**David González Álvaro,
Universidad Politécnica de Madrid**

**Title:** Some properties of biquotient vector bundles

**Abstract:** Given a biquotient G//H, we say that a vector bundle
is a biquotient vector bundle if it is isomorphic to one of the form G
×_{H} V, where H acts diagonally via a representation
V. These bundles admit metrics of non-negative sectional) curvature,
thus the obvious question is which vector bundles over G//H are
actually biquotient vector bundles. In the particular case of a
homogeneous space G/H, homogeneous vector bundles are known to satisfy
two nice properties:

(1) The tangent bundle of a homogeneous space is homogeneous (and hence admits a metric of non-negative curvature)

(2) For every homogeneous vector bundle one can find another homogeneous vector bundle such that their Whitney sum is trivial. This property, combined with K-theoretical tools, implies that, up to stabilization, every complex vector bundle is homogeneous (and hence admits a metric of non-negative curvature) provided rank G - rank H < 1.
In this talk we will give examples showing that these properties fail to hold for biquotient vector bundles over arbitrary biquotients. Thus, new ideas are required to extend the results above on the existence of non-negatively curved metrics on bundles over homogeneous spaces to arbitrary biquotients. This is joint work with Jason DeVito.