$\mathbb{A}^{1}$-homotopy invariants of topological Fukaya categories of surfaces. (9th June 2017) - Dyckerhoff, Tobias
Abstract : We provide an explicit formula for localizing$\mathbb{A}^{1}$ -homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential$\mathbb{Z}$ -graded category is defined as global sections of a constructible cosheaf of dg categories on any spine of the surface. Our theorem utilizes this sheaf-theoretic description to reduce the calculation of invariants to the local case when the surface is a boundary-marked disk. At the heart of the proof lies a theory of localization for topological Fukaya categories which is a combinatorial analog of Thomason–Trobaugh's theory of localization in the context of algebraic$K$ -theory for schemes.