On The betti numbers of oriented Grassmannians and independent semi-invariants of binary forms

Abstract

Abstract We present a complete functional formula expressing the ith ℤ2-Betti number of the oriented Grassmann manifold of oriented 3-dimensional vector subspaces in Euclidean n-space for i from the range determined by the characteristic rank of the canonical oriented 3-dimensional vector bundle over this manifold. The same formula explicitly exhibits the number of linearly independent semi-invariants of degree 3 of a binary form of degree n − 3. Using the approach and data presented in this note, analogous results can be obtained for the oriented Grassmann manifold of oriented 4-dimensional vector subspaces in Euclidean n-space and semi-invariants of degree 4 of a binary form of degree n − 4.