Which singular tangent bundles are isomorphic?

TL;DR

The paper investigates when singular tangent bundles, particularly $b$-tangent and $b^m$-tangent bundles, are isomorphic to the standard tangent bundle or to each other. It develops a concrete gluing-data framework to model these bundles, and shows that even versus odd $m$ yields distinct isomorphism behavior, with even $m$ giving ${}^{b^{m}}T M \\cong T M$ and odd $m$ giving ${}^{b^{m}}T M \\cong {}^{b}T M$, closely tied to a desingularization picture for $b^m$-symplectic forms. A key combinatorial invariant is the graph $G_{M,Z}$, whose two-colorability governs obstructions to isomorphism and KO-class equality; Stiefel-Whitney and Pontrjagin data are analyzed in terms of a line bundle $L$ linked to $Z$, while a Poincaré–Hopf-type theorem for $b$-tangent bundles connects the Euler class to the topology of $(M,Z)$ and the dynamics of $b$-vector fields. The paper culminates with explicit results for $b$-spheres: ${}^{b}T\mathbb{S}^n \cong T\mathbb{S}^n$ holds iff $n$ is odd, while no such isomorphism exists for even $n$, derived via obstruction theory and a lifting argument. The discussion extends to edge structures and folded-cotangent bundles, suggesting broader implications for geometry and dynamics beyond the $b$-case.

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 Poincaré-Hopf theorem for the $b^m$-tangent bundle, offering new insights into the interplay between singular structures and topological invariants.

Figures (7)

• Figure 1: Images of a $b$-manifold $M$ (in black) with critical set $Z$ (in red) and an open cover in the assumptions of \ref{['sec:isomorphism-bundles']}. The collar neighbourhoods $V_\gamma$ are represented in green, while the open sets $U_\alpha$ are drawn in orange.
• Figure 2: Depiction of the local defining function for $Z$ used in \ref{['subsec:gluing-bm-tangent']}. This choice can be further adapted to additional requirements: we may assume $f \neq 1, -1$ only inside the open set $(- \varepsilon, \varepsilon)$. Moreover, this choice can be made so that $f'(0) = 1$ always.
• Figure 3: Example of a $b$-manifold $M$ with critical set $Z$ (in red). On the right, we have its associated graph. Observe one edge can go from one node to itself: in particular, this graph is not two-colorable.
• Figure 4: Plots with various level sets of the function $f_\delta = \frac{x }{3} + y^2/2$, whose gradient is $X_\delta$, for different values of $\delta$. The black line represents the hypersurface $Z = \{x = 0\}$, and the black point represents the zero $p_\delta = (\delta, 0)$.
• Figure 5: Description of the tangent planes $\mathrm{T}_{p_\text{n}} \mathbb{S}^{n - 1}$ and $\mathrm{T}_{p_\text{s}} \mathbb{S}^{n - 1}$. The reader should observe that, under the identification with the plane $\mathbb{R}^{n} \cap \{x_{n - 1} = 0\}$, the induced orientations in both planes are different.

Theorems & Definitions (44)

• theorem 2.1: Serre serre_faisceaux_1955
• example 2.2
• example 2.3
• remark 2.4
• example 2.5
• example 2.6
• remark 3.1
• theorem 3.2
• proof
• corollary 3.3