The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Abstract

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 [30,29] and the authors [15], before discussing a new approach using family Floer cohomology [11] 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.A classical problem in symplectic topology is to describe exact Lagrangian submanifolds inside a cotangent bundle. The main conjecture, usually attributed to Arnol'd [4], asserts that any (compact) submanifold of this kind should be Hamiltonian isotopic to the zero-section. In this sharp form, the result is known only for S 2 and RP 2 , and the proof uses methods which are specifically four-dimensional (both cases are due to Hind [17]; concerning the state of the art for surfaces of genus > 0, see [18]). In higher dimensions, work has concentrated on trying to establish topological restrictions on exact Lagrangian submanifolds. There are many results dealing with assorted partial aspects of this question ([26, 37, 38, 7, 35] and others), using a variety of techniques. Quite recently, more categorical methods have been added to the toolkit, and these have led to a result covering a fairly general situation. The basic statement (which we will later generalise somewhat) is as follows:Theorem 0.1 (Nadler, Fukaya-Seidel-Smith). Let Z be a closed, simply connected manifold which is spin, and M = T * Z its cotangent bundle. Suppose that L ⊂ M is an exact closed Lagrangian submanifold, which is also spin, and additionally has vanishing Maslov class. Then: (i) the projection L ֒→ M → Z has degree ±1; (ii) pullback by the projection is an isomorphism H * (Z; K) ∼ = H * (L; K) for any coefficient field K; and (iii) given any two Lagrangian submanifolds L 0 , L 1 ⊂ M with these properties, which meet transversally, one has |L 0 ∩ L 1 | ≥ dim H * (Z; K). Remarkably, three ways of arriving at this goal have emerged, which are essentially independent of each other, but share a basic philosophical outlook. One proof is due to Nadler [29], building on earlier work of Nadler and Zaslow [30] (the result in [29] is formulated for K = C, but it seems that the proof goes through for any K). Another one is given in [15], and involves, among other things, tools from [36] (for technical reasons, this actually works only for char(K) = 2). The third one, which is collaborative work of the three authors of this paper, is not complete at the time of writing, mostly because it relies on ongoing developments in general Floer homology theory. In spite of this, we included a description of it, to round off the overall picture.The best starting point may actually be the end of the proof, which can be taken to be roughly the same in all three cases. Let L be as in Theorem 0.1. The Floe...

Exact Lagrangian submanifolds in simply-connected cotangent bundles

“…In view of the correspondence between critical points of p R and q, and the isotopy invariance of Floer cohomology in T * N (which makes it irrelevant whether one takes the cotangent fibre at x j or y j−r ), the E 1 page now takes on the form (6). In fact, the last-mentioned observation also shows that all columns on this page are isomorphic, up to a shift.…”

“…Having derived the spectral sequence in this purely algebraic framework, we then quote, without proof, the geometric results from [22, Chapter 3] which explain how this applies to Lefschetz fibrations. An informal overview of the geometric side of the story is also given in [6].…”

“…Nevertheless, there are many similarities on a philosophical level; notably, the use of the Fukaya category of T * N , enlarged by admitting certain non-compact Lagrangian submanifolds, and of decompositions of the diagonal. We postpone a more detailed comparison (and a discussion of the situation when π 1 (N ) = 0) to [6]. This paper is organized as follows.…”

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

“…Very little is known about to what extent this conjecture is true. However, work by M. Abouzaid and T. Kragh [33], going back to a series of work by M. Abouzaid [1], K. Fukaya, P. Seidel, and I. Smith [20], [21], and D. Nadler [39], shows that the following strong partial result holds: The canonical projection of the cotangent bundle restricts to a homotopy equivalence L ֒→ T * N → N . Combining this statement with Theorem 7.1 proven below, it thus follows that the nearby Lagrangian conjecture holds in the case of (T * T 2 , dλ); see Theorem B.…”

A symplectic prolegomenon

“…Several of the first applications were variations on this theme: the following are taken from [47,62,80].…”