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Georgios Dimitroglou Rizell

Publications and source records attributed to Georgios Dimitroglou Rizell.

At least 37 records · Page 2Linked to original sources
arXiv2020

Refined disk potentials for immersed Lagrangian surfaces

We define a refined Gromov-Witten disk potential of self-transverse monotone immersed Lagrangian surfaces in a symplectic 4-manifold as an element in a capped version of the Chekanov--Eliashberg dg-algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point. We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5-sphere.

math.SG↗
arXiv2020

On Legendrian products and twist spuns

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In order to do this, we prove a few structural results on the bilinearised Legendrian contact homology and augmentation variety of a twist spun. In addition, we show that the threefold Bohr-Sommerfeld covers of the Clifford torus and Chekanov torus are not twist spuns.

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arXiv2020

Linking of Lagrangian Tori and Embedding Obstructions in Symplectic 4-Manifolds

We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a first corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ which are homologous to the zero section. As a second corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the second part of the paper, we study linking of Lagrangian tori in $(\mathbb{R}^4, ω)$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.

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arXiv2020

Bulky Hamiltonian isotopies of Lagrangian tori with applications

We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e.~the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J.~Gutt and M.~Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in terms of quantitative considerations; more specifically, we show that a torus is Hamiltonian isotopic to the Clifford torus whenever one can find a symplectic embedding of a sufficiently large ball in its complement.

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arXiv2019

Positive Legendrian isotopies and Floer Theory

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds admitting a filling of a Liouville domain with non-zero symplectic homology is strongly orderable in the sense of Liu.

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arXiv2019

The classification of Lagrangians nearby the Whitney immersion

The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number $+1.$ This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal cubic curve inside the projective plane, and the latter Weinstein manifold is thus the `standard' neighbourhood of Lagrangian immersions of this type. We classify the Lagrangians inside such a neighbourhood which are homologically essential, and which either are embedded or immersed with a single double point; they are shown to be Hamiltonian isotopic to either product tori, Chekanov tori, or rescalings of the Whitney immersion.

math.SG↗
arXiv2019

The stable Morse number as a lower bound for the number of Reeb chords

Assume that we are given a closed chord-generic Legendrian submanifold $Λ\subset P \times \mathbb R$ of the contactisation of a Liouville manifold, where $Λ$ moreover admits an exact Lagrangian filling $L_Λ \subset \mathbb R \times P \times \mathbb R$ inside the symplectisation. Under the further assumptions that this filling is spin and has vanishing Maslov class, we prove that the number of Reeb chords on $Λ$ is bounded from below by the stable Morse number of $L_Λ$. Given a general exact Lagrangian filling $L_Λ$, we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of $L_Λ$, following Ono-Pajitnov's implementation in Floer homology of invariants due to Sharko. This improves previously known bounds in terms of the Betti numbers of either $Λ$ or $L_Λ$.

math.SG↗
arXiv2018

Floer theory for Lagrangian cobordisms

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 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.

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arXiv2018

An energy-capacity inequality for Legendrian submanifolds

We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $\mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy is measured in terms of the oscillation of the contact Hamiltonian, together with the maximal factor by which the contact form is shrunk during the isotopy. The main tool used is a Mayer--Vietoris sequence for Lagrangian Floer homology, obtained by "neck-stretching" and "splashing."

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arXiv2016

Lagrangian isotopy of tori in $S^2 \times S^2$ and $\mathbb{C}P^2$

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone $S^2 \times S^2$. The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for $T^*\mathbb{T}^2$, i.e.~it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

