A certain Grothendieck topology assigned to a metric space gives rise sheaf cohomology theory which sees the coarse structure of space. Already constant coefficients produce interesting groups. In degree 0, they see number ends this paper, resolution via cochains is developed. It serves be valuable tool for computing cohomology. addition, homotopy invariance with established. This property can ...

We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly mentioned.

Let G be a topological group. By a G-space X we mean a topological space X together with a left action of G on X. Under the assumption that G is a compact Lie group, a discrete group or an abelian locally compact group, we constructed in [5] (see also [6]) an equivariant homology and cohomology theory, defined on the category of all G-pairs and G-maps, which both satisfy all seven equivariant E...

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups, see Theorem 4.4. Moreover, we show that the stable cohomology of the n-fold wreath product Gn = Z/p o · · · oZ/p of cyclic groups Z/p is detected by elementary abelian p-subgroups and we describe the resulting coho...

Abstract We investigate the topological nilpotence degree, in sense of Henn–Lannes–Schwartz, a connected Noetherian unstable algebra R . When is mod p cohomology ring compact Lie group, Kuhn showed how this invariant controlled by centralizers elementary abelian -subgroups. By replacing -subgroups with components Lannes’ T -functor, and utilizing techniques algebras over Steenrod algebra, we ar...

We define a group structure on the set of compact “minimal” paths in R2. We classify all finitely generated subgroups of this group G: they are free products of free abelian groups and surface groups. Moreover, each such group occurs in G. The subgroups of G isomorphic to surface groups arise from certain topological 1-forms on the corresponding surfaces. We construct examples of such 1-forms f...