Transferred Chern classes in Morava $K$-theory

Abstract

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)).

Morava K-theory rings for the modular groups in Chern classes

“…be the associated transfer homomorphism induced by the stable transfer map [1], [15], [10]. We will need the following transfer formula from [7].…”

“…It is proved in [19] that K(s) * (BG) is evenly generated and for s = 2 is generated by Euler classes and transferred Euler classes. One consequence of our main theorem below is that this is true for any s. We obtain generators for the ideal R above by using the formula for transferred Euler class from [7] and follow a certain plan, which proved to be sufficient to handle the 2-groups D, SD, QD, Q [9], [6] and modular p-groups [4]. For a discussion of the ring structure of all other groups of order 32 see [19], [20].…”

All extensions of $C_2$ by $C_{2^n} \times C_{2^n}$ are good for the Morava $K$-theory

“…See [2], [4], [16], [18] for detailed discussion and examples. In this particular case, for various examples of groups of order 32, the multiplicative structure of K Ã ðBGÞ is also determined in [2], [4] using transfer methods of [5], [6].…”

Morava K-theory rings of the extensions of C2 by the products of cyclic 2-groups

“…In the works by McClure and Snaith, Hunton and others cohomology groups of homotopy orbit spaces have been constructed, however it is especially interesting to express the ring structure purely in terms of transferred characteristic classes. Thus we are led to consider for finite coverings the interaction of transfers and Chern classes along the lines taken in [6]. Initial results for some examples given in [7], [3], [4] and [5] are the first ones to describe multiplicative structure completely in terms of Euler classes and transferred Euler classes 2.…”

“…We will need the following transfer formula from [6], which does not work for Morava K-theory at s = 1. Therefore we restrict to s > 1.…”