Group cohomology

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 formula_1 representing "n"-simplices, topological properties of the space may be computed, such as the set of cohomology groups formula_2. 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 formula_3 can be thought of as the singular cohomology of the circle S1, and similarly for formula_4 and formula_5
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.
Motivation.
A general paradigm in group theory is that a group "G" should be studied via its group representations. A slight generalization of those representations are the "G"-modules: a "G"-module is an abelian group "M" together with a group action of "G" on "M", with every element of "G" acting as an automorphism of "M". We will write "G" multiplicatively and "M" additively.
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 formula_7 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 formula_8 is to precisely measure this difference.
The group cohomology functors formula_9 in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence.
Definitions.
The collection of all "G"-modules is a category (the morphisms are group homomorphisms "f" with the property formula_10 for all "g" in "G" and "x" in "M"). Sending each module "M" to the group of invariants formula_11 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 formula_2, "the "n"-th cohomology group of "G" with coefficients in "M"". Furthermore, the group formula_13 may be identified with formula_11.
Cochain complexes.
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 formula_15, let formula_16 be the group of all functions from formula_1 to "M" (here formula_18 means formula_19). This is an abelian group; its elements are called the (inhomogeneous) "n"-cochains. The coboundary homomorphisms
One may check that formula_21 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 Ext"n" and formal definition of group cohomology.
Interpreting "G"-modules as modules over the group ring formula_25 one can note that
i.e., the subgroup of "G"-invariant elements in "M" is identified with the group of homomorphisms from formula_3, 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 formula_3, 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 formula_30-resolution (e.g. a free formula_30-resolution) of the trivial formula_30-module formula_3:
e.g., one may always take the resolution of group rings, formula_35 with morphisms
Recall that for formula_30-modules "N" and "M", Hom"G"("N", "M") is an abelian group consisting of formula_30-homomorphisms from "N" to "M". Since formula_39 is a contravariant functor and reverses the arrows, applying formula_39 to "F" termwise and dropping formula_41 produces a cochain complex formula_42:
The cohomology groups formula_44 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 formula_46, that is, functions formula_47 that obey
The coboundary operator formula_49 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 formula_51, is given by reparameterizing so that
and so on. Thus
as in the preceding section.
Group homology.
Dually to the construction of group cohomology there is the following definition of group homology: given a "G"-module "M", set "DM" to be the submodule generated by elements of the form "g"·"m" − "m", "g" ∈ "G", "m" ∈ "M". Assigning to "M" its so-called "coinvariants", the quotient
is a right exact functor. Its left derived functors are by definition the group homology
The covariant functor which assigns "MG" to "M" is isomorphic to the functor which sends "M" to formula_56 where formula_3 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:
Specifically, the homology groups "Hn"("G", "M") can be computed as follows. Start with a projective resolution "F" of the trivial formula_30-module formula_60 as in the previous section. Apply the covariant functor formula_61 to "F" termwise to get a chain complex formula_62:
Then "H""n"("G", "M") are the homology groups of this chain complex, formula_64 for "n" ≥ 0.
Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups.
The group homology formula_65 of abelian groups "G" with values in a principal ideal domain "k" is closely related to the exterior algebra formula_66.
Low-dimensional cohomology groups.
"H" 1.
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 "H"1("G","M") = Hom("G", "M"), the group of group homomorphisms "G" → "M".
Consider the case of formula_67 where formula_68 denotes the non-trivial formula_69-structure on the group of integers. Then crossed homomorphisms constitute all maps formula_70 satisfying formula_71 and formula_72 for some integer "a". Principal crossed homomorphisms satisfy additionally formula_73 hence
"H" 2.
If "M" is a trivial "G"-module (i.e. the action of "G" on "M" is trivial), the second cohomology group "H"2("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, "H"2("G","M") classifies the isomorphism classes of all extensions formula_75 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, formula_76 as the only extension of formula_69 by formula_3 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:
Basic examples.
Group cohomology of a finite cyclic group.
