Fredrik Cumlin: Verdier Duality: Generalization of Poincaré Duality via Sheaf Cohomology

Abstract.

Poincare duality is a relationship of the structure of the homology and cohomology groups of orientable manifolds. This paper discusses a possible generalization of this relationship to a wider set of topological spaces, deriving the so-called Verdier duality. To this end, we will discuss chain complexes over a general abelian category, and derive the homotopy category and the derived category thereof; the derived category arises from a localization of the homotopy category. This allows us to derive relationships that do not necessarily hold at the level of chain complexes. We further show a way to induce functors in the derived category.

We continue by introducing sheaves and study their properties as a category. We discuss several ways in which continuous functions induce functors between categories of sheaves. Further, we derive a way to induce these functors in the derived category of sheaves, yielding relationships at the level of complexes. We finish by presenting Verider duality and show that Poincare duality is a special case of this relationship.