Formality of the constructible derived category for spheres: A combinatorial and a geometric approach

TL;DR

This paper proves the formality of the constructible derived category of sheaves on spheres using combinatorial quiver representations and geometric differential forms, providing new insights into their algebraic and geometric structures.

Contribution

It introduces two methods—combinatorial and geometric—to establish the formality of the constructible derived category on spheres, extending results to multiple stratifications.

Findings

Formality of the constructible derived category for 2-spheres with principal ideal domain coefficients.

Generalization of formality results to spheres with multiple stratification points.

Proof of formality for n-spheres over real or complex coefficients using differential forms.

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.