Characteristic Classes on Grassmann Manifolds. arXiv:0809.0808
Preprint, arXiv:0809.0808 [math.FA] (2008).
Summary: In this paper, we use characteristic classes of the canonical vector bundles and the Poincaré 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 Poincaré dualality: \(H^q(G(k,n),{\bf R}) \to H_{k(n-k)-q}(G(k,n),{\bf R})\) can be given explicitly.
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 55R10 | Fiber bundles in algebraic topology |
| 55U30 | Duality in applied homological algebra and category theory (aspects of algebraic topology) |
| 57T15 | Homology and cohomology of homogeneous spaces of Lie groups |
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