Abstract. The main result of this paper is the nilpotency fomula φ 4 i = 0, ∀i ≥ 1 for N. Ray classes φ i in the torsion of the symplectic bordism ring MSp *

Abstract. Morava K-theory rings of classifying spaces of the dihedral, semidihedral and generalized quaternion groups are presented in terms of Chern classes.

For finite coverings we elucidate the interaction between transferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of various homotopy orbit spaces. In turn these results provide universal examples for computing the stable Euler classes (i.e. T r * (1)) and transferred Chern classes for p-fold covers. Applications to the classifying spaces of p-groups are given.

Let η be a complex n-plane bundle over the total space of a cyclic covering of prime index p. We show that for k ∈ {1, 2, ..., np} \ {p, 2p, ..., np} the k-th Chern class of the transferred bundle differs from a certain transferred class ω k of η by a polynomial in the Chern classes cp, ..., cnp of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in K(s) * B(Z/p × U (n)).

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