This article is expanded from a talk given by the author in (the preliminary version of) the Student Algebraic Topology seminar in University of Michigan, Ann Arbor. The rst part will be a review of ordinary sheaf cohomology which is done in such a way that generalization to generalized sheaf cohomology theory becomes automatic. The second part is a quick introduction of the approach to general...

We define cohomology groups Ĥn(G;M), n ∈ Z, for an arbitrary group G and G-module M , using the concept of satellites. These cohomology groups generalize the Farrell-Tate groups for groups of finite virtual cohomological dimension and form a connected sequence of functors, characterized by a natural universal property. The classical Tate cohomology groups of finite groups have been generalized ...

Abstract The van Est map is a from Lie groupoid cohomology (with respect to sheaf taking values in representation) algebroid cohomology. We generalize the allow for more general sheaves, namely sheaves of sections (smooth or holomorphic) $G$-module, where $G$-modules are structures, which differentiate representations. Many geometric structures involving groupoids and stacks classified by not r...

Let G be a simple, simply connected algebraic group defined and split over the prime field F p > 3. Let F be a finite dimensional rational G-module. As shown in [1] , for d suitably large, the Eilenberg-Mac Lane cohomology groups H*(G(Fpd), V) achieve a stable value H*en(G, V)9 the so-called generic cohomology of V. This generic cohomology can in turn be determined from the rational cohomology ...

It is well known that the homotopy category of connected CW-complexes X whose homotopy groups n;(X) are trivial for i> 1 is equivalent to the category of groups. One of the objects of this paper is to prove a similar equivalence for the connected CW-complexes X whose homotopy groups are trivial for i > n + 1 (where n is a fixed non-negative integer). For n = 1 the notion of crossed module inven...

Let G be a compact Lie group. Set Λ• = H∗(G) and S • = H(BG). The coefficients are in R or C. Suppose G acts on a reasonable space X. In the paper [GKM] Goresky, Kottwitz and MacPherson established a duality between the ordinary cohomology which is a module over Λ• and equivariant cohomology which is a module over S • . This duality is on the level of chains, not on the level of cohomology. The...

We characterize Hopf spaces with finitely generated cohomology as algebra over the Steenrod algebra. We “deconstruct” the original space into an Hspace Y with finite mod p cohomology and a finite number of p-torsion EilenbergMac Lane spaces. One reconstructs X from Y by taking extensions by principal H-fibrations. We give a precise description of homotopy commutative H-spaces in this setting an...

‎‎we exhibit an explicit construction for the second cohomology group‎ ‎$h^2(l‎, ‎a)$ for a lie ring $l$ and a trivial $l$-module $a$‎. ‎we show how the elements of $h^2(l‎, ‎a)$ correspond one-to-one to the‎ ‎equivalence classes of central extensions of $l$ by $a$‎, ‎where $a$‎ ‎now is considered as an abelian lie ring‎. ‎for a finite lie‎ ‎ring $l$ we also show that $h^2(l‎, ‎c^*) cong m(l)$‎...