Algebraic topology of special Lagrangian manifolds

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