A1-homotopy invariants of topological Fukaya categories of surfaces

TL;DR

This paper derives an explicit formula for localizing A^1-homotopy invariants of topological Fukaya categories of surfaces, reducing complex calculations to simpler local cases using sheaf theory and localization techniques.

Contribution

It introduces a sheaf-theoretic approach to compute A^1-homotopy invariants of topological Fukaya categories, including a localization theory analogous to algebraic K-theory.

Findings

Explicit formula for invariants of Fukaya categories of surfaces.

Reduction of calculations to boundary-marked disks.

Development of a localization theory for topological Fukaya categories.

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.