This paper offers an algorithmic solution to the problem of obtaining “economical” formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain an explicit combinatorial description of all Steenrod kth powers exclusive...

We obtain combinatorial model categories of parametrised spectra, together with systems base change Quillen adjunctions associated to maps parameter spaces. work simplicial objects and use Hovey's sequential symmetric stabilisation machines. By means a Grothendieck construction for categories, we produce controlling the totality stable homotopy theory. The global category spectra is equipped mo...

let $a$ be an abelian topological group and $b$ a trivial topological $a$-module. in this paper we define the second bilinear cohomology with a trivial coefficient. we show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

Given an infinite field k and a simplicial complex Δ, common theme in studying the f- h-vectors of Δ has been consideration Hilbert series Stanley–Reisner ring k[Δ] modulo generic linear system parameters Θ. Historically, these computations have restricted to special classes complexes (most typically triangulations spheres or manifolds). We provide compact topological expression hd−1a(Δ), dimen...

let $s$ be an inverse semigroup and let $e$ be its subsemigroup of idempotents. in this paper we define the $n$-th module cohomology group of banach algebras and show that the first module cohomology group $hh^1_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is zero, for every odd $ninmathbb{n}$. next, for a clifford semigroup $s$ we show that $hh^2_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is a banach space,...

Let Γ be a finitely generated group and Γ′ = {Γi | i ∈ I} be a family of its subgroups. We utilize the notion of tuple (Γ,Γ′, X,V ′) that makes the statements and arguments for the pair (Γ,Γ′) parallel to the non-relative case, and define the snake metric dς on the set of edges of a simplicial complex. The language of tuples and snake metrics seems to be convenient for dealing with relative hyp...

We introduce a notion of crossed simplicial group, which generalizes Connes' notion of the cyclic category. We show that this concept has several equivalent descriptions and give a complete classification of these structures. We also show how many of Connes' results can be generalized and simplified in this framework. A simplicial set (resp. group) is a family of sets (resp. groups) {Gn}n>0 tog...