Duality and Local Group Cohomology

Pairings, duality, amenability and bounded cohomology

We give a new perspective on the homological characterisations of amenability given by Johnson in the context of bounded cohomology and by Block and Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterisations. We apply these ideas to give a new proof of non- vanishing for the bounded cohomology of a free group.

Euler classes and Bredon cohomology for groups with restricted families of finite subgroups

We explore some of the special features with respect to Bredon cohomology of groups having all its finite subgroups either nilpotent or p-groups or cyclic p-groups. We get some results on dimensions and also a formula for the equivariant Euler class for certain groups. We consider the generalization for Bredon cohomology of the properties of being duality or Poincaré duality and study their behavior under p-power index extensions with coefficients in a field of characteristic p.

Symmetric cohomology of groups and Poincaré duality

Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_*^\varkappa(G,M)$ for which $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_λ(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_λ(G,M)$ is a variation of the group cohomology introduced by Zarelua, which in many cases is isomorphic to the symmetric cohomology of groups defined by Staic. The groups $H_*^\varkappa(G,M)$ come together with transformations from Tate cohomology. We find conditions under which these transformations are isomorphisms.

On the Duality between l^1-Homology and Bounded Cohomology

We modify the definition of l^1-homology and argue why our definition is more adequate than the classical one. While we cannot reconstruct the classical l^1-homology from the new definition for various reasons, we can reconstruct its Hausdorffification so that no information concerning semi-norms is lost. We obtain an axiomatic characterization of our l^1-homology as a universal delta-functor and prove that it is pre-dual to our definition of bounded cohomology. We thus answer a question raised by Loeh in her thesis. Moreover, we prove Gromov's theorem and the Matsumoto-Morita conjecture in ou…

Relative cohomology theory for profinite groups

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 …

Pontryagin对偶与profinite模层

Pontryagin duality and sheaves of profinite modules

本文研究profinite群的Pontryagin对偶理论，填补了在离散模系数上同调研究中缺失的对偶构造这一理论空白。