A Cohomological Transfer Map for Profinite Groups

✍ Scribed by Oliver Schirokauer

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 251 KB
• Volume: 195
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7076

✦ Synopsis

Let G be a group, A a G-module, and H a subgroup of G. The standard Ž . Ž . 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 A ª E ª G ª 1 splits if, for every prime number p, there exists a homomorphism ␥ from a p p-Sylow subgroup S of G to E such that ␤ (␥ is the identify on S . Of particular p p p importance in our proof is the fact that the composition of the restriction map Ž .