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. This paper is going to have continuation in which will be considered the stable complex structures equivariant under given torus action on homogeneous spaces of positive Euler characteristic