The paper deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree $2$ coefficients $\mathbb{Q}/\mathbb{Z}(1)$, that is taking values in Brauer group. Our main tool \'etale cohomology sheaves on simplicial schemes. We get a description these for \emph{every} groups, particular non reductive imperfect

Via the BGG-correspondence a simplicial complex ∆ on [n] is transformed into a complex of coherent sheaves L̃(∆) on the projective space P. In general we compute the support of each of its cohomology sheaves. When the Alexander dual ∆ is Cohen-Macaulay there is only one such non-zero cohomology sheaf. By considering when this sheaf can be an a’th syzygy sheaf in a locally free resolution, we get...

2 Background Material 2 2.1 Homotopy Theory of Simplicial Vector Spaces . . . . . . . . . 2 2.1.1 The Dold-Kan correspondence . . . . . . . . . . . . . . 2 2.1.2 Homotopy theory . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Tensor Products and the Eilenberg-Zilber Theorem . . 5 2.2 Representation Theory of Finite Groups . . . . . . . . . . . . 6 2.2.1 Coinvariants and Invariants of kG-modul...

Suppose that G is a finite group. We look at the problem of expressing the classifying space BG, up to mod p cohomology, as a homotopy colimit of classifying spaces of smaller groups. A number of interesting tools come into play, such as simplicial sets and spaces, nerves of categories, equivariant homotopy theory, and the transfer.

in this paper, we verify the solvability of free product of finite cyclic groups with topological methods. we use cayley graphs and everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. in particular, using these methods, we prove that the commutator subgroup of $mathbb{z}_{m}*mathbb{z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

We prove that the de Rham $ L^{\phi} -cohomology of a Riemannian manifold M admitting convenient triangulation X is isomorphic to simplicial \ell^{\phi} under some assumptions on Young function \phi . This result implies quasi-isometry invariance first cohomology.

England Abstract A stable homology theory is defined for completely distributive CSL algebras in terms of the point-neighbourhood homology of the partially ordered set of meet-irreducible elements of the invariant projection lattice. This specialises to the simplicial homology of the underlying simplicial complex in the case of a digraph algebra. These groups are computable and useful. In parti...

This thesis deals with the homology and cohomology groups of simplicial complexes, and especially with the two duality theorems which willdemonstrate links between the two notions. For this we will need some results, and we will therefore prove the long exact sequence for the reduced relative homology group; and to link the homology and cohomolgy to our geometric notion of a simplicial complex ...

For a simplicial complex on {1, 2, . . . , n} we define enriched homology and cohomology modules. They are graded modules over k[x1, . . . , xn] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein∗ complexes , and also orientable homology manifolds in terms of the enriched modules. We ...

For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation of the ‘coarse’ underlying space together with the local isotropy groups of M. We prove that, for any local system on M, this complex S(M) has the same cohomology as M. The use of S(M) in explicit calculations is illustrated in the example of the ‘teardrop’ orbifold.