THE GYSIN HOMOMORPHISM FOR GENERALIZED FLAG BUNDLES

Positivity in the Cohomology of Flag Bundles

Graham deduces this from a more general result about varieties with finitely many unipotent orbits, which is proved using induction and a calculation in the rank-one case. The goal of this note is to give a short, geometric proof of Graham’s positivity theorem, based on a transversality argument. Here I only discuss type A, but other types work as well. (For a type-uniform version, a change of ...

T-duality for Principal Torus Bundles and Dimensionally Reduced Gysin Sequences

We reexamine the results on the global properties of T-duality for principal circle bundles in the context of a dimensionally reduced Gysin sequence. We will then construct a Gysin sequence for principal torus bundles and examine the consequences. In particular, we will argue that the T-dual of a principal torus bundle with nontrivial H-flux is, in general, a continuous field of noncommutative,...

Conley index and stable sets for flows on flag bundles

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected noncompact semi-simple Lie group with finite center. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an addit...

The Class of the Diagonal in Flag Bundles

Positivity in the cohomology of flag bundles (after Graham)

Graham deduces this from a more general result about varieties with finitely many unipotent orbits, which is proved using induction and a calculation in the rank-one case. (In fact, A. Knutson points out that Graham’s proof yields a stronger result: as a polynomial in all the negative roots, cuv is a nonnegative combination of squarefree monomials.) The goal of this note is to give a short, geo...