Characteristic Classes on Grassmann Manifolds

Details Characteristic Classes on Grassmann Manifolds Characteristic Classes on Grassmann Manifolds

2008-09-04

arxiv.org/abs/0809.0808v1

Mathematics

Functional Analysis

38 pages

Scientific paper

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.

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