The symplectic geometry of cotangent bundles from a categorical viewpoint

Abstract

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.

27 pages, 2 figures. Version 4 -- application added (to exact Lagrangians in Euclidean space which are standard at infinity), results sharpened for cotangent bundles of tori

References in corpus (2)

• Lagrangian isotopies in Stein manifolds
• Isotopies of high genus Lagrangian surfaces

Cited by in corpus (4)

• Floer homology on the universal cover, a proof of Audin's conjecture and other constraints on Lagrangian submanifolds
• Constraints on exact Lagrangians in cotangent bundles of manifolds fibred over the circle
• The Picard-Lefschetz theory of complexified Morse functions
• Complexifications of Morse functions and the directed Donaldson-Fukaya category