Searcharxiv⌕ Search

arXiv subjects

Daniel Alvarez-Gavela

Publications and source records attributed to Daniel Alvarez-Gavela.

12 recordsLinked to original sources
arXiv2026

Framed bordism of Lagrangian homotopy spheres via generating functions

This note combines a result of B\"okstedt and Waldhausen concerning the so-called derivative map on tubes with the existence theorem for generating functions of tube type for nearby Lagrangian homotopy spheres due to Abouzaid, Courte, Guillermou and Kragh to obtain a restriction on the smooth structure of nearby Lagrangian homotopy spheres. Concretely, if a homotopy $n$-sphere $L$ admits a Lagrangian embedding in the cotangent bundle of some other homotopy $n$-sphere $M$, then the difference $[L]-[M]$ in $\theta_n/bP_{n+1}$ is a multiple of the Hopf element $\eta \in \pi^1_s$. In particular it follows that $[L]-[M]$ is 2-torsion in $\theta_n/bP_{n+1}$, hence if $n$ is even then $L\# L$ is diffeomorphic to $M \# M$. As another application, if a homotopy $8$-sphere $L$ admits a Lagrangian embedding in $T^*S^8$, then $L$ is diffeomorphic to $S^8$. The results presented in this note are subsumed by a joint work with Abouzaid, Courte and Kragh which treats the general case in which $M$ is an arbitrary smooth manifold. When $M$ is a homotopy sphere the situation is significantly simpler and the purpose of this note is to give a concise exposition of the main result in this special case.

math.SG↗
arXiv2025

On arborealization, Maslov data, and lack thereof

For a Weinstein manifold, we compare and contrast the properties of admitting an arboreal skeleton and admitting Maslov data. Both properties are implied by the existence of a polarization, for which a basic obstruction is that the odd Chern classes are 2-torsion. In a similar spirit we establish cohomological obstructions to the existence of arboreal skeleta and to the existence of Maslov data, exhibiting concrete examples to illustrate their failure. For instance, we show that complements of smooth anti-canonical divisors in complex projective space may fail to admit either arboreal skeleta or Maslov data. We also exhibit an example of a Weinstein manifold which admits Maslov data but does not admit an arboreal skeleton.

math.SG↗
arXiv2024

Flexibility of singularities and beyond

We survey a selection of Yasha Eliashberg's contributions to the philosophy of the h-principle, with a focus on the simplification of singularities and its applications.

math.GT↗
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.

math.SG↗
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.

math.SG↗
arXiv2021

Caustics of Lagrangian homotopy spheres with stably trivial Gauss map

For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $π_n U_n/O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.

math.SG↗
arXiv2020

A Legendrian Turaev torsion via generating families

We introduce a Legendrian invariant built out of the Turaev torsion of generating families. This invariant is defined for a certain class of Legendrian submanifolds of 1-jet spaces, which we call of Euler type. We use our invariant to study mesh Legendrians: a family of 2-dimensional Euler type Legendrian links whose linking pattern is determined by a bicolored trivalent ribbon graph. The Turaev torsion of mesh Legendrians is related to a certain monodromy of handle slides, which we compute in terms of the combinatorics of the graph. As an application, we exhibit pairs of Legendrian links in the 1-jet space of any orientable closed surface which are formally equivalent, cannot be distinguished by any natural Legendrian invariant, yet are not Legendrian isotopic. These examples appeared in a different guise in the work of the second author with J. Klein on pictures for $K_3$ and the higher Reidemeister torsion of circle bundles.

math.SG↗
arXiv2017

The simplification of singularities of Lagrangian and Legendrian fronts

We establish a full $h-$principle ($C^0-$close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy.

math.SG↗
arXiv2017

Refinements of the holonomic approximation lemma

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working one pure partial derivative at a time is central to the discussion. We give applications of our results to flexible symplectic and contact topology.

math.GT↗
arXiv2016

Gaussian curvature in codimension > 1

The Gaussian curvature $K$ is a fundamental geometric quantity discovered by Gauss in the case of surfaces embedded in $\mathbb{R}^3$. One can naturally extend the definition of the Gaussian curvature to arbitrary submanifolds of $\mathbb{R}^k$ so that the extrinsic interpretation of $K$, the Theorema Egregium and the Gauss-Bonnet Theorem still hold. We give a concise exposition of these classical facts.

math.DG↗