Which singular tangent bundles are isomorphic?

TL;DR

This paper studies when logarithmic and $b$-tangent bundles are isomorphic to tangent bundles, especially on spheres, and proves a Poincaré-Hopf theorem for $b^m$-tangent bundles, linking singular geometry with topology.

Contribution

It characterizes conditions for isomorphisms of singular tangent bundles and establishes a Poincaré-Hopf theorem for $b^m$-tangent bundles, advancing understanding of singular structures in geometry.

Findings

Conditions for isomorphism of singular tangent bundles on spheres

A Poincaré-Hopf theorem for $b^m$-tangent bundles

Insights into the relationship between singular structures and topological invariants

Abstract

Logarithmic and $b$-tangent bundles provide a versatile framework for addressing singularities in geometry. Introduced by Deligne and Melrose, these modified bundles resolve singularities by reframing singular vector fields as well-behaved sections of these singular bundles. This approach has gained significant attention in symplectic geometry, particularly through its applications to the study of Poisson manifolds that are symplectic away from a hypersurface ($b^m$-symplectic forms). In this article, we investigate the conditions under which these singular tangent bundles are isomorphic to the tangent bundle or other singular bundles, analyzing in detail the case of spheres. Furthermore, we establish a Poincar\'e-Hopf theorem for the $b^m$-tangent bundle, offering new insights into the interplay between singular structures and topological invariants.