We approach Mackenzie’s LA-groupoids from a supergeometric point of view by introducing Q-groupoids. A Q-groupoid is a groupoid object in the category of Q-manifolds, and there is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. Using this approach, we associate to every LA-groupoid a double complex whose cohomology simultaneously generalizes Lie groupoid coh...

By a &-graph we will mean a collection of k-element subsets of some fixed set V. A A:-graph can be regarded as a (k l)-chain on 2y, the simplicial complex of all subsets of V, over the coefficient group Z/2 , the additive group of integers modulo 2. The induced group structure on the {k 1)chains leads to natural definitions of the coboundary Ô of a chain, the cochain complex of C = {Ck, ô) and ...

A simplicial complex L on n vertices determines a subcomplex TL of the ntorus, with fundamental group the right-angled Artin groupGL. Given an epimorphism χ : GL → Z, let T L be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of T L with coefficients in a field k, viewed as module...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-going generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a reach combinatorial structure, e.g., it is a manifold with corners provided that the action is locally st...

This short research announcement briefly describes the simplicial method underlying the GAP package SCO for computing the socalled orbifold cohomology of topological resp. smooth orbifolds. SCO can be used to compute the lower dimensional group cohomology of some infinite groups. Instead of giving a complete formal definition of an orbifold, we start with a simple construction, general enough t...

the aim of this paper is to study co-prolongations of central extensions. we construct the obstruction theory forco-prolongations and classify the equivalence classes of these by kernels of homomorphisms between 2-dimensional cohomology groups of groups.

We study the relations between the fundamental group and the homological operations on integer homology. For the “rational” fundamental group (Malcev completion) see [2, 3, 4, 11, 16, 17]. This work is an attempt to understand the invariant of the fundamental group of the complement of a complex hyperplane arrangement that was used in [13]. Note that this invariant necessarily vanishes over Q (...

to the morphism f ·p−1, is a bijection provided that X and Y are locally fibrant in the sense that all of their stalks are Kan complexes. Here, π(Z, Y ) denotes simplicial homotopy classes of maps, and the colimit is indexed over homotopy classes represented by hypercovers p : Z → X. A hypercover is a map which is a trivial fibration of simplicial sets in all stalks, and is therefore invertible...