Formality of the Constructible Derived Category for Spheres: A Combinatorial and a Geometric Approach

Abstract

We describe the constructible derived category of sheaves on the n-sphere, stratified in a point and its complement, as a dg module category of a formal dg algebra. We prove formality by exploring two different methods. As a combinatorial approach, we reformulate the problem in terms of representations of quivers and prove formality for the 2-sphere, for coefficients in a principal ideal domain. We give a suitable generalization of this formality result for the 2-sphere stratified in several points and their complement. As a geometric approach, we give a description of the underlying dg algebra in terms of differential forms, which allows us to prove formality for n-spheres, for real or complex coefficients.