Suppose G is a finitely presented group that hyperbolic relative to P finite collection of proper subgroups G. Our main theorem states if the second cohomology H2(P,ZP) free abelian for each P?P, then H2(G,ZG) abelian. The problem reduces case when and members are 1-ended. In case, we prove “Cusped Space” this pair has semistable fundamental at ?. This provides starting point in our proof theor...

We introduce a generalization of the celebrated Lovász theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such...

We describe a variant construction of the unstable Adams spectral sequence for space $Y$, associated to any free simplicial resolution $H^*(Y;R)$ $R=\mathbb{F}_p$ or $\mathbb{Q}$. use this differentials and filtration in terms appropriate systems higher cohomology operations.

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...

We propose a method for calculating cohomology operations for finite simplicial complexes. Of course, there exist well–known methods for computing (co)homology groups, for example, the “reduction algorithm” consisting in reducing the matrices corresponding to the differential in each dimension to the Smith normal form, from which one can read off (co)homology groups of the complex [Mun84], or t...