representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups
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.
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
in general measure the extent to which taking invariants doesn't respect exact sequences.
, 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.
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.
of abelian groups G with values in a principal ideal domain k is closely related to the exterior algebra
[c] 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.
for some arbitrary choice of integer t. Principal crossed homomorphisms must additionally satisfy
with the given nontrivial action is the infinite dihedral group, which is a split extension and so trivial inside the
letters can be readily computed by comparing group cohomology with its interpretation in topology.
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
There is a way to compute the semi-direct product of groups using the topology of fibrations and properties of Eilenberg-Maclane spaces.
is induced by the norm map: Tate cohomology enjoys similar features, such as long exact sequences, product structures.
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.
A special case occurring in algebra and number theory is when G is profinite, for example the absolute Galois group of a field.
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.
It instead has the structure of a pointed set – exactly the same situation arises in the 0th homotopy group,
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 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.
) 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.
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 (Rotman 1995, p. 358); Hopf and Beno Eckmann in Switzerland; Hans Freudenthal in the Netherlands (Weibel 1999, p. 807); and Dmitry Faddeev in the Soviet Union (Arslanov 2011, p. 29, Faddeev 1947).
The situation was chaotic because communication between these countries was difficult during World War II.
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.
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.
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.