  In this paper we give a survey of various results about the topology of
oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of
these results are new. We give self-contained proofs here. One often encounters
these spaces when studying submanifolds of manifolds with calibrated
geometries. For the sake of completeness we decided to collect them here in a
self-contained way to be easily accessible for future usage in calibrated
geometry. As an application we deduce existence of certain special 3 and 4
dimensional submanifolds of G_2 manifolds with special properties, which appear
in the first named author's work with S. Salur about G_2 dualities.

Differential Geometry;Algebraic Topology