Homogeneous spaces, curvature and cohomology

Morse functions and cohomology of homogeneous spaces

Morse functions are useful tool in revealing the geometric formation of its domain manifolds M. They define the handle decompositions of M from which the additive homologies H * (M) may be constructed. In these lectures two further questions were emphasized. (1) How to find a Morse function on a given manifold? (2) From Morse functions can one derive the multiplicative coho-mology rather than t...

On the Cohomology of Generalized Homogeneous Spaces

We observe that work of Gugenheim and May on the cohomology of classical homogeneous spaces G/H of Lie groups applies verbatim to the calculation of the cohomology of generalized homogeneous spaces G/H, where G is a finite loop space or a p-compact group and H is a “subgroup” in the homotopical sense. We are interested in the cohomology H∗(G/H ; R) of a generalized homogeneous space G/H with co...

Equivariant Quantum Cohomology of Homogeneous Spaces

We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety X = G/P . As in the case when X is a Grassmannian ([Mi1]), this formula implies an algorithm to compute the structure constants of the equivariant quantum cohomology algebra of X.

Quantum cohomology of minuscule homogeneous spaces

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov-Witten invariants can always be interpreted as classical intersection numbers on auxiliary homogeneous varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In pa...

On Riemannian Curvature of Homogeneous Spaces