Algebraic topology of G2 manifolds

In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G2G2. 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 G2G2 manifolds with special properties, which appear in the first named author’s work with S. Salur about G2G2dualities.