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Applications of Homological Algebra Introduction to Perverse Sheaves Spring 2007 P. Achar

Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Goresky­MacPherson Stratifications and Perversities
April 12, 2007
Intersection cohomology complexes are in some sense the most important perverse sheaves--essentially all
perverse sheaves that come up in practice are direct sums of intersection cohomology complexes. But as
pointed out in the last set of notes, it is not clear that intersection cohomology complexes are actually
constructible. In fact, we need to modify the definition of "stratification" to make sure that they are.
Definition 1. Let X be a topological space equipped with a stratification S, and let n = max{dim S | S
S}. S is called a topological stratification (or a Goresky­MacPherson stratification) if it satisfies
the following additional condition: for any point x in a stratum of dimension k with k < n, there is a
neighborhood U together with a homeomorphism
U Rk
× C
C = (Y × [0, )/(Y × {0})
(the "cone" on Y ), and where Y is a topologically stratified space of dimension n - k - 1. Moreover, the set
of strata T of Y should be in bijection with the strata of X that are larger than S in the partial order on
strata: specifically, if S S is such that S S but S = S , then there must exist a stratum T of Y such
that the homeomorphism U Rk


Source: Achar, Pramod - Department of Mathematics, Louisiana State University


Collections: Mathematics