Топтық когомология және гомология: алгебралық топологиялық құралдар

Tools for studying groups based on techniques from algebraic topology
homology and cohomology of a group

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G module M to elucidate the properties of the group. By treating the G module as a kind of topological space with elements of representing n simplices, topological properties of the space may be computed, such as the set of cohomology groups The cohomology groups in turn provide insight into the structure of the group G and G module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G module, G acts on a nonabelian G group; in effect, a generalization of a module to non Abelian coefficients. These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group G is the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of can be thought of as the singular cohomology of the circle S1, and similarly for and

A great deal is known about the cohomology of groups, including interpretations of low dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.

The collection of all G modules is a category (the morphisms are group homomorphisms f with the property for all g in G and x in M). Sending each module M to the group of invariants yields a functor from the category of G modules to the category Ab of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors. Their values are abelian groups and they are denoted by , "the n th cohomology group of G with coefficients in M". Furthermore, the group may be identified with .

The first cohomology group is the quotient of the so called crossed homomorphisms, i. e. maps (of sets) f : G → M satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so called principal crossed homomorphisms, i. e. maps f : G → M given by f(g) = gm−m for some fixed m ∈ M. This follows from the definition of cochains above. If the action of G on M is trivial, then the above boils down to H1(G,M) = Hom(G, M), the group of group homomorphisms G → M, since the crossed homomorphisms are then just ordinary homomorphisms and the coboundaries (i. e. the principal crossed homomorphisms) must have image identically zero: hence there is only the zero coboundary. On the other hand, consider the case of where denotes the non trivial structure on the additive group of integers, which sends a to a for every ; and where we regard as the group By considering all possible cases for the images of , it may be seen that crossed homomorphisms constitute all maps satisfying and for some arbitrary choice of integer t. Principal crossed homomorphisms must additionally satisfy for some integer m: hence every crossed homomorphism sending 1 to an even integer is principal, and therefore:

with the group operation being pointwise addition: , noting that is the identity element.

For the finite cyclic group of order with generator , the element in the associated group ring is a divisor of zero because its product with , given bygivesThis property can be used to construct the resolution of the trivial module via the complexgiving the group cohomology computation for any module Note the augmentation map gives the trivial module its structure byThis resolution gives a computation of the group cohomology since there is the isomorphism of cohomology groupsshowing that applying the functor to the complex above (with removed since this resolution is a quasi isomorphism), gives the computationforFor example, if , the trivial module, then , , and , hence

Cocycles for the group cohomology of a cyclic group can be given explicitly using the Bar resolution. We get a complete set of generators of cocycles for odd as the mapsgiven byfor odd, , a primitive th root of unity, a field containing th roots of unity, andfor a rational number denoting the largest integer not greater than Also, we are using the notationwhere is a generator for Note that for non zero even indices the cohomology groups are trivial.

Given a set the associated free group has an explicit resolution of the trivial module which can be easily computed. Notice the augmentation maphas kernel given by the free submodule generated by the set , so. Because this object is free, this gives a resolutionhence the group cohomology of with coefficients in can be computed by applying the functor to the complex , givingthis is because the dual mapsends any module morphismto the induced morphism on by composing the inclusion. The only maps which are sent to are multiples of the augmentation map, giving the first cohomology group. The second can be found by noticing the only other mapscan be generated by the basis of maps sending for a fixed , and sending for any .

The group cohomology of free groups generated by letters can be readily computed by comparing group cohomology with its interpretation in topology. Recall that for every group there is a topological space , called the classifying space of the group, which has the propertyIn addition, it has the property that its topological cohomology is isomorphic to group cohomologygiving a way to compute some group cohomology groups. Note could be replaced by any local system which is determined by a mapfor some abelian group In the case of for letters, this is represented by a wedge sum of circles which can be showed using the Van Kampen theorem, giving the group cohomology

For an integral lattice of rank (hence isomorphic to ), its group cohomology can be computed with relative ease. First, because , and has , which as abelian groups are isomorphic to , the group cohomology has the isomorphismwith the integral cohomology of a torus of rank .

In the following, let M be a G module.

The Hochschild–Serre spectral sequence relates the cohomology of a normal subgroup N of G and the quotient G/N to the cohomology of the group G (for (pro )finite groups G). From it, one gets the inflation restriction exact sequence.

There is a way to compute the semi direct product of groups using the topology of fibrations and properties of Eilenberg Maclane spaces. Recall that for a semi direct product of groups there is an associated short exact sequence of groupsUsing the associated Eilenberg Maclane spaces there is a Serre fibrationwhich can be put through a Serre spectral sequence. This gives an pagewhich gives information about the group cohomology of from the group cohomology groups of Note this formalism can be applied in a purely group theoretic manner using the Lyndon–Hochschild–Serre spectral sequence.

Algebraic K theory is closely related to group cohomology: in Quillen's + construction of K theory, K theory of a ring R is defined as the homotopy groups of a space Here is the infinite general linear group. The space has the same homology as i. e., the group homology of GL(R). In some cases, stability results assert that the sequence of cohomology groups

becomes stationary for large enough n, hence reducing the computation of the cohomology of the infinite general linear group to the one of some Such results have been established when R is a field or for rings of integers in a number field. The phenomenon that group homology of a series of groups stabilizes is referred to as homological stability. In addition to the case just mentioned, this applies to various other groups such as symmetric groups or mapping class groups.

Given a topological group G, i. e., a group equipped with a topology such that product and inverse are continuous maps, it is natural to consider continuous G modules, i. e., requiring that the action

is a continuous map. For such modules, one can again consider the derived functor of A special case occurring in algebra and number theory is when G is profinite, for example the absolute Galois group of a field. The resulting cohomology is called Galois cohomology.