Characteristic Classes on Grassmann Manifolds

In this paper, we use characteristic classes of the canonical vector bundles and the Poincar' {e} dualality to study the structure of the real homology and cohomology groups of oriented Grassmann manifold $G(k,n)$. Show that for $k=2$ or $n\le 8$, the cohomology groups $H^{\ast }(G(k,n),\mathbf{R})$ are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincar' {e} dualality: $H^q(G(k,n),{\bf R}) \to H_{k(n-k)-q}(G(k,n),{\bf R})$ can be given explicitly.