Low-dimensional bounded cohomology and extensions of groups

Bounded cohomology of groups was first studied by Gromov in 1982 in his seminal paper M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982), no. 56, 5–99. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic “non-positively curved” groups. On the other hand, there is a well-known interpretation of ordinary group cohomology in dimension $2$ and $3$ in terms of group extensions. The aim of this paper is to make this interpretation available for bounded group cohomology. This will involve quasihomomorphisms as defined and studied by K. Fujiwara and M. Kapovich, On quasihomomorphisms with noncommutative targets, Geom. Funct. Anal. 26 (2016), no. 2, 478–519.

Cohomology of Presheaves of Monoids

The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of H -extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth.

Cohomology of Groups

This chapter introduces the basic ingredients of the cohomology of groups and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: integral homology of finite groups such as the Mathieu groups, homology of crystallographic groups, homology of nilpotent groups, homology of Coxeter groups, transfer homomorphism, homological perturbation theory, mod-p comology rings of small finite p-groups, Lyndon-Hocshild-Serre spectral sequence, Bokstein operation, Steenrod squares, Stiefel-Whitney classes, Lie algebras, the modular isomorphism problem, and Bredon homology.

FREE FINITE GROUP ACTIONS ON RATIONAL HOMOLOGY 3-SPHERES

We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.

On central automorphisms of crossed modules

A crossed module $(T,G,\partial)$ consist of a group homomorphism $\partial:T\rightarrow G$ together with an action $(g,t)\rightarrow{}^{\,g}t$ of $G$ on $T$ satisfying $\partial(^{\,g}t)=g\partial(t)g^{-1}$ and $\,^{\partial(s)}t=sts^{-1}$, for all $g\in G$ and $s,t\in T$. The term crossed module was introduced by J. H. C. Whitehead in his work on combinatorial homotopy theory. Crossed modules and its applications play very important roles in category theory, homotopy theory, homology and cohomology of groups, algebra, K-theory etc. In this paper, we define Adeny-Yen crossed module map and central automorphisms of crossed modules. If $C^*$ is the set of all central automorphisms of crossed module $(T,G,\partial)$ fixing $Z(T,G,\partial)$ element-wise, then we give a necessary and sufficient condition such that $C^*=I_{nn}(T,G,\partial).$ In this case, we prove $Aut_C(T,G,\partial)\cong Hom((T,G,\partial), Z(T,G,\partial))$. Moreover, when $Aut_C(T,G,\partial)\cong Z(I_{nn}(T,G,\partial)))$, we obtain some results in this respect.