Relative cohomology theory for profinite groups

TL;DR

This paper develops a relative cohomology theory for profinite groups, establishing duality, cup products, and decomposition results, with applications to 3-manifold groups and their profinite completions.

Contribution

It introduces a new relative cohomology framework for profinite groups, including duality and decomposition tools, connecting discrete and profinite duality theories.

Findings

Defined profinite relative cohomology and cup products

Established profinite Poincaré duality pairs

Applied to decompositions of 3-manifold groups

Abstract

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincar\'e duality pairs. We use the theory of groups acting on profinite trees to give Mayer-Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincar\'e duality group relative to the profinite completions of the fundamental groups of its boundary components.