# Geometric and algebraic presentations of Weinstein domains

@article{Lazarev2019GeometricAA, title={Geometric and algebraic presentations of Weinstein domains}, author={Oleg Lazarev}, journal={arXiv: Symplectic Geometry}, year={2019} }

We prove that geometric intersections between Weinstein handles induce algebraic relations in the wrapped Fukaya category, which we use to study the Grothendieck group. We produce a surjective map from middle-dimensional singular cohomology to the Grothendieck group, show that the geometric acceleration map to symplectic cohomology factors through the categorical Dennis trace map, and introduce a Viterbo functor for $C^0$-close Weinstein hypersurfaces, which gives an obstruction for Legendrians…

## 6 Citations

Localization and flexibilization in symplectic geometry

- Mathematics
- 2021

We introduce the critical Weinstein category the result of stabilizing the category of Weinstein sectors and inverting subcritical morphisms and construct localizing ‘P-flexibilization’ endofunctors…

Prime-localized Weinstein subdomains

- Mathematics
- 2020

For any high-dimensional Weinstein domain and finite collection of primes, we construct a Weinstein subdomain whose wrapped Fukaya category is a localization of the original wrapped Fukaya category…

Lagrangian cobordism functor in microlocal sheaf theory

- Mathematics
- 2021

Given a Lagrangian cobordism L of Legendrian submanifolds from Λ− to Λ+, we construct a functor Φ ∗ L : Sh c Λ+ (M) → ShcΛ−(M) ⊗C−∗(Ω∗Λ−) C−∗(Ω∗L) between sheaf categories of compact objects with…

$C^0$-limits of Legendrian knots and contact non-squeezing

- Mathematics
- 2022

Take a sequence of contactomorphisms of a contact three-manifold that C-converges to a homeomorphism. If the images of a Legendrian knot limit to a smooth knot under this sequence, we show that it is…

Homological Mirror Symmetry for local SYZ singularities

- Mathematics
- 2021

Gross and Siebert identified a class of singular Lagrangian torus fibrations which arise when smoothing toroidal degenerations, and which come in pairs that are related by mirror symmetry. We…

Nonexistence of exact Lagrangian tori in affine conic bundles over $\mathbb{C}^n$

- Mathematics
- 2021

Let M Ă C be a smooth affine hypersurface defined by the equation xy ` ppz1, ̈ ̈ ̈ , zn ́1q “ 1, where p is a Brieskorn-Pham polynomial and n ě 2. We prove that if L ãÑ M is an exact Lagrangian…

## References

SHOWING 1-10 OF 33 REFERENCES

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

- Mathematics
- 2017

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points…

Symplectic cohomology and duality for the wrapped Fukaya category

- Mathematics
- 2012

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that…

Symplectic structures from topological Fukaya categories

- Mathematics
- 2016

The topological Fukaya category is the global sections of a certain sheaf of categories on a Lagrangian skeleton. Nadler has given a combinatorial description of this sheaf in the case of generic…

A geometric criterion for generating the Fukaya category

- Mathematics
- 2010

Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the…

On partially wrapped Fukaya categories

- MathematicsJournal of Topology
- 2019

We define a new class of symplectic objects called ‘stops’, which, roughly speaking, are Liouville hypersurfaces in the boundary of a Liouville domain. Locally, these can be viewed as pages of a…

Legendrian fronts for affine varieties

- MathematicsDuke Mathematical Journal
- 2019

In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein…

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

- Mathematics
- 2008

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the…

From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds

- Mathematics
- 2012

A beautiful and comprehensive introduction to this important field. --Dusa McDuff, Barnard College, Columbia University This excellent book gives a detailed, clear, and wonderfully written treatment…

Estimating the number of Reeb chords using a linear representation of the characteristic algebra

- Mathematics
- 2014

Given a chord-generic horizontally displaceable Legendrian submanifold $\Lambda\subset P\times \mathbb R$ with the property that its characteristic algebra admits a finite-dimensional matrix…