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Characteristic classes of direct image bundles for covering maps. (English) Zbl 0628.55010

Let \(\lambda\) be an e-dimensional complex representation of the discrete, index n subgroup K of G. If \(f: K\to G\) denotes the inclusion homomorphism, let \(N_ f\) denote the multiplicative transfer in \(H^*(,{\mathbb{Z}})\) defined by L. Evens, and let \(M_ k=(k!)^{- 1}\prod_{p}p^{[k/p-1]}\). The authors’ purpose is to obtain an explicit formula for the Chern classes of the induced representation \(f_ !\lambda\) in terms of those of \(\lambda\) and the manner in which K embeds as a subgroup of G. The general formula is too complicated to be stated in short review; however the following special cases both give the flavour of the theory and are particularly useful in applications.
(1) If K is normal and \(n=p\) is prime, then \(c.(f_ !\lambda)=N_ f(c.\lambda)+\sum^{e-1}_{d=0}[(1-\mu^{p-1})^{e-d}-1] N_ f(c_ d\lambda)\). Here \(\mu\) is a generator of \(H^ 2(G/K,{\mathbb{Z}})\), and to simplify notation we identify \(\mu\) with its inflation.
(2) \(c.(f_ !\lambda)=c.(f_ !1_ K)^ e N_ f(c.\lambda)\), where \(\lambda\) is a Galois representation and there are perhaps restrictions on the pair G, K.
(3) \(M_ k (s_ k(f_ !\lambda)-f_*(s_ k\lambda))=0\), where \(s_ k\) is the kth Newton polynomial in the Chern classes. This is the so- called Riemann-Roch formula for induced representations.
The method of proof is first to take \(e=1\) and apply a certain graph construction, which is explained in Section 25. One then applies a splitting principle (11.1) and a technical result about multiplicative transfer (8.6) in order to obtain the general case. It is interesting to note that the 1-dimensional formula was originally obtained by L. Evens using an embedding of G in the wreath product \(K\wr S_ n\), and that the authors’ splitting principle, the proof of which is very condensed, may be replaced by one proved for flat bundles by O. Staffelbach, a student of B. Eckmann. (This works for finite or profinte coefficients.)
The theory has already found several interesting applications, for example to the study of the Chern subring of \(H^*(G,{\mathbb{Z}})\) for groups of low rank, and it is conceivable that, translated to K-theory, the formula will yield a proof of Atiyah’s famous conjecture on admissible filtrations for R(G). Furthermore, although I have stated results for Chern classes of representations, there is a corresponding theory for Stiefel-Whitney classes of real representations, and for Chow ring valued characteristic classes for suitable algebraic bundles.
Reviewer: C.B.Thomas

MSC:

55R40 Homology of classifying spaces and characteristic classes in algebraic topology
20J05 Homological methods in group theory
14C40 Riemann-Roch theorems
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