Relative cohomology theory for profinite 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 Poincaré 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 Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components.• the restriction of T to C π pGq commutes with inverse limits in the sense thatOne defines notions of connected sequences of continuous functors, and continuous cohomological and homological functors in the natural way. A continuous cohomological functor T ‚ is co-effaceable if T i pJq " 0 for all i ą 0 and all injective modules J P D π pGq. Similarly for effaceability.

Virtual pro-p properties of 3-manifold groups

“…for any finite p-primary G-module M . Here the last isomorphism is induced by Pontrjagin duality (see for example [RZ00b, Proposition 6.3.6]), together with the identity HompM, Z p pχq˚q " pZ p pχq b M qT he theory of the (co)homology of a profinite group relative to a collection of subgroups was defined and studied in [Wil17a]. We will state a restricted version of the definition here and leave it to the curious reader to read further.…”

“…We will state a restricted version of the definition here and leave it to the curious reader to read further. Definition 1.3 (Definition 2.1 of [Wil17a]). Let G be a pro-p group and S " tS 0 , .…”

“…Since cup products are natural with respect to maps of pairs [Wil17a, Proposition 3.1], the cup product on H 1 pG, Sq b H 1 pGq is the restriction of the cup product on H 1 pG, Sq b H 1 pGq. Furthermore Equation 3.6 in Section 3.2 of [Wil17a] gives a commutative pentagon…”

“…S b´1 u. The map of pairs pG, S 1 q Ñ pG, Sq and Propositions 2.4 and 2.6 of [Wil17a] give a commutative diagram (with coefficients in I m pχq on the top row and I m pρq on the bottom row):…”

“…In this paper we will carry out this program in the realm of pro-p groups. The notions of 'relative (co)homology' and 'Poincaré duality pair' for profinite groups were defined and developed in the author's paper [Wil17a]. In the first part of this paper we will classify those pro-p group pairs which are Poincaré duality pairs of dimension 2.…”