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...

Using local cohomology and algebraic D-Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, A. Adolphson and S. Sperber.

Lecture 1 (Pflaum): Title: Relative cohomology and its pairings Relative cyclic cohomology theory and its pairings turned out to be a powerful tool to explain crucial properties of certain invariants in global analysis such as for example the divisor flow. In this talk, the homological foundations for pairings in relative cyclic cohomology will be explained. Moreover, the relative Chern-charact...

In this study, we compute simplicial cohomology groups with different coefficients of a connected sum of certain minimal simple surfaces by using the universal coefficient theorem for cohomology groups. The method used in this paper is a different way to compute digital cohomology groups of minimal simple surfaces. We also prove some theorems related to degree properties of a map on digital sph...

Using local cohomology and algebraic D-Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, A. Adolphson and S. Sperber.

A relative Rota-Baxter algebra is a generalization of algebra. Relative algebras are closely related to dendriform algebras. In this paper, we introduce bimodules over that fits with the representations We define cohomology coefficients in bimodule and then study abelian extenfsions terms second group. Finally, consider homotopy classify skeletal above-defined cohomology.

Let G be a finite group and F be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative F-projective resolution for Z when F is the family of all subgroups H ≤ G with rkH ≤ rkG − 1. We answer this question negatively by calculating the relative group cohomology FH(G,F2) where G = Z/2 × Z/2 and F is the family o...

Let G be the Klein four-group and let k an arbitrary field of characteristic 2. A classification indecomposable kG-modules is known. We calculate relative cohomology groups Hχi(G,N) for every kG-module N, where χ set proper subgroups in G. This extends work Pamuk Yalcin to with non-trivial coefficients. also show that all cup products strictly positive degree Hχ*(G,k) are trivial.

We compute the BRST cohomology of the holomorphic part of the N=2 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies b...