arXiv subjects

Fabián Torres-Ardila

Publications and source records attributed to Fabián Torres-Ardila.

5 recordsLinked to original sources

arXiv2025

The Geometry of Loop Spaces II: Characteristic Classes

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in $H^{2k-1}(LM)$ associated to the residue Chern character on the loop space $LM$ of a Riemannian manifold $M^{2k-1}$. These WCS classes are associated to the $L^2$ connection and the Sobolev $s=1$ connections on $LM.$ The WCS classes detect several families of 5-manifolds whose isometry group has infinite fundamental group. These manifolds are the total spaces of the circle bundles associated to a multiple $pω, |p|\gg 0$, of the Kähler form $ω$ over an integral Kähler surface.

math.DG↗

arXiv2024

The Geometry of Loop Spaces II: Corrections

This paper contains corrections to Madea, Rosenberg, Torres-Ardila, "The Geometry of Loop Spaces II: Characteristic Classes," Advances in Math. (287), 2016, 485-518. The main change is that results about $π_1({\rm Diff}(M))$ are replaced by results about $π_1({\rm Isom}(M))$, where Diff$(M)$, Isom$(M)$ refer to the diffeomorphism and isometry group of the manifold $M$.

math.DG↗

arXiv2014

The Geometry of Loop Spaces I: $H^s$-Riemannian Metrics

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. We compute the connection forms of these metrics and the higher symbols of their curvature forms, which take values in pseudodifferential operators. These calculations are used in a followup paper "The Geometry of Loop Spaces II: Characteristic Classes" to construct Chern-Simons classes on the tangent bundle TLM which detect nontrivial elements in the diffeomorphism group of certain Sasakian 5-manifolds associated to Kaehler surfaces.

math.DG↗

arXiv2012

Secondary Characteristic Classes on Loop Spaces

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. The connection and curvature forms of these metrics take values in pseudodifferential operators. We develop a theory of Wodzicki-Chern-Simons classes using the s=0, 1 connections and the Wodzicki residue. These classes distinguish the smooth homotopy type of some circle actions on M = S^2 x S^3, and imply that the fundamental group of Diff(M) is infinite.

math.DG↗

arXiv2010

Characteristic Classes and Zeroth Order Pseudodifferential Operators

We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the identity. The leading order Chern classes are nonzero in general, and they detect elements of the de Rham cohomology groups of the classifying space BZ.

math.DG↗