Biquotient vector bundles with no inverse

TL;DR

This paper demonstrates that, unlike certain vector bundles, biquotient bundles often lack inverses, revealing limitations in previous approaches to curvature on homogeneous spaces and identifying specific dimensions with such properties.

Contribution

The authors show that biquotient bundles generally do not have inverses, contrasting with previous results for homogeneous spaces, and identify dimensions where inverses always or never exist.

Findings

In each dimension n≥4 (except n=5), there exist biquotients with bundles lacking inverses.

For n≥6 (except n=7), infinitely many biquotients have no invertible non-trivial bundles.

Every biquotient bundle over certain low-dimensional biquotients has an inverse within the class.

Abstract

In previous work, the second author and others have found conditions on a homogeneous space $G/H$ which imply that, up to stabilization, all vector bundles over $G/H$ admit Riemannian metrics of non-negative sectional curvature. One important ingredient of their approach is Segal's result that the set of vector bundles of the form $G\times_H V$ for a representation $V$ of $H$ contains inverses within the class. We show that this approach cannot work for biquotients $G/\!\!/ H$, where we consider vector bundles of the form $G\times_{H} V$. We call such vector bundles biquotient bundles. Specifically, we show that in each dimension $n\geq 4$ except $n=5$, there is a simply connected biquotient of dimension $n$ with a biquotient bundle which does not contain an inverse within the class of biquotient bundles. In addition, we show that for $n\geq 6$ except $n=7$, there are infinitely many homotopy types of biquotients with the property that no non-trivial biquotient bundle has an inverse. Lastly, we show that every biquotient bundle over every simply connected biquotient $M^n = G/\!\!/ H$ with $G$ simply connected and with $n\in \{2,3,5\}$ has an inverse in the class of biquotient bundles.