Zero-loop open strings in the cotangent bundle and Morse homotopy

Abstract

0. Introduction. Many important works in symplectic geometry and topology are regarded as the symplectization or the quantization of the corresponding results in ordinary geometry and topology. One outstanding example is the celebrated Arnold conjecture which concerns the number of fixed points of a symplectic diffeomorphism or that of intersection points of two Lagrangian submanifolds. The homological version of the conjecture has been proved in various cases (see [Fll-5], [02,3,6], [On] and [PSS], and [07] for a survey and references on the Arnold conjecture and Floer homology). The estimate (in its homological version) predicted by the Arnold conjecture can be regarded as the symplectization or the quantization of the Morse inequality, and conversely the latter can be considered as the semi-classical limit and so a consequence of the former. From now on, we will use the term "quantization" for the similar process that appear below. To illustrate this statement, we consider the cotangent bundle of a given compact manifold and the graphs of exact one forms. The graph of an exact one form becomes a Lagrangian submanifold of the cotangent bundle with respect to the canonical sym-plectic structure. Then Floer's result on the Lagrangian intersections [Fll,3] will imply the Morse inequality. The Lagrangian intersection theory is indeed the intersection theoretic version of the Morse theory, while the Lefsechtz intersection theory is that of the degree theory of generic vector fields. The principle that the symplectic topology and geometry of the cotangent bundle (or more generally that of symplectic manifolds) is the quantization of the ordinary topology and geometry of the base, is a general principle which can be applied to many other situations. The equivalence of the two often holds, when there occurs the absence of the quantum contribution (or the non-existence of the bubbling phenomena). In this paper, we will provide another example of this principle in which we prove that the rational homotopy type of a compact manifold M can be described by the moduli space of pseudo holomorphic disks with appropriate Lagrangian boundary conditions in its cotangent bundle T*M. The precise statement of our result is in Section 1. Our result paves the way to applying the A 00-structure introduced by the first author [Fu2] to the study of the estimate, in terms of the rational homotopy invariant of the base manifold, of the number of intersections of the zero section in the cotangent bundle and its Hamiltonian deformation. This enables us to go one step further, beyond the existing homological estimate in the literature, towards the proof of the original Arnold conjecture which states that the number of the intersections will be greater than or equal to the Morse number of M. Viterbo [V] and Eliashberg-Gromov [EG] have also studied this kind of estimate using the generating functions of Lagrangian submanifolds.

Homogeneous coordinate rings and mirror symmetry for toric varieties

“…There are essentially no differences between the case we are studying, and that of closed Lagrangians which is explained in Fukaya and Oh [8].…”

Constructible sheaves and the Fukaya category

“…We next recall the moduli space of gradient trees from Fukaya-Oh [9]. A based metric ribbon tree is a quadruple (T, i, v 0 , λ) of the following data.…”

Knot and braid invariants from contact homology I

“…Showing that the differentials of these DGAs actually count the appropriate holomorphic disks is beyond the scope of this paper. The analytical details, which use an approach due to Fukaya and Oh [10] of counting gradient flow trees, are the subject of work in progress. As in the theory of Legendrian knots in standard contact R 3 , however, the combinatorial proof of the invariance of our DGAs under isotopy gives evidence for the validity of our "computation" of relative contact homology.…”