A cohomological transfer map for profinite groups

[摘要] Let G be a group, A a G-module, and H a subgroup of G. The standard cohomological transfer map from H*(H, A) to H*(G, A) is defined in the case that H is of finite index in G and is given explicitly in each dimension by a formula involving a sum over a set of representives for H \ G. In this paper, we obtain a new transfer in the case that G is a profinite group, A is an abelian protorsion group on which G acts continuously, H is a closed subgroup of G, and the cohomology is continuous. We do this by developing a theory of integration for continuous functions from a compact space to a projective limit of discrete modules and replacing the finite sum in the formula for the standard transfer with an integral. As an application of the new transfer, we prove a profinite version of the well-known result that for A abelian and G finite, an extension 0 --> A --> E B over right arrow G --> 1 splits if, for every prime number p, there exists a homomorphism yp from a p-Sylow subgroup S-p of G to E such that beta circle gamma(p) y, is the identify on S-p. Of particular importance in our proof is the fact that the composition of the restriction map from Hr(G, A) to H*(H, A) and the transfer which we introduce is equal to multiplication by [G: H], where the index in this case is a suitably defined element in (Z) over cap. (C) 1997 Academic Press.

[发布日期] 1997-09-01