Intersection cohomology

In mathematics, intersection cohomology is a theory from algebraic topology, initially developed by Goresky and MacPherson, to apply to spaces with singularities.

The cohomology groups of a topological manifold have an interesting symmetry called Poincaré duality. In particular,

H^k (X) ≅ H_n−k (X),

where n is the dimension of a closed, orientable manifold. Unfortunately, many interesting spaces have singularities; that is, places where the space does not look like Rⁿ. Intersection cohomology is a modified definition of cohomology which recovers the property of Poincaré duality for a much larger category of spaces, Witt spaces; this includes all algebraic varieties.

Verdier duality approach

A good way to understand intersection cohomology starts from the pairing between cohomology with compact support H_c and ordinary cohomology: the notion of Verdier duality. The Verdier dual of the space X is an object D_X in the derived category of sheaves on X, that represents the functor of taking cohomology with compact support and taking the dual over the base ring (taken in the right derived sense). It is easiest to explain when cohomology is taken over a field F. In the derived category, cohomology can be interpreted as chain homotopy classes of maps

H^k (X, F) = [F[−k],F] = [F, F[k]]

where F[−k] is the complex with the constant sheaf F concentrated in degree k, and [—, —] are the chain homotopy classes of maps. The Verdier dual allows us to interpret homology in the derived category as well:

[F[−k], D_X] = H_k (X, F).

The left hand side is by definition the dual of the cohomology with compact support, so this equation says that homology is dual to cohomology with compact support.

It also follows that for an oriented manifold M, the Verdier dual is given by

D_M = F[−n].

Ordinary Poincaré duality of a manifold can then be interpreted as the perfect pairing

[F[−k], F] ⊗ [F[k−n], F[−n]] → [F[−n], F[−n]] → F.

Intersection cohomology can now be understood as choosing a "root" IC(X) of the Verdier dual in the sense that in the derived category

IC(X) ⊗ IC(X) ≅ D_X.

IC(X) is thus conceptually similar to a theta characteristic of an algebraic curve. There is an algorithm for constructing IC(X) from Deligne given that X is an even-dimensional manifold outside a set of real codimension 2, starting from the ansatz

IC(U) = F[dim (U)/2]

on the manifold part U of X. The intersection cohomology of X is then defined as

IHC^k(X) = [F[−k], IC(X)].