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 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.
Read more about this topic: Intersection Homology
Famous quotes containing the words singular and/or intersection:
“It is singular how soon we lose the impression of what ceases to be constantly before us. A year impairs, a lustre obliterates. There is little distinct left without an effort of memory, then indeed the lights are rekindled for a momentbut who can be sure that the Imagination is not the torch-bearer?”
—George Gordon Noel Byron (17881824)
“You can always tell a Midwestern couple in Europe because they will be standing in the middle of a busy intersection looking at a wind-blown map and arguing over which way is west. European cities, with their wandering streets and undisciplined alleys, drive Midwesterners practically insane.”
—Bill Bryson (b. 1951)