Characteristic classes on Grassmannians

In this paper, we study the geometry and topology on the oriented Grassmann manifolds. In particular, we use characteristic classes and the Poincare duality to study the homology groups of Grassmann manifolds. We show that for k = 2 or n ≤ 8, the cohomology groups H*(G(k, n), R) are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincare duality: H^q (G(k, n), R) → Hk(n−k) −q (G(k, n), R) can be expressed explicitly.