Intersection Homology - Perversities


Intersection homology groups IpHi(X) depend on a choice of perversity p, which measures how far cycles are allowed to deviate from transversality. (The origin of the name "perversity" was explained by Goresky (2010).) A perversity p is a function from integers ≥2 to integers such that

  • p(2) = 0
  • p(k + 1) − p(k) is 0 or 1

The second condition is used to show invariance of intersection homology groups under change of stratification.

The complementary perversity q of p is the one with

Intersection homology groups of complementary dimension and complementary perversity are dually paired.


  • The minimal perversity has p(k) = 0. Its complement is the maximal perversity with q(k) = k − 2.
  • The (lower) middle perversity m is defined by m(k) = integer part of (k − 2)/2. Its complement is the upper middle perversity, with values the integer part of (k − 1)/2. If the perversity is not specified, then one usually means the lower middle perversity. If a space can be stratified with all strata of even dimension (for example, any complex variety) then the intersection homology groups are independent of the values of the perversity on odd integers, so the upper and lower middle perversities are equivalent.

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