A1-homotopy invariants of topological Fukaya categories of surfaces

Abstract

We provide an explicit formula for localizing $A^1$-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.

32 pages, v3: references added, comments welcome, v4: strengthened main result, submitted

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