The integral Pontrjagin homology of the based loop space on a flag manifold

The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the based loop space on a flag manifold is torsion free. This results in a description of the integral homology. In addition, the integral Pontrjagin structure is determined.Comment: revised version, 17 pages, to appear in Osaka Journal of Mathematic

Complex cobordism classes of homogeneous spaces

We consider compact homogeneous spaces G/H of positive Euler characteristic endowed with an invariant almost complex structure J and the canonical action \theta of the maximal torus T ^{k} on G/H. We obtain explicit formula for the cobordism class of such manifold through the weights of the action \theta at the identity fixed point eH by an action of the quotient group W_G/W_H of the Weyl groups for G and H. In this way we show that the cobordism class for such manifolds can be computed explicitly without information on their cohomology. We also show that formula for cobordism class provides an explicit way for computing the classical Chern numbers for (G/H, J). As a consequence we obtain that the Chern numbers for (G/H, J) can be computed without information on cohomology for G/H. As an application we provide an explicit formula for cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular interesting examples.Comment: improvements in subsections 7.1 and 7.2; some small comments are added or revised and some typos correcte