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Noé Bárcenas

Publications and source records attributed to Noé Bárcenas.

9 recordsLinked to original sources
arXiv2024

The ideal-valued index of fibrations with total space a $G_{2}$ flag manifold

Using the cohomology of the $G_2$-flag manifolds $G_2/U(2)_{\pm}$, and their structure as a fiber bundle over the homogeneous space $G_2/SO(4)$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of such fiber bundles, for the $\mathbb{Z}_2$ action given by complex conjugation. Also, considering the tautological bundle $γ$ over $\widetilde{G}_{4}(\mathbb{R}^{7})$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of the pullback bundle of $sγ$ along the composition of the fiber bundle $ G_2/U(2)_{\pm} \to G_2/SO(4)$, the embedding between $G_2/SO(4)$ and $\widetilde{G}_{3}(\mathbb{R}^{7})$, and the map that takes the orthogonal complement of a subspace. Here $sγ$ means the associated sphere bundle of $γ$. Furthermore, we derive a general formula for the $n$-fold product bundle $sγ^n$ for which we make the same computations. We finish our work with an application of our computations in a problem concerning discrete geometry.

math.AT↗
arXiv2023

Rigidity of actions on metric spaces close to three dimensional manifolds

In this article we propose a metric variation on the C^0-version of the Zimmer program for three manifolds. After a reexamination of the isometry groups of geometric three-manifolds, we consider homomorphisms defined on higher rank lattices to them and establish a dichotomy betweeen finite image or infinite volume of the quotient. Along the way, we enumerate classification results for actions of finite groups on three manifolds where available, and we give an answer to a metric variation on topological versions of the Zimmer program for aspherical three manifolds, as asked by Weinberger and Ye, which are based on the dichotomy stablished in this work and known topological rigidity phenomena for three manifolds. Using results by John Pardon and Galaz-García-Guijarro, the dichotomy for homomorphisms of higher rank lattices to isometry groups of three manifolds implies that a C 0 -isometric version of the Zimmer program is also true for singular geodesic spaces closely related to three dimensional manifolds, namely three dimensional geometric orbifolds and Alexandrov spaces. A topological version of the Zimmer Program is seen to hold in dimension 3 for Alexandrov spaces using Pardon's ideas.

math.DG↗
arXiv2020

A probabilistic approach to exhaustion in the infinite-genus case

We explore the use of Costa and Farber's model for random simplicial complexes to give probabilistic evidence for exhaustion via rigid expansions on random simplicial complexes which are analogous of curve complexes. This has potential applications to action rigidity following Ivanov's meta-conjecture.

math.GT↗
arXiv2018

On topological rigidity of Alexandrov $3$-spaces

In this note we prove the Borel Conjecture for closed, irreducible and sufficiently collapsed three-dimensional Alexandrov spaces. We also pose several questions related to characterization of fundamental groups of three-dimensional Alexandrov spaces, finite groups acting on them and rigidity results.

math.MG↗
arXiv2018

Analytical Signatures and Proper Actions

In this short note we compare Mishchenko's definition of noncommutative signature for a manifold with proper $G$-action of a discrete, countable group $G$ with the (more analytical) counterpart defined by Higson and Roe in the series of articles "Mapping Surgery to Analysis". A generalization of the coarse index is also addressed.

math.KT↗
arXiv2018

Positive scalar curvature and low-degree group homology

Let $Γ$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $\mathrm{B} Γ$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $Γ$ with coefficients in the $\mathbb{C}Γ$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey's work on explicitly inverting this Chern character in low homological degrees.

math.KT↗
arXiv2016

Stable finiteness properties of infinite discrete groups

Let $G$ be an infinite discrete group. A classifying space for proper actions of $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are contractible for all finite subgroups $H$ of $G$. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper $G$-spectra and study its finiteness properties. We investigate when $G$ admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper $G$-spectra and to classical finiteness properties of the Weyl groups of finite subgroups of $G$. Finally, if the group $G$ is virtually torsion-free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of $G$, thus providing the first geometric interpretation of the virtual cohomological dimension of a group.

math.AT↗
arXiv2016

Multiplicative Structures and the Twisted Baum-Connes Assembly map

Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-theory groups whenever this assembly maps are isomorphisms.

math.KT↗