arXiv subjects

Alfonso Garmendia

Publications and source records attributed to Alfonso Garmendia.

9 recordsLinked to original sources

arXiv2025

A singular Serre-Swan theorem via tepui fibrations

The classical Serre-Swan theorem asserts that any finitely generated projective module over the algebra $C^\infty(M)$ of smooth functions of a manifold $M$ can be realized as the sections of a vector bundle over $M$. In this article, we extend this theorem beyond the projective case by introducing a notion of singular vector bundle whose sections can realize all finitely generated $C^\infty(M)$-modules, up to invisible elements. We introduce tepui fibrations as the underlying geometric objects of these singular vector bundles, and show how these tepui fibrations can model singular foliations, their holonomy groupoids, and singular subalgebroids.

math.DG↗

arXiv2025

PB-groupoids vs VB-groupoids

In this paper we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid. This recovers as particular cases various constructions involving structural Lie 2-groups, Lie groupoids and Lie groups. Moreover, given any VB-groupoid, we detect which frames are compatible with the underlying structure and show that the set of such frames has a natural PB-groupoid structure, with the action of the general linear 2-groupoid of the appropriate ranks. We use this construction, as well as that of the associated bundle induced by a 2-representation, to extend the classical correspondence between vector bundles and principal bundles to a new correspondence between VB-groupoids and PB-groupoids. We conclude with several examples and applications.

math.DG↗

arXiv2025

E-structures and almost regular Poisson manifolds

In recent years, $b$-symplectic manifolds have become important structures in the study of symplectic geometry, serving as Poisson manifolds that retain symplectic properties away from a hypersurface. Inspired by this rich landscape, $E$-structures were introduced by Nest and Tsygan in \cite{NT2} as a comprehensive framework for exploring generalizations of $b$-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by \cite{MS21}. We also examine the closely related concept of almost regular Poisson manifolds, as studied in \cite{AZ17}, which reveals a natural Poisson groupoid associated with these structures. In this article, we investigate the intricate relationship between $E$-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the $E$-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.

math.SG↗

arXiv2023

Principal bundle groupoids, their gauge group and their nerve

We consider groupoids in the category of principal bundles, which we call principal bundles (PB) groupoids. Inspired by work by Th. Nikolaus and K. Waldorf, we generalise bundle gerbes over manifolds to bundle gerbes over groupoids and discuss a functorial correspondence between PB groupoids and bundle gerbes over groupoids. From a PB groupoid over a fibre product groupoid, we build a bundle gerbe over another fibre product groupoid. Conversely, from a bundle gerbe over a Lie groupoid, we build a PB groupoid. It has a trivial base and from any PB groupoid with trivial base, we build a bundle gerbe over a Lie groupoid. In that case, the resulting bundle gerbe is isomorphic as a groupoid to a partial quotient of the PB groupoid. We describe the nerves of PB groupoids and their partial quotients, which are simplicial objects in the category of principal bundles. Applying this construction enables us to define the inner transformation group of the nerve of a partial quotient groupoid and to describe the transformations of the corresponding bundle gerbe.

math.DG↗

arXiv2021

Groupoids and singular foliations

This doctoral thesis has two objectives. The first objective is to introduce a notion of equivalence for singular foliations that preserves their transverse geometry and is compatible with the notions of Morita equivalence of the holonomy groupoids and the transverse equivalence for regular foliations that appeared in the 1980's. The second one is to describe the structures behind quotients of singular foliations and to connect these results with their associated holonomy groupoids. It also wants to give an introduction to the notion of singular foliations as given by Androulidakis and Skandalis in arXiv:math/0612370, as well as to their relation with Lie groupoids and Lie algebroids.

math.DG↗

arXiv2021

Integration of singular foliations via paths

We give a new construction of the holonomy and fundamental groupoids of a singular foliation. In contrast with the existing construction of Androulidakis and Skandalis, our method proceeds by taking a quotient of an infinite dimensional space of paths. This strategy is a direct extension of the classical construction for regular foliations and mirrors the integration of Lie algebroids via paths (per Crainic and Fernandes). In this way, we obtain a characterization of the holonomy and fundamental groupoids of a singular foliation that more clearly reflects the homotopic character of these invariants. As an application of our work, we prove that the constructions of the fundamental and holonomy groupoid of a foliation have functorial properties.

math.DG↗

arXiv2020

Quotients of singular foliations and Lie 2-group actions

Every singular foliation has an associated topological groupoid, called holonomy groupoid (see arXiv:math/0612370). In this note we exhibit some functorial properties of this assignment: if a foliated manifold $(M,\mathcal{F}_M)$ is the quotient of a foliated manifold $(P,\mathcal{F}_P)$ along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analog statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.

math.DG↗

arXiv2018

On the Inner Automorphisms of a Singular Foliation

A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give an alternative proof of the (surprisingly non-trivial) fundamental fact that the time-one flow of an element of a singular foliation (i.e. its exponential) is an automorphism of the singular foliation.

math.DG↗

arXiv2018

Hausdorff Morita Equivalence of singular foliations

We introduce a notion of equivalence for singular foliations - understood as suitable families of vector fields - that preserves their transverse geometry. Associated to every singular foliation there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's.

math.DG↗