, to Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
—this duality was understood in terms of intersection theory
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
. An element of
- Hj(X)
is represented by a j-dimensional cycle. If an i-dimensional and an (n − i)-dimensional cycle are in general position
General position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
, 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)-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 Rn—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
- IHi(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)-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 stratificationTopologically stratified space
In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way...
, 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
- for each i and for each point x of Xi − Xi−1, there exists a neighborhood
of x in X, a compact (n − i − 1)-dimensional stratified space L, and a filtration-preserving homeomorphism 
. Here 
is the open cone on L. - Xn−1 = Xn−2
- X − Xn−1 is dense in X.
If X is a topological pseudomanifold, the i-dimensional stratum of X is the space Xi − Xi−1.
Examples:
- If X is an n-dimensional simplicial complexSimplicial complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
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.
Perversities
Intersection homology groups IpHi(X) depend on a choice of perversity p, which measures how far cycles are allowed to deviate from transversality. (The origin of the name "perversity" was explained by .)A perversity p is a function from integers ≥2 to integers such that
- p(2) = 0
- p(k + 1) − p(k) is 0 or 1
The second condition is used to show invariance of intersection homology groups under change of stratification.
The complementary perversity q of p is the one with
Intersection homology groups of complementary dimension and complementary perversity are dually paired.
Examples:
- The minimal perversity has p(k) = 0. Its complement is the maximal perversity with q(k) = k − 2.
- The (lower) middle perversity m is defined by m(k) = integer part of (k − 2)/2. Its complement is the upper middle perversity, with values the integer part of (k − 1)/2. If the perversity is not specified, then one usually means the lower middle perversity. If a space can be stratified with all strata of even dimension (for example, any complex variety) then the intersection homology groups are independent of the values of the perversity on odd integers, so the upper and lower middle perversities are equivalent.
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 Δi to X (a singular simplex) is called allowable if
is contained in the i − k + p(k) skeleton of ΔiThe complex Ip(X) 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 singularitiesResolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V...
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. 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 thatwhere ICp(X) is a certain complex of 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 ICp(X) is given by starting with the constant sheaf on the open set X−Xn−2 and repeatedly extending it to larger open sets X−Xn−k and then truncating it in the derived category; more precisely it is given by Deligne's formula
where τ≤p is a truncation functor in the derived category, and ik is the inclusion of X−Xn−k into X−Xn−k−1 and CX−Xn−2 is the constant sheaf on X−Xn−2. (Warning: there is more than one convention for the way that the perversity enters Deligne's construction: the numbers p(k)−n are sometimes written as p(k).)
By replacing the constant sheaf on X−Xn−2 with a local system, one can use Deligne's formula to define intersection cohomology with coefficients in a local system.
Properties of the complex IC(X)
The complex ICp(X) has the following properties- On the complement of some closed set of codimension 2, we have
is 0 for i+m≠ 0, and for i=−m the groups form the constant local system C 
is 0 for i + m < 0- If i > 0 then
is zero except on a set of codimension at least a for the smallest a with p(a) ≥ m − i - If i>0 then
is zero except on a set of codimension at least a for the smallest a with q(a) ≥ (i)
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
Verdier duality
In mathematics, Verdier duality is a generalization of the Poincaré duality of manifolds to locally compact spaces with singularities. Verdier duality was introduced by , as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck...
takes ICp to ICq shifted by n = dim(X) in the derived category.
See also
- Borel–Moore homology
- Topologically stratified spaceTopologically stratified spaceIn topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way...
- Intersection theoryIntersection theoryIn mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...