Cohomology with coefficients in a non-Abelian group, a sheaf of non-Abelian groups, etc. The best known examples are the cohomology of groups, topological spaces and the more general example of the cohomology of sites (i.e. topological categories; cf. Topologized category) in dimensions 0, 1. A unified approach to non-Abelian cohomology can be based on the following concept. Let 
, 
be groups, let 
be a set with a distinguished point 
, let 
be the holomorph of 
(i.e. the semi-direct product of 
and 
; cf. also Holomorph of a group), and let 
be the group of permutations of 
that leave 
fixed. Then a non-Abelian cochain complex is a collection
where 
, 
are homomorphisms and 
is a mapping such that
Define the 
-dimensional cohomology group by
and the 
-dimensional cohomology set (with distinguished point) by
where 
and the factorization is modulo the action 
of the group 
.
Examples.
1) Let 
be a topological space with a sheaf of groups 
, and let 
be a covering of 
; one then has the Čech complex
where 
are defined as in the Abelian case (see Cohomology),
Taking limits with respect to coverings, one obtains from the cohomology sets 
, 
, the cohomology 
, 
, of the space 
with coefficients in 
. Under these conditions, 
. If 
is the sheaf of germs of continuous mappings with values in a topological group 
, then 
can be interpreted as the set of isomorphism classes of topological principal bundles over 
with structure group 
. Similarly one obtains a classification of smooth and holomorphic principal bundles. In a similar fashion one defines the non-Abelian cohomology for a site; for an interpretation see Principal 
-object.
2) Let 
be a group and let 
be a (not necessarily Abelian) 
-group, i.e. an operator group with group of operators 
. Denote the action of an operator 
on an element 
by 
. Define a complex 
by the formulas
The group 
is the subgroup 
of 
-fixed points in 
, while 
is the set of equivalence classes of crossed homomorphisms 
, interpreted as the set of isomorphism classes of principal homogeneous spaces (cf. Principal homogeneous space) over 
. For applications and actual computations of non-Abelian cohomology groups see Galois cohomology. Analogous definitions yield the non-Abelian cohomology of categories and semi-groups.
3) Let 
be a smooth manifold, 
a Lie group and 
the Lie algebra of 
. The non-Abelian de Rham complex 
is defined as follows: 
is the group of all smooth functions 
; 
, 
, is the space of exterior 
-forms on 
with values in 
;
The set 
is the set of classes of totally-integrable equations of the form 
, 
, modulo gauge transformations. An analogue of the de Rham theorem provides an interpretation of this set as a subset of the set 
of conjugacy classes of homomorphisms 
. In the case of a complex manifold 
and a complex Lie group 
, one again defines a non-Abelian holomorphic de Rham complex and a non-Abelian Dolbeault complex, which are intimately connected with the problem of classifying holomorphic bundles [3]. Non-Abelian complexes of differential forms are also an important tool in the theory of pseudo-group structures on manifolds.
For each subcomplex of a non-Abelian cochain complex there is an associated exact cohomology sequence. For example, for the complex 
of Example 2 and its subcomplex 
, where 
is a 
-invariant subgroup of 
, this sequence is
If 
is a normal subgroup of 
, the sequence can be continued up to the term 
, and if 
is in the centre it can be continued to 
. This sequence is exact in the category of sets with a distinguished point. In addition, a tool is available ( "twisted" cochain complexes) for describing the pre-images of all — not only the distinguished — elements (see [1], [6], [3]). One can also construct a spectral sequence related to a double non-Abelian complex, and the corresponding exact boundary sequence.
Apart from the 0- and 
-dimensional non-Abelian cohomology groups just described, there are also 
-dimensional examples. A classical example is the 
-dimensional cohomology of a group 
with coefficients in a group 
; the definition is as follows. Let 
denote the set of all pairs 
, where 
, 
are mappings such that
here 
is the inner automorphism generated by the element 
. Define an equivalence relation in 
by putting 
if there is a mapping 
such that
and
The equivalence classes thus obtained are the elements of the cohomology set 
. They are in one-to-one correspondence with the equivalence classes of extensions of 
by 
(see Extension of a group).
The correspondence 
gives a mapping 
of the set 
into the set of all homomorphisms
let 
for 
. If one fixes 
, the centre 
of 
takes on the structure of a 
-module and so the cohomology groups 
are defined. It turns out that 
is non-empty if and only if a certain class in 
is trivial. Moreover, under this condition the group 
acts simplely transitively on the set 
.
This definition of a two-dimensional cohomology can be generalized, carrying it over to sites (see [2], where the applications of this concept are also presented). A general algebraic scheme that yields a two-dimensional cohomology is outlined in [4]; just as in the special case described above, computation of two-dimensional cohomology reduces to the computation of one-dimensional non-Abelian and ordinary Abelian cohomology.
References
| [1] | J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303 |
| [2] | J. Giraud, "Cohomologie non abélienne" , Springer (1971) MR0344253 Zbl 0226.14011 |
| [3] | A.L. Onishchik, "Some concepts and applications of the theory of non-Abelian cohomology" Trans. Moscow Math. Soc. , 17 (1979) pp. 49–98 Trudy Moskov. Mat. Obshch. , 17 (1967) pp. 45–88 |
| [4] | A.K. Tolpygo, "Two-dimensional cohomologies and the spectral sequence in the nonabelian theory" Selecta Math. Sov. , 6 (1987) pp. 177–197 MR0548342 Zbl 0619.18006 |
| [5] | P. Dedecker, "Three-dimensional nonabelian cohomology for groups" , Category theory, homology theory and their applications (Battelle Inst. Conf.) , 2 , Springer (1968) pp. 32–64 |
| [6] | J. Frenkel, "Cohomology non abélienne et espaces fibrés" Bull. Soc. Math. France , 85 : 2 (1957) pp. 135–220 |
| [7] | H. Goldschmidt, "The integrability problem for Lie equations" Bull. Amer. Math. Soc. , 84 : 4 (1978) pp. 531–546 MR0517116 Zbl 0439.58025 |
| [8] | T.A. Springer, "Nonabelian  in Galois cohomology" A. Borel (ed.) G.D. Mostow (ed.) , Algebraic groups and discontinuous subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 164–182 MR209297 Zbl 0193.48902 |
