An Integral Riemann-roch Formula for Flat Line Bundles

Let p be a unitary representation of the subgroup K of the finite group 0, with inclusion map / . Then, if /, and /# denote the transfer maps for representation theory and cohomology respectively, Knopfmacher [8] has proved that, for all k ^ 1, there exist positive integers Mk such that Mk(k\chk(fiP))=U(Mkk\ohk(P)). Here ch& denotes the fcth component of the Chern character, so that k! chfc is an integral class, the kth Newton polynomial in the Chern classes. In the same paper he mentions the conjecture that the best value for Mk is {k\)~ JJ Ztfc/tf-)!, where the product is taken over all primes I; this is known to be true for k = 1,2 [8, Theorem 6]. The conjectured value is suggested by the rational Riemann-Roch theorem; one interesting point being that, in contrast to Knopfmacher's much larger estimate, (Mk, p) = 1 for p prime and k < p — 1. Hence for representations of ^-groups, the equation above would hold in low dimensions without the factor Mki and could be used to investigate the integral cohomology of the group. In this paper we shall show that Knopfmacher's estimate may be replaced by