The Homology Of Grassmann Manifolds

In this paper,we use Characteristic class and diffrential geometry to study the homology of Grassmann manifolds G(2,N)and G(4,8).The other low dimensional Grassmann manifolds can be treated similarly.We show that the cohomology of G(2,2n+2)can be generated by the Euler classes of the canonical vector bundles E(2,2n+2)and F(2,2n+2).The generators of the homology of G(2,2n+2)are also determined.While the cohomology of G(2,2n+3)can be generated by the Euler class of the vector bundle E(2,2n+3).Then we show that the cohomology of G(4,8)can be generated by the Euler classes of the vector bundles E(4,8)and F(4,8)and the first Pontryagin class of E(4,8).