**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 *I**p*(*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.

Read more about this topic: Intersection Homology

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