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arXiv2016

Legendrian Ambient Surgery and Legendrian Contact Homology

Let $L \subset Y$ be a Legendrian submanifold of a contact manifold, $S\subset L$ a framed embedded sphere bounding an isotropic disc $D_S \subset Y \setminus L$, and use $L_S$ to denote the manifold obtained from $L$ by a surgery on $S$. Given some additional conditions on $D_S$ we describe how to obtain a Legendrian embedding of $L_S$ into an arbitrarily small neighbourhood of $L \cup D_S \subset Y$ by a construction that we call Legendrian ambient surgery. In the case when the disc is subcritical, we produce an isomorphism of the Chekanov-Eliashberg algebra of $L_S$ with a version of the Chekanov-Eliashberg algebra of $L$ whose differential is twisted by a count of pseudo-holomorphic discs with boundary-point constraints on $S$. This isomorphism induces a one-to-one correspondence between the augmentations of the Chekanov-Eliashberg algebras of $L$ and $L_S$.

math.SG↗
arXiv2016

Noncommutative augmentation categories

To a differential graded algebra with coefficients in a noncommutative algebra, by dualisation we associate an $A_\infty$-category whose objects are augmentations. This generalises the augmentation category of Bourgeois and Chantraine to the noncommutative world.

math.SG↗
arXiv2016

Uniqueness of extremal Lagrangian tori in the four-dimensional disc

The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold $L$ of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on $L$. They also showed that this quantity is bounded from above by $π/n$ for a Lagrangian torus inside the $2n$-dimensional unit disc equipped with the standard symplectic form. A Lagrangian torus for which this upper bound is attained is called extremal. We show that an extremal Lagrangian torus inside the four-dimensional unit disc is contained in the boundary $\partial D^4=S^3$, and is hence Hamiltonian isotopic to the product torus $S^1_{1/\sqrt{2}} \times S^1_{1/\sqrt{2}} \subset S^3$. This provides an answer to a question by L. Lazzarini in the four-dimensional case.

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arXiv2016

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

Given a chord-generic horizontally displaceable Legendrian submanifold $Λ\subset P\times \mathbb R$ with the property that its characteristic algebra admits a finite-dimensional matrix representation, we prove an Arnold-type lower bound for the number of Reeb chords on $Λ$. This result is a generalization of the results of Ekholm-Etnyre-Sullivan and Ekholm-Etnyre-Sabloff which hold for Legendrian submanifolds whose Chekanov-Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds $Λ$ of $\mathbb C^{n}\times \mathbb R$, $n \ge 1$, whose characteristic algebras admit finite-dimensional matrix representations, but whose Chekanov-Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct a Legendrian submanifold $Λ\subset \mathbb C^{n}\times \mathbb R$ with the property that the characteristic algebra of $Λ$ does not satisfy the rank property. Finally, in the case when a Legendrian submanifold $Λ$ has a non-acyclic Chekanov-Eliashberg algebra, using rather elementary algebraic techniques we obtain lower bounds for the number of Reeb chords of $Λ$. These bounds are slightly better than the number of Reeb chords that is possible to achieve with a Legendrian submanifold whose Chekanov-Eliashberg algebra is acyclic.

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arXiv2016

Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications

Let $\mathbb{R} \times (P \times \mathbb{R})$ be the symplectisation of the contactisation of an exact symplectic manifold $P$, and let $\mathbb{R} \times Λ$ be a cylinder over a Legendrian submanifold in the contactisation. We show that a pseudo-holomorphic polygon in $P$ having boundary on the projection of $Λ$ can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on $\mathbb{R} \times Λ$. It follows that Legendrian contact homology may be equivalently defined by counting either of these objects. Using this result, we give a proof of Seidel's isomorphism of the linearised Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling.

math.SG↗
arXiv2015

Nontriviality results for the characteristic algebra of a DGA

Assume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn.

math.RA↗
arXiv2015

Exotic spheres and the topology of symplectomorphism groups

We show that, for certain families $ϕ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $ϕ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the $A_m$-Milnor fibre. The nontriviality is detected by considering the action of symplectomorphisms on the space of parametrised Lagrangian submanifolds. We find related examples of symplectic mapping classes for $T^*(S^n\times S^1)$ and of an exotic symplectic structure on $T^*(S^n\times S^1)$ standard at infinity.

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