Parametrized ring-spectra and the nearby Lagrangian conjecture

Abstract

Abstract:We prove that any closed connected exact Lagrangian manifold L in a connected cotangent bundle T * N is up to a finite covering space lift a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra FL parametrized by the manifold N . The homology of FL will be (twisted) symplectic cohomology of T * L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL and the product combined with intersection product on N induces a product on this spectral sequence. This product structure and its relation to the intersection product on L is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that L → N is always a homotopy equivalence.

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

“…Combining this theorem with the results by Abouzaid-Kragh [1], [33], we obtain a positive answer to the nearby Lagrangian conjecture for (T * T 2 , dλ), i.e. this establishes Theorem B.…”

“…By the result in [33], the latter is indeed always the case. Furthermore, after a fiber-wise rescaling in T * T 2 (which induces a Hamiltonian isotopy of exact Lagrangian submanifolds), we may assume that any exact Lagrangian submanifold is contained in the bounded subset T * 1/4π S 1 × T * 1/4π S 1 .…”

“…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

“…Abouzaid and Kragh [2,61] have proved that L and L must be homotopy equivalent, and in a few cases-L = S 4n+1 or L = (S 1 × S 8n−1 ); see [1,33]-it is known that T * L remembers aspects of the smooth structure on L, i.e., T * L ∼ = ω T * (L#Σ) for certain homotopy spheres Σ. Question 2.3.…”

The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

“…This issue was highlighted by T. Kragh in [Kra07], and clarified in [Abo11], [Kra11] and [AS13]. Namely, if one defines the boundary operator of the Floer complex using standard conventions, the Floer complex of H is isomorphic to the Morse complex of S with a system of local coefficients on H 1 (T, M ).…”