In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several long exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.In both cases, Λ + = ∅. Thus we obtain a new proof of the following result.Corollary 1.4 ([23]). If Λ ⊂ P × R admits an augmentation, then there is no exact Lagrangian cobordism from Λ to ∅, i.e. there is no exact Lagrangian "cap" of Λ.Remark 1.5. Assume that Λ − admits an exact Lagrangian filling L inside the symplectisation, and that ε − is the augmentation induced by this filling. It follows that ε + is the augmentation induced by the filling L ⊙ Σ of Λ + obtained as the concatenation of L and Σ. Using Seidel's isomorphismsto replace the relevant terms in the long exact sequences (1) and (3), we obtain the long exact sequence for the pair (L ⊙ Σ, L) and the Mayer-Vietoris long exact sequence for the decomposition L ⊙ Σ = L ∪ Σ, respectively. This fact was already observed and used by the fourth author in [49].

We prove that the wrapped Fukaya category of any 2ndimensional Weinstein manifold (or, more generally, Weinstein sector) W is generated by the unstable manifolds of the index n critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer homology and the fairly simple observation that Floer homology vanishes for Lagrangian submanifolds that can be disjoined from the isotropic skeleton of the Weinstein manifold. Note that we do not need any additional assumptions on this skeleton. By applying our generation result to the diagonal in the product W × W , we obtain as a corollary that the open-closed map from the Hochschild homology of the wrapped Fukaya category of W to its symplectic cohomology is an isomorphism, proving a conjecture of Seidel. We work mainly in the "linear setup" for the wrapped Fukaya category, but we also sketch the minor modifications which we need to extend the proofs to the "quadratic setup" and to the "localisation setup". This is necessary for dealing with Weinstein sectors and for the applications.

We study the relation of an embedded Lagrangian cobordism between two closed, orientable Legendrian submanifolds of R 2n+1 . More precisely, we investigate the behavior of the Thurston-Bennequin number and (linearized) Legendrian contact homology under this relation. The result about the Thurston-Bennequin number can be considered as a generalization of the result of Chantraine which holds when n = 1. In addition, we provide a few constructions of Lagrangian cobordisms and prove that there are infinitely many pairs of exact Lagrangian cobordant and not pairwise Legendrian isotopic Legendrian n-tori in R 2n+1 .

Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of "compactification" and "transversality" with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory.

Abstract. We show that an exact Lagrangian cobordism L ⊂ R×P ×R from a Legendrian submanifold Λ ⊂ P ×R to itself satisfies H i (L; F) = H i (Λ; F) for any field F in the case when Λ admits a spin exact Lagrangian filling and the concatenation of any spin exact Lagrangian filling of Λ and L is also spin. The main tool used is Seidel's isomorphism in wrapped Floer homology. In contrast to that, for loose Legendrian submanifolds of C n × R, we construct examples of such cobordisms whose homology groups have arbitrary high ranks. In addition, we prove that the front S m -spinning construction preserves looseness, which implies certain forgetfulness properties of it.