Categorical non-properness in wrapped Floer theory

TL;DR

This paper demonstrates that in broad settings, the wrapped Fukaya category of Weinstein manifolds is either non-proper or trivial, extending observed phenomena from explicit computations to a general theoretical framework.

Contribution

It establishes a general non-properness or triviality result for wrapped Fukaya categories of Weinstein and Liouville manifolds, including criteria for non-exact cases.

Findings

Wrapped Fukaya category of Weinstein manifolds is either non-proper or zero.

Non-compact exact Lagrangians are either non-proper objects or zero.

The phenomenon extends beyond explicit examples to broad theoretical classes.

Abstract

In all known explicit computations on Weinstein manifolds, the self-wrapped Floer homology of non-compact exact Lagrangian is always either infinite-dimensional or zero. We show that a global variant of this observed phenomenon holds in broad generality: the wrapped Fukaya category of any positive-dimensional Weinstein (or non-degenerate Liouville) manifold is always either non-proper or zero, as is any quotient thereof. Moreover any non-compact connected exact Lagrangian is always either a "(both left and right) non-proper object" or zero in such a wrapped Fukaya category, as is any idempotent summand thereof. We also examine criteria under which the argument persists or breaks if one drops exactness, which is consistent with known computations of non-exact wrapped Fukaya categories which are smooth, proper, and non-vanishing (e.g., work of Ritter-Smith).