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 where 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 Collections: Mathematics