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 $n\leq 8$, the cohomology groups $H^*(G(k,n),{ \bf R})$ 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.
Categories: math.FA