Complex cobordism classes of homogeneous spaces

TL;DR

This paper derives explicit formulas for the cobordism classes and Chern numbers of certain homogeneous spaces with invariant almost complex structures, enabling computations without cohomology data.

Contribution

It provides a novel explicit formula for cobordism classes of homogeneous spaces using Weyl group weights, bypassing cohomology calculations.

Findings

Explicit cobordism class formulas for G/H with invariant structures.

Method to compute Chern numbers without cohomology.

Application to flag and Grassmann manifolds.

Abstract

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.