Group cohomology with coefficients in a crossed module

We compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

On bounded cohomology of amalgamated products of groups

We investigate the structure of the singular part of the second bounded cohomology group of amalgamated products of groups by constructing an analog of the initial segment of the Mayer-Vietoris exact cohomology sequence for the spaces of pseudocharacters.

Non-Abelian Cohomology with Coefficients in Crossed Bimodules

When the coefficients are crossed bimodules, Guin's non-abelian cohomology [Guin, C. R. Acad. Sci. Paris 301: 337–340, 1985], [Guin, J. Pure Appl. Algebra 50: 109–137, 1988] is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.

Non-Abelian Cohomology of Groups

Following Guin's approach to non-abelian cohomology [Guin, Pure Appl. Algebra 50: 109–137, 1988] and, using the notion of a crossed bimodule, a second pointed set of cohomology is defined with coefficients in a crossed module, and Guin's six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2.

Whitney-sums (fibre-joins) in over space theory and obstruction theory for cohomology with local coefficients

SynopsisThe generalised Whitney sum (fibre-join) and the h-fibre-join can be defined in topM, the category of spaces over M. We note here some general properties of these constructions, and, as a specific example, we consider the relation between them and the extensions to the topM category of the top h-fibre-sequences F∗ΩB→E ∪ CF→B determined by top fibrations F→E→B. As an application we obtain the truncated local coefficient cohomology sequence for a top fibration which is topM principal fibration: this situation applies, for example, to the various stages of the Postnikov decomposition of a non-simply connected space X, and in this case we have M = K1(π1(X)).