Ordinary RO(G)RO\left( G \right)RO(G)-graded cohomology (original) (raw)

An Introduction to the Cohomology of Groups

H n (G, M) where n = 0, 1, 2, 3,. . ., called the nth homology and cohomology of G with coefficients in M. To understand this we need to know what a representation of G is. It is the same thing as ZG-module, but for this we need to know what the group ring ZG is, so some preparation is required. The homology and cohomology groups may be defined topologically and also algebraically. We will not do much with the topological definition, but to say something about it consider the following result: THEOREM (Hurewicz 1936). Let X be a path-connected space with π n X = 0 for all n ≥ 2 (such X is called 'aspherical'). Then X is determined up to homotopy by π 1 (x). If G = π 1 (X) for some aspherical space X we call X an Eilenberg-MacLane space K(G, 1), or (if the group is discrete) the classifying space BG. (It classifies principal G-bundles, whatever they are.) If an aspherical space X is locally path connected the universal cover˜X is contractible and X = ˜ X/G. Also H n (X) and H n (X) depend only on π 1 (X). If G = π 1 (X) we may thus define H n (G, Z) = H n (X) and H n (G, Z) = H n (X) and because X is determined up to homotopy equivalence the definition does not depend on X. As an example we could take X to be d loops joined together at a point. Then π 1 (X) = F d is free on d generators and π n (X) = 0 for n ≥ 2. Thus according to the above definition H n (F d , Z) = Z if n = 0 Z d if n = 1 0 otherwise. Also, the universal cover of X is the tree on which F d acts freely, and it is contractible. The theorem of Hurewicz tells us what the group cohomology is if there happens to be an aspherical space with the right fundamental group, but it does not say that there always is such a space.

Pointed admissible G-covers and G-equivariant cohomological field theories

For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduces to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the 'quotient' by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the 'quotient' by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov-Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen-Ruan orbifold Gromov-Witten invariants of [X/G] as well as the ring H • (X, G) of Fantechi and Göttsche.

Equivariant homology and cohomology of groups

Topology and its Applications, 2005

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.

The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups

Pure and Applied Mathematics Quarterly, 2005

After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-G-set-version, the inverselimit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the zero-th equivariant stable cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant stable cohomotopy groups of finite proper equivariant CW-complexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We formulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological K-theory of the classifying space for proper actions.

Application of group cohomology to space constructions

Transactions of the American Mathematical Society, 1987

From a short exact sequence of crossed modules 1 → K → H → H ¯ → 1 1 \to K \to H \to \bar H \to 1 and a 2 2 -cocycle ( ϕ , h ) ∈ Z 2 ( G ; H ) (\phi ,\,h) \in {Z^2}(G;\,H) , a 4 4 -term cohomology exact sequence H a b 1 ( G ; Z ) → H ( ϕ ¯ , h ¯ ) 1 ( G ; H ¯ , Z ¯ ) → δ ⋃ { H ψ 2 ( G ; K ) : ψ o u t = ϕ o u t } → H a b 2 ( G ; Z ) H_{ab}^1(G;Z) \to H_{(\bar {\phi }, \bar {h})}^1 (G; \bar {H}, \bar {Z}) \stackrel {\delta }{\to } \bigcup \{ H_\psi ^2 (G;K) : \psi _{\mathrm {out}} = \phi _{\mathrm {out}} \} \to H_{ab}^2(G;\,Z) is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and Z Z is the center of the crossed module H H . The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique.

Cohomology for Generalized Bredon Coefficient Systems and Higher K-Theory

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2013

Let C be a generalized based category (see definition 1.2). In this paper, we construct a cohomology theory in the category B R .C/ of contravariant functors: C ! R-Mod where R is a commutative ring with identity, which generalizes Bredon cohomology involving finite, profinite or discrete groups.

Higher non-abelian cohomology of groups

Glasgow Mathematical Journal, 2002

The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substantially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.

On the equivariant 2-type of a G-space

Journal of Pure and Applied Algebra, 1998

A classical theorem of Mac Lane and Whitehead states that the homotopy type of a topological space with trivial homotopy at dimensions 3 and greater can be reconstructed from its 711 and 712, and a cohomology class ks ~H~(rri, 7~). More recently, Moerdijk and Svensson suggested the possibility of using Bredon cohomology to extend this result to the equivariant case, that is, for spaces X equipped with an action by a fixed group G. In this paper we carry out this suggestion and prove an analogue of the classical result in the equivariant case.