arXiv subjects

Georgios Dimitroglou Rizell

Publications and source records attributed to Georgios Dimitroglou Rizell.

40 records · Page 3Linked to original sources

arXiv2015

Floer homology and Lagrangian concordance

We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of bilinearised Legendrian contact cohomology. This is used to prove the existence of non-invertible exact Lagrangian concordances in all dimensions. In addition, using a result of Eliashberg-Polterovich, we completely classify exact Lagrangian concordances from the Legendrian unknot to itself in the tight contact-three sphere: every such concordance is the trace of a Legendrian isotopy. We also discuss a high dimensional topological result related to this classification.

math.SG↗

arXiv2014

On homological rigidity and flexibility of exact Lagrangian endocobordisms

We show that an exact Lagrangian cobordism $L\subset \mathbb R \times P \times \mathbb R$ from a Legendrian submanifold $Λ\subset P\times \mathbb R$ to itself satisfies $H_i(L;\mathbb F)=H_i(Λ;\mathbb F)$ for any field $\mathbb F$ 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 $\mathbb{C}^n \times \mathbb R$, 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.

math.SG↗

arXiv2013

Exact Lagrangian caps and non-uniruled Lagrangian submanifolds

We make the elementary observation that the Lagrangian submanifolds of $\mathbb{C}^n$, for each $n \ge 3$, constructed by Ekholm, Eliashberg, Murphy and Smith are non-uniruled and moreover have infinite relative Gromov width. The construction of these submanifolds use exact Lagrangian caps, which obviously are non-uniruled in themselves. This property is also used to show that if a Legendrian submanifold inside a 1-jet space admits an exact Lagrangian cap then its Legendrian contact homology DGA is acyclic.

math.SG↗

arXiv2011

Knotted Legendrian surfaces with few Reeb chords

For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$ into $J^1(R^2)=R^5$ which lie in pairwise distinct Legendrian isotopy classes and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the Legendrian contact homology DGA does not admit any augmentation over $Z/2Z$, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in $J^1(S^2)$ from a similar perspective.

math.SG↗