Biquotient vector bundles with no inverse

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.