Generation for Lagrangian cobordisms in Weinstein manifolds

Abstract

We prove that Lagrangian cocores and Lagrangian linking disks of a stopped Weinstein manifold generate the Lagrangian cobordism infinity-category. As a geometric consequence, we see that any brane (after stabilization) admits a Lagrangian cobordism to a disjoint union of some standard collection of branes (cocores, linking disks, and a zero object). For example, when our stopped Weinstein manifold is a point stopped by itself, we find that any exact brane in Euclidean space admits a Lagrangian cobordism to a disjoint union of cotangent fibers and a zero object. (This is a stronger statement than one could obtain from purely Fukaya-categorical generation results.) Our methods are constructive. For example, when our Weinstein manifold is a point, after stabilization we can resolve the conormal to a compact manifold A of R^n by a sequence of cotangent fibers; the resulting filtration realizes, after passage to the wrapped Fukaya category, the Morse cochain complex of A associated to (and hence filtered by) a generic ``distance to a point'' function; the associated gradeds are the reduced homologies of the Morse attaching spheres. There is also an algebraic consequence. Lagrangian cobordism theory is conjectured (in analogue to classical cobordism theory) to be linear over a ring spectrum L controlling Lagrangian cobordisms between cotangent fibers in Euclidean spaces. Our main theorem gives strong evidence for this conjecture: The infinity-category of Lagrangians and their cobordisms in R^infinity is equivalent to a full subcategory of modules over L. We conclude by proving a \pi_0-level theorem that gives further evidence of the above conjecture: We exhibit a \pi_0-level symmetric monoidal structure compatible with the linear structure of L-modules.