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Yakov Eliashberg

Publications and source records attributed to Yakov Eliashberg.

At least 19 recordsLinked to original sources
arXiv2026

Weinstein manifolds as cotangent buildings

We introduce the framework of cotangent buildings to complement and refine that of Weinstein handlebodies. While Weinstein handlebodies are suitable for a ``bottom-up" analysis of the Weinstein structure, cotangent buildings also enable a ``top down" analysis. Further, cotangent buildings include a precise control over the interaction of any subcollection of the various building blocks, each of which is modeled on the cotangent bundle of a manifold with corners. Our main result is that any Weinstein manifold is Weinstein homotopic to one admitting the structure of a cotangent building.

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arXiv2025

Honda-Huang's work on contact convexity revisited

Following the overall strategy of the paper ``Convex hypersurfaces in contact topology" by Ko Honda and Yang Huang on contact convexity in high dimensions, we present a simplified proof of their main result.

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arXiv2022

Geomorphology of Lagrangian ridges

We prove an "h-principle without pre-conditions" for the elimination of tangencies of a Lagrangian submanifold with respect to a Lagrangian distribution. The main result states that such tangencies can always be completely removed at the cost of allowing the Lagrangian to develop certain non-smooth points, called Lagrangian ridges, modeled on the corner $\{p=|q|\} \subset \mathbb{R}^2$ together with its products and stabilizations. This result plays an essential role in the arborealization program.

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arXiv2022

Arboreal models and their stability

This is the first in a series of papers by the authors on the arborealization program. The main goal of the paper is the proof of uniqueness of arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. The parametric version of the stability result implies that the space of germs of symplectomorphisms that preserve a canonical model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically.

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arXiv2021

The space of tight contact structures on ${\mathbb R}^3$ is contractible

It was proven in the first author's paper "Contact 3-manifolds twenty years since J. Martinet's work" (Ann. Inst. Fourier, 42(1992), 165--192) that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. It was also claimed there without a proof that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on $S^3$, fixed at a point, is contractible. We prove this result in the current paper.

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arXiv2020

Making cobordisms symplectic

We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary.

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arXiv2017

Contact orderability up to conjugation

We study in this paper the remnants of the contact partial order on the orbits of the adjoint action of contactomorphism groups on their Lie algebras. Our main interest is a class of non-compact contact manifolds, called convex at infinity.

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arXiv2017

Weinstein manifolds revisited

This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds.

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arXiv2016

Flexible Lagrangians

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.

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arXiv2015

Symplectic homology product via Legendrian surgery

This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken in [BEE1] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here we express the product and $\BV$-operator of symplectic homology in that context.

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arXiv2014

Existence and classification of overtwisted contact structures in all dimensions

We establish a parametric extension $h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the $3$-dimensional result from \cite{Eli89}. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.

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arXiv2013

Constructing exact Lagrangian immersions with few double points

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding $S^1\times S^2\to\R^6_\st$ with vanishing Maslov class.

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arXiv2013

Stein structures: existence and flexibility

This survey on the topology of Stein manifolds is an extract from our recent joint book. It is compiled from two short lecture series given by the first author in 2012 at the Institute for Advanced Study, Princeton, and the Alfred Renyi Institute of Mathematics, Budapest.

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arXiv2013

Lagrangian caps

We establish an $h$-principle for exact Lagrangian embeddings with concave Legendrian boundary. We prove, in particular, that in the complement of the unit ball $B$ in the standard symplectic $\R^{2n}, 2n\geq 6$, there exists an embedded Lagrangian $n$-disc transversely attached to $B$ along its Legendrian boundary.

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arXiv2011

Effect of Legendrian Surgery

The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in our forthcoming paper. In the Appendix written by S. Ganatra and M. Maydanskiy it is shown that the results of this paper imply P. Seidel's conjecture equating symplectic homology with Hochschild homology of a certain Fukaya category.

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