Generating Families for Lagrangians in $mathbb{R}^{2n}$ and the Hatcher-Waldhausen map

In this paper we construct a generating family quadratic at infinity for any exact Lagrangian in $mathbb{R}^{2n}$ equal to $mathbb{R}^n$ outside a compact set. Such generating families are related to the space $mathcal{M}_infty$ considered by Eliashberg and Gromov. We show that this space is the homotopy fiber of the Hatcher-Waldhausen map, and thus serves as a geometric model for this space. This relates the understanding of exact Lagrangians (and Legendrians) to algebraic K-theory of spaces. We then use this fibration sequence to produce new results (restrictions) on this type of Lagrangian. In particular we show how Bokstedts result that the Hatcher-Waldhausen map is a rational homotopy equivalence proves the new result that the stable Lagrangian Gauss map for our Lagrangian relative infinity is homotopy trivial.

Simple Homotopy Equivalence of Nearby Lagrangians

Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.

Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also obtained by Nadler). The ideas in that paper were extended by Abouzaid who proved that vanishing Maslov class alone implies homotopy equivalence. In this paper we present a short proof of the fact that any exact Lagrangian with vanishing Maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism. When the fundamental group of the base is pro-finite this implies homotopy equivalence.

Lagrangian exotic spheres

Let k>2. We prove that the cotangent bundles of oriented homotopy (2k-1)-spheres S and S are symplectomorphic only if the classes defined by S and S agree up to sign in the quotient group of oriented homotopy spheres modulo those which bound parallelizable manifolds. We also show that if the connect sum of real projective space of dimension (4k-1) and a homotopy (4k-1)-sphere admits a Lagrangian embedding in complex projective space, then twice the homotopy sphere framed bounds. The proofs build on previous work of Abouzaid and the authors, in combination with a new cut-and-paste argument, which also gives rise to some interesting explicit exact Lagrangian embeddings into plumbings. As another application, we show that there are re-parameterizations of the zero-section in the cotangent bundle of a sphere which are not Hamiltonian isotopic (as maps, rather than as submanifolds) to the original zero-section.

The Viterbo Transfer as a Map of Spectra

Let $L$ and $N$ be two smooth manifolds of the same dimension. Let $jcolon Lto T^*N$ be an exact Lagrange embedding. We denote the free loop space of $X$ by $Lambda X$. C. Viterbo constructed a transfer map $(Lambda j)^! colon H^*(Lambda L) to H^*(Lambda N)$. This transfer was constructed using finite dimensional approximation of Floer homology. In this paper we define a family of finite dimensional approximations and realize this transfer as a map of Thom spectra: $(Lambda j)_! colon (Lambda N)^{-TN} to (Lambda L)^{-TL+eta}$, where $eta$ is a virtual vector bundle classified by the tangential information of $j$.

On the immersion classes of nearby Lagrangians

We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.

Fibrancy of Symplectic Homology in Cotangent Bundles

We describe how the result in [1] extends to prove the existence of a Serre type spectral sequence converging to the symplectic homology SH_*(M) of an exact Sub-Liouville domain M in a cotangent bundle T*N. We will define a notion of a fiber-wise symplectic homology SH_*(M,q) for each point q in N, which will define a graded local coefficient system on N. The spectral sequence will then have page two isomorphic to the homology of N with coefficients in this graded local system.

Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture

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.

Orientations and Connective Structures on 2-vector Bundles

In work by Ausoni, Dundas and Rognes a half magnetic monopole is discovered and describes an obstruction to creating a determinant K(ku) to ku*. In fact it is an obstruction to creating a determinant gerbe map from K(ku) to K(Z,3). We describe this obstruction precisely using monoidal categories and define the notion of oriented 2-vector bundles, which removes this obstruction so that we can define a determinant gerbe. We also generalize Brylinskis notion of a connective structure to 2-vector bundles, in a way compatible with the determinant gerbe.