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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-moduleM 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.
Given such a G-module M, it is natural to consider the submodule of G-invariant elements:
Now, if N is a G-submodule of M (i.e., a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The purpose of the first group cohomology is to precisely measure this difference.
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 definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful. For , let be the group of all functions from to M (here means ). This is an abelian group; its elements are called the (inhomogeneous) n-cochains. The coboundary homomorphisms
One may check that so this defines a cochain complex whose cohomology can be computed. It can be shown that the above-mentioned definition of group cohomology in terms of derived functors is isomorphic to the cohomology of this complex
Here the groups of n-cocycles, and n-coboundaries, respectively, are defined as
The functors Extn and formal definition of group cohomology
Interpreting G-modules as modules over the group ring one can note that
i.e., the subgroup of G-invariant elements in M is identified with the group of homomorphisms from , which is treated as the trivial G-module (every element of G acts as the identity) to M.
Therefore, as Ext functors are the derived functors of Hom, there is a natural isomorphism
These Ext groups can also be computed via a projective resolution of , the advantage being that such a resolution only depends on G and not on M. We recall the definition of Ext more explicitly for this context. Let F be a projective -resolution (e.g. a free -resolution) of the trivial -module :
e.g., one may always take the resolution of group rings, with morphisms
The cohomology groups of G with coefficients in the module M are defined as the cohomology of the above cochain complex:
This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of , that is, functions that obey
The coboundary operator is now naturally defined by, for example,
The relation to the coboundary operator d that was defined in the previous section, and which acts on the "inhomogeneous" cochains , is given by reparameterizing so that
and so on. Thus
as in the preceding section.
Dually to the construction of group cohomology there is the following definition of group homology: given a G-moduleM, set DM to be the submodulegenerated by elements of the form g·m - m, g ? G, m ? M. Assigning to M its so-called coinvariants, the quotient
The covariant functor which assigns MG to M is isomorphic to the functor which sends M to where is endowed with the trivial G-action. Hence one also gets an expression for group homology in terms of the Tor functors,
Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted "co-" switches:
superscripts correspond to cohomology H* and invariants XG while
subscripts correspond to homology H* and coinvariants XG := X/G.
Specifically, the homology groups Hn(G, M) can be computed as follows. Start with a projective resolutionF of the trivial -module as in the previous section. Apply the covariant functor to F termwise to get a chain complex:
Then Hn(G, M) are the homology groups of this chain complex, for n >= 0.
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(a) = am-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 homomorphismsG -> M.
Consider the case of where denotes the non-trivial -structure on the group of integers. Then crossed homomorphisms constitute all maps satisfying and for some integer a. Principal crossed homomorphisms satisfy additionally hence
If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H2(G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H2(G,M) classifies the isomorphism classes of all extensions of G by M, in which the action of G on E (by inner automorphisms), endows (the image of) M with isomorphic G-module structure.
In the example as above, as the only extension of by with the given nontrivial action is the infinite dihedral group.
An example of a second group cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:
In the following, let M be a G-module.
Long exact sequence of cohomology
In practice, one often computes the cohomology groups using the following fact: if
can be described in terms of inhomogeneous cochains as follows. If is represented by an n-cocycle then is represented by where is an n-cochain "lifting" (i.e. is the composition of with the surjective map M -> N).
Group cohomology depends contravariantly on the group G, in the following sense: if f : H -> G is a group homomorphism, then we have a naturally induced morphism Hn(G, M) -> Hn(H, M) (where in the latter, M is treated as an H-module via f). This map is called the restriction map. If the index of H in G is finite, there is also a map in the opposite direction, called transfer map,
In degree 0, it is given by the map
Given a morphism of G-modules M -> N, one gets a morphism of cohomology groups in the Hn(G, M) -> Hn(G, N).
Similarly to other cohomology theories in topology and geometry, such as singular cohomology or de Rham cohomology, group cohomology enjoys a product structure: there is a natural map called cup product:
for any two G-modules M and N. This yields a graded anti-commutative ring structure on where R is a ring such as or For a finite group G, the even part of this cohomology ring in characteristic p, carries a lot of information about the group the structure of G, for example the Krull dimension of this ring equals the maximal rank of an abelian subgroup .
For example, let G be the group with two elements, under the discrete topology. The real projective space is a classifying space for G. Let the field of two elements. Then
More generally, one can attach to any G-module M a local coefficient system on BG and the above isomorphism generalizes to an isomorphism
Cohomology of finite groups
Higher cohomology groups are torsion
The cohomology groups Hn(G, M) of finite groups G are all torsion for all n≥1. Indeed, by Maschke's theorem the category of representations of a finite group is semi-simple over any field of characteristic zero (or more generally, any field whose characteristic does not divide the order of the group), hence, viewing group cohomology as a derived functor in this abelian category, one obtains that it is zero. The other argument is that over a field of characteristic zero, the group algebra of a finite group is a direct sum of matrix algebras (possibly over division algebras which are extensions of the original field), while a matrix algebra is Morita equivalent to its base field and hence has trivial cohomology.
If the order of G is invertible in a G-module M (for example, if M is a -vector space), the transfer map can be used to show that for A typical application of this fact is as follows: the long exact cohomology sequence of the short exact sequence (where all three groups have a trivial G-action)
yields an isomorphism
Tate cohomology groups combine both homology and cohomology of a finite group G:
Tate cohomology of finite cyclic groups, is 2-periodic in the sense that there are isomorphisms
A necessary and sufficient criterion for a d-periodic cohomology is that the only abelian subgroups of G are cyclic. For example, any semi-direct product has this property for coprime integers n and m.
Algebraic K-theory and homology of linear groups
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.
which we recognise as the statement that i.e. that is a cocycle taking values in We can ask whether we can eliminate the phases by redefining
This we recognise as shifting by a coboundary The distinct projective representations are therefore classified by Note that if we allow the phases themselves to be acted on by the group (for example, time reversal would complex-conjugate the phase), then the first term in each of the coboundary operations will have a acting on it as in the general definitions of coboundary in the previous sections. For example,
Cohomology of topological groups
Given a topological groupG, 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.
Non-abelian group cohomology
Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G.
The zeroth cohomology of G with coefficients in A is defined to be the subgroup
of elements of A fixed by G.
The first cohomology of G with coefficients in A is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map to be a 1-cocycle is that and if there is an a in A such that . In general, is not a group when A is non-abelian. It instead has the structure of a pointed set - exactly the same situation arises in the 0th homotopy group, which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point.
Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let
be a short exact sequence of G-groups, then there is an exact sequence of pointed sets
In 1941, while studying (which plays a special role in groups), Heinz Hopf discovered what is now called Hopf's integral homology formula (Hopf 1942), which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group:
From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a model for the topological classifying spaceBG as discussed above. In practice, this meant using topology to produce the chain complexes used in formal algebraic definitions. From a module-theoretic point of view this was integrated into the Cartan-Eilenberg theory of homological algebra in the early 1950s.
Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of spontaneous symmetry breaking phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter--short-range entangled states with symmetry. Short-range entangled states with symmetry are also known as symmetry-protected topological states.
^Recall that the tensor product is defined whenever N is a right -module and M is a left -module. If N is a left -module, we turn it into a right -module by setting ag = g-1a for every g ? G and every a ? N. This convention allows to define the tensor product in the case where both M and N are left -modules.
^For example, the two are isomorphic if all primes p such that G has p-torsion are invertible in k. See (Knudson 2001), Theorem A.1.19 for the precise statement.