  In this paper, we prove various results on the topology of the Grassmannian
of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We
give self-contained proofs. These spaces come up when studying submanifolds of
manifolds with calibrated geometries. We collect these results here for the
sake of completeness. As applications of our algebraic topological study we
present some results on special Lagrangian-free embeddings of surfaces and
3-manifolds into the Euclidean 4 and 6-space.
