Examples of biquotients whose tangent bundle is not a biquotient vector bundle

TL;DR

The article investigates when the tangent bundle of a biquotient $M\cong G/\\!/ H$ is a biquotient vector bundle. It establishes that, in general, $TM$ need not belong to $\operatorname{BVec}(G/\\!/ H)$ for any presentation, producing infinite families (notably certain $\mathbb{C}P^k$, $\mathbb{H}P^k$, and higher-dimensional examples) where $TM$ fails the biquotient-vector-bundle criterion. The authors leverage Hirzebruch’s genus and obstruction theory to show nonexistence results and construct stable-biquotient tangents under a cohomological vanishing condition, with consequences for constructing nonnegative-curvature metrics on tangent bundles. They also provide rigorous low-dimensional classifications: for many dimension $\le 5$ biquotients, $TM$ is a biquotient bundle for every reduced presentation with $G$ simply connected, with specific exceptions such as $\mathbb{C}P^2\#\mathbb{C}P^2$, $\mathbb{C}P^2$, and $S^4$ that admit presentations where the tangent bundle is not a biquotient bundle. Altogether, the paper sharply delineates when the tangent bundle of biquotients can (or cannot) be realized as a biquotient vector bundle, and it highlights when stable-biquotient-tangent conclusions suffice for geometric applications.

Abstract

A biquotient vector bundle is any vector bundle over a biquotient $G/\!\!/ H$ of the form $G\times_{H} V$ for an $H$-representation $V$. Over most biquotients, biquotient vector bundles are the only vector bundles known to admit metrics of non-negative sectional curvature, and hence they play a crucial role in the context of the converse to the Soul Theorem of Cheeger and Gromoll. In this article, we study the question of when the tangent bundle of $G/\!\!/ H$ is a biquotient vector bundle. We find infinite families of examples of biquotients $M\cong G/\!\!/ H$ for which the tangent bundle is not a biquotient vector bundle for any presentation of $M$ as a biquotient. In addition, we find infinite families of manifolds which arise as biquotients in two ways: one for which the tangent bundle is a biquotient bundle, and one for which it is not. Some of these results depend on an observation of Hirzebruch which relates the signature and Euler characteristic of an almost complex manifold. We include a proof of this fact as it seems to be missing from the literature.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Hirzebruch
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4