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Register a new userH-principles for regular Lagrangians
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Authors
Oleg Lazarev
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We prove an existence h-principle for regular Lagrangians with Legendrian boundary in arbitrary Weinstein domains of dimension at least six; this extends a previous result of Eliashberg, Ganatra, and the author for Lagrangians in flexible domains. Furthermore, we show that all regular Lagrangians come from our construction and describe some related decomposition results. We also prove a regular version of Eliashberg and Murphys h-principle for Lagrangian caps with loose negative end. As an application, we give a new construction of infinitely many regular Lagrangian disks in the standard Weinstein ball.
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We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed $n$-manifolds of dimension $n>2$ can be realized as exact Lagrangian submanifolds of $T^*S^n$ with possibly exotic Weinstein symplectic structures. These Weinstein structures on $T^* S^n$, infinitely many of which are distinct, are formed by a single handle attachment to the standard $2n$-ball along the Legendrian boundaries of flexible Lagrangians. We also formulate a number of open problems.
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.
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.
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.
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.
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