$\mathbb{A}^{1}$-homotopy invariants of topological Fukaya categories of surfaces. (9th June 2017)

• Record Type: Journal Article
• Authors: Dyckerhoff, Tobias
• Is Part Of: Compositio mathematica. Volume 153:Number 8(2017)
• Page Start: 1673
• Page End: 1705
• Publication Date: 2017-06-09
• Subjects: 18G55 (primary); topological Fukaya categories, -- ∞-categories, -- Segal spaces, -- K-theory; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007205
• Languages: English
• ISSNs: 0010-437X

Abstract

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.