Characteristic Classes on Grassmann Manifolds

TL;DR

This paper investigates the structure of real homology and cohomology groups of oriented Grassmann manifolds using characteristic classes, providing explicit descriptions for certain cases.

Contribution

It demonstrates how characteristic classes and Poincaré duality can explicitly describe the cohomology of Grassmann manifolds for specific dimensions.

Findings

Cohomology groups are generated by the first Pontrjagin class and Euler classes for certain cases.

Explicit Poincaré duality maps are provided for these cases.

The structure of homology and cohomology groups is clarified using characteristic classes.

Abstract

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.