In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. Analogous to
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s, 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
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
,
homological algebra,
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
and
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
, as well as in applications to
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
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 S
1, 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.
Motivation
A general paradigm in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
is that a
group ''G'' should be studied via its
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s. A slight generalization of those representations are the
''G''-modules: a ''G''-module is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''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
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 group cohomology functors
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
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
.
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
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 Ext''n'' 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 (e.g. a
free ) of the trivial