The Poincaré duality and the Gysin homomorphism for flag manifolds (Q1370797)

Let \(G\) be a connected complex reductive Lie group, \(B\) a fixed Borel subgroup in \(G\) and \(G/B\) the corresponding complete flag manifold of \(G\). Considering the Bruhat decomposition and parabolic subgroups in \(G\), there are obtained the partial flag manifolds of \(G\). The author describes the Poincaré duality of \(G/P\) in terms of the Weyl group of \(G\). Then he uses this description in order to obtain the Gysin homomorphisms between the partial flag manifolds. In the case where \(G=GL_n(\mathbb{C})\), the author obtains the Schubert cell decomposition corresponding to the Bruhat decomposition, the description of the Gysin homomorphism and presents some relations with the Chern classes.