A1-homotopy invariants of topological Fukaya categories of surfaces

Understand

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.