Intersection Homology - Stratifications

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

• for each i and for each point x of X_i − X_i−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.
• X_n−1 = X_n−2
• X − 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.

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.