Intersection Homology - Singular Intersection Homology

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 Δ_i to X (a singular simplex) is called allowable if

is contained in the i − k + p(k) skeleton of Δ_i

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.