For the finite cyclic group formula_80 of order formula_81 with generator formula_82, the element formula_83 in the associated group ring has a multiplicative inverse formula_84 given byformula_85sinceformula_86This property can be used to construct the resolution of the trivial formula_87-module formula_88 via the complexformula_89giving the group cohomology computation for any formula_87-module formula_91. Note the augmentation map gives the trivial module formula_88 its formula_87-structure byformula_94This resolution gives a computation of the group cohomology since there is the isomorphism of cohomology groupsformula_95showing that applying the functor formula_96 to the complex above (with formula_88 removed since this resolution is a quasi-isomorphism), gives the computationformula_98forformula_99For example, if formula_100, the trivial module, then formula_101, formula_102, and formula_103, henceformula_104
Cohomology of free groups.
Using a resolution.
Given a set formula_105 the associated free group formula_106 has an explicit resolution of the trivial module formula_107 which can be easily computed. Notice the augmentation mapformula_108has kernel given by the free submodule formula_109 generated by the set formula_110, soformula_111.Because this object is free, this gives a resolutionformula_112hence the group cohomology of formula_113 with coefficients in formula_107 can be computed by applying the functor formula_115 to the complex formula_116, givingformula_117this is because the dual mapformula_118sends any formula_87-module morphismformula_120to the induced morphism on formula_109 by composing the inclusion. The only maps which are sent to formula_122 are formula_88-multiples of the augmentation map, giving the first cohomology group. The second can be found by noticing the only other mapsformula_124can be generated by the formula_88-basis of maps sending formula_126 for a fixed formula_127, and sending formula_128 for any formula_129.
Using topology.
The group cohomology of free groups formula_130 generated by formula_131 letters can be readily computed by comparing group cohomology with its interpretation in topology. Recall that for every group formula_113 there is a topological space formula_133, called the classifying space of the group, which has the propertyformula_134In addition, it has the property that its topological cohomology is isomorphic to group cohomologyformula_135giving a way to compute some group cohomology groups. Note formula_88 could be replaced by any local system formula_137 which is determined by a mapformula_138for some abelian group formula_139. In the case of formula_140 for formula_131 letters, this is represented by a wedge sum of formula_131 circles formula_143 which can be showed using the Van-Kampen theorem, giving the group cohomologyformula_144
Properties.
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
is a short exact sequence of "G"-modules, then a long exact sequence is induced:
The so-called connecting homomorphisms,
can be described in terms of inhomogeneous cochains as follows. If formula_148 is represented by an "n"-cocycle formula_149 then formula_150 is represented by formula_151 where formula_152 is an "n"-cochain formula_153 "lifting" formula_154 (i.e. formula_154 is the composition of formula_152 with the surjective map "M" → "N").
Functoriality.
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").
Products.
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 formula_160 where "R" is a ring such as formula_3 or formula_162 For a finite group "G", the even part of this cohomology ring in characteristic "p", formula_163 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 formula_164.
For example, let "G" be the group with two elements, under the discrete topology. The real projective space formula_165 is a classifying space for "G". Let formula_166 the field of two elements. Then
a polynomial "k"-algebra on a single generator, since this is the cellular cohomology ring of formula_5
Künneth formula.
If, "M" = "k" is a field, then "H*"("G"; "k") is a graded "k"-algebra and the cohomology of a product of groups is related to the ones of the individual groups by a Künneth formula:
For example, if "G" is an elementary abelian 2-group of rank "r", and formula_166 then the Künneth formula shows that the cohomology of "G" is a polynomial "k"-algebra generated by "r" classes in "H"1("G"; "k").,
Homology vs. cohomology.
As for other cohomology theories, such as singular cohomology, group cohomology and homology are related to one another by means of a short exact sequence
where "A" is endowed with the trivial "G"-action and the term at the left is the first Ext group.
Amalgamated products.
Given a group "A" which is the subgroup of two groups "G"1 and "G"2, the homology of the amalgamated product formula_173 (with integer coefficients) lies in a long exact sequence
The homology of formula_175 can be computed using this:
This exact sequence can also be applied to show that the homology of the formula_177 and the special linear group formula_178 agree for an infinite field "k".
Change of group.
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.
Cohomology of the classifying space.
Group cohomology is closely related to topological cohomology theories such as sheaf cohomology, by means of an isomorphism
The expression "formula_133" at the left is a classifying space for formula_113. It is an Eilenberg–MacLane space formula_182, i.e., a space whose fundamental group is formula_113 and whose higher homotopy groups vanish). Classifying spaces for formula_184 and formula_185 are the 1-sphere S1, infinite real projective space formula_186 and lens spaces, respectively. In general, "formula_133" can be constructed as the quotient formula_188, where formula_189 is a contractible space on which formula_113 acts freely. However, "formula_133" does not usually have an easily amenable geometric description.
