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## ring of cohomology De Rham grassmannian

For grassmannian manifold G_2(R^{2^{r}+2}), how can i show H^{2}(G_2(R^{2^{r}+2}))=Z_2+Z_2 ? in ring of cohomology Whit coefficients Z_2!

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You seem to get cohomology with integer coefficients, not deRham cohomology. This also looks like homework, since the cohomology groups of Grassmannians are very well understood. – Ben McKay Nov 13 2011 at 17:29
I agree that the question seems very much like homework, but I'll give a hint nevertheless: get a hold Milnor and Stasheff and read about Stiefel-Whitney classes. – Donu Arapura Nov 13 2011 at 18:21