Non-Abelian cohomology

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) MR0181643 Zbl 0143.05901 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