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

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

2008-11-02

arxiv.org/abs/0811.0191v1

Mathematics

Algebraic Topology

Scientific paper

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.

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