  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.
