Geometric and algebraic presentations of Weinstein domains

TL;DR

This paper explores the relationship between geometric structures of Weinstein domains and their algebraic invariants, establishing new connections and criteria for generation and equivalence in symplectic topology.

Contribution

It introduces a geometric approach to algebraic relations in the wrapped Fukaya category and extends criteria for Weinstein domain generation.

Findings

Surjective map from middle-dimensional cohomology to Grothendieck group

Factorization of the acceleration map through the Dennis trace

Obstruction for Legendrians to be $C^0$-close

Abstract

We prove that geometric intersections between Weinstein handles induce algebraic relations in the wrapped Fukaya category, which we use to study the Grothendieck group. We produce a surjective map from middle-dimensional singular cohomology to the Grothendieck group, show that the geometric acceleration map to symplectic cohomology factors through the categorical Dennis trace map, and introduce a Viterbo functor for $C^0$-close Weinstein hypersurfaces, which gives an obstruction for Legendrians to be $C^0$-close. We show that symplectic flexibility is a geometric manifestation of Thomason's correspondence between split-generating subcategories and subgroups of the Grothendieck group, which we use to upgrade Abouzaid's split-generation criterion to a generation criterion for Weinstein domains. Thomason's theorem produces exotic presentations for certain categories and we give geometric analogs: exotic Weinstein presentations for standard cotangent bundles and Legendrians whose Chekanov-Eliashberg algebras are not quasi-isomorphic but are derived Morita equivalent.