More generally, one can attach to any formula_113-module formula_91 a local coefficient system on formula_133 and the above isomorphism generalizes to an isomorphism
Further examples.
Semi-direct products of groups.
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 formula_196 there is an associated short exact sequence of groupsformula_197Using the associated Eilenberg-Maclane spaces there is a Serre fibrationformula_198which can be put through a Serre spectral sequence. This gives an formula_199-pageformula_200which gives information about the group cohomology of formula_113 from the group cohomology groups of formula_202. Note this formalism can be applied in a purely group-theoretic manner using the Lyndon–Hochschild–Serre spectral sequence.
Cohomology of finite groups.
Higher cohomology groups are torsion.
The cohomology groups "H""n"("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 formula_203-vector space), the transfer map can be used to show that formula_204 for formula_205 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.
Tate cohomology groups combine both homology and cohomology of a finite group "G":
where formula_209 is induced by the norm map:
Tate cohomology enjoys similar features, such as long exact sequences, product structures. An important application is in class field theory, see class formation.
Tate cohomology of finite cyclic groups, formula_211 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 formula_213 has this property for coprime integers "n" and "m".
Applications.
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 formula_214 Here formula_215 is the infinite general linear group. The space formula_216 has the same homology as formula_217 "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 formula_219. 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 formula_220 stabilizes is referred to as homological stability. In addition to the case formula_221 just mentioned, this applies to various other groups such as symmetric groups or mapping class groups.
Projective representations and group extensions.
In quantum mechanics we often have systems with a symmetry group formula_222 We expect an action of formula_113 on the Hilbert space formula_224 by unitary matrices formula_225 We might expect formula_226 but the rules of quantum mechanics only require
where formula_228 is a phase. This projective representation of formula_113 can also be thought of as a conventional representation of a group extension formula_230 of formula_113 by formula_232 as described by the exact sequence
Requiring associativity 
leads to
which we recognise as the statement that formula_236 i.e. that formula_237 is a cocycle taking values in formula_238 We can ask whether we can eliminate the phases by redefining
which changes
This we recognise as shifting formula_237 by a coboundary formula_242 The distinct projective representations are therefore classified by formula_243 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 formula_244 acting on it as in the general definitions of coboundary in the previous sections. For example, formula_245
Extensions.
Cohomology of topological 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 formula_247. 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 formula_249 to be a 1-cocycle is that formula_250 and formula_251 if there is an "a" in "A" such that formula_252. In general, formula_253 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, formula_254 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
History and relation to other fields.
The low-dimensional cohomology of a group was classically studied in other guises, well before the notion of group cohomology was formulated in 1943–45. The first theorem of the subject can be identified as Hilbert's Theorem 90 in 1897; this was recast into "Emmy Noether's equations" in Galois theory (an appearance of cocycles for formula_257). The idea of "factor sets" for the extension problem for groups (connected with formula_258) arose in the work of Otto Hölder (1893), in Issai Schur's 1904 study of projective representations, in Otto Schreier's 1926 treatment, and in Richard Brauer's 1928 study of simple algebras and the Brauer group. A fuller discussion of this history may be found in .
In 1941, while studying formula_259 (which plays a special role in groups), Heinz Hopf discovered what is now called Hopf's integral homology formula , which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group:
where formula_261 and "F" is a free group.
Hopf's result led to the independent discovery of group cohomology by several groups in 1943-45: Samuel Eilenberg and Saunders Mac Lane in the United States ; Hopf and Beno Eckmann in Switzerland; and Hans Freudenthal in the Netherlands . The situation was chaotic because communication between these countries was difficult during World War II.
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 space "BG" 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.
The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) . Some refinements in the theory post-1960 have been made, such as continuous cocycles and John Tate's redefinition, but the basic outlines remain the same. This is a large field, and now basic in the theories of algebraic groups.
The analogous theory for Lie algebras, called Lie algebra cohomology, was first developed in the late 1940s, by Claude Chevalley and Eilenberg, and Jean-Louis Koszul . It is formally similar, using the corresponding definition of "invariant" for the action of a Lie algebra. It is much applied in representation theory, and is closely connected with the BRST quantization of theoretical physics.
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.