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

is contained in the ik + 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:

    Commerce is unexpectedly confident and serene, alert, adventurous, and unwearied. It is very natural in its methods withal, far more so than many fantastic enterprises and sentimental experiments, and hence its singular success.
    Henry David Thoreau (1817–1862)

    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)