Fukaya category of infinite-type surfaces

TL;DR

This paper constructs a Fukaya category for infinite-type surfaces using gradient sectorial Lagrangians, expanding the understanding of symplectic invariants in non-compact, infinite-genus settings.

Contribution

It introduces a new Fukaya category for infinite-type surfaces, describing its generators and showing it differs from finite-type surface categories.

Findings

Constructed a Fukaya category for infinite-type surfaces.

Described generators in terms of the surface's end structure.

Proved the category is not quasi-equivalent to finite-type limits.

Abstract

In this paper, we construct a Fukaya category of any infinite type surface whose objects are gradient sectorial Lagrangians. This class of Lagrangian submanifolds is introduced by one of the authors in [Oh21b] which can serve as an object of a Fukaya category of any Liouville manifold that admits an exhausting proper Morse function, in particular on the Riemann surface of infinite type. We describe a generating set of the Fukaya category in terms of the end structure of the surface when the surface has countably many limit points in its ideal boundary, the latter of which can be described in terms of a subset of the Cantor set. We also show that our Fukaya category is not quasi-equivalent to the limit of the Fukaya category of surfaces of finite type appearing in the literature.