Abstract :
LetGbe a group,AaG-module, andHa subgroup ofG. The standard cohomological transfer map fromH*(H, A) toH*(G, A) is defined in the case thatHis of finite index inGand is given explicitly in each dimension by a formula involving a sum over a set of representives forH\G. In this paper, we obtain a new transfer in the case thatGis a profinite group,Ais an abelian protorsion group on whichGacts continuously,His a closed subgroup ofG, 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 forAabelian andGfinite, an extensionimagesplits if, for every prime numberp, there exists a homomorphism γpfrom ap-Sylow subgroupSpofGtoEsuch that β ring operator γpis the identify onSp. Of particular importance in our proof is the fact that the composition of the restriction map fromH*(G, A) toH*(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 image.
