homotopy invariants of topological Fukaya categories of surfaces

We prove a Mayer-Vietoris theorem for localizing A 1 -homotopy invariants of topological Fukaya categories of stable marked framed surfaces. Following a proposal of Kontsevich, this differential 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 nature to reduce the calculation of invariants to the local case when the surface is a boundary-marked disk. As an application, we compute the periodic cyclic homology over fields of characteristic 0. 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 of schemes.