View source for 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|Topologized category]]) in dimensions 0, 1. A unified approach to non-Abelian cohomology can be based on the following concept. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669001.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669002.png" /> be groups, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669003.png" /> be a set with a distinguished point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669004.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669005.png" /> be the holomorph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669006.png" /> (i.e. the semi-direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669008.png" />; cf. also [[Holomorph of a group|Holomorph of a group]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n0669009.png" /> be the group of permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690010.png" /> that leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690011.png" /> fixed. Then a non-Abelian cochain complex is a collection

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690014.png" /> are homomorphisms and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690015.png" /> is a mapping such that

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Define the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690017.png" />-dimensional cohomology group by

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and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690019.png" />-dimensional cohomology set (with distinguished point) by

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690021.png" /> and the factorization is modulo the action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690022.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690023.png" />.

===Examples.===


1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690024.png" /> be a topological space with a sheaf of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690025.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690026.png" /> be a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690027.png" />; one then has the Čech complex

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690029.png" /> are defined as in the Abelian case (see [[Cohomology|Cohomology]]),

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Taking limits with respect to coverings, one obtains from the cohomology sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690034.png" />, the cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690036.png" />, of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690037.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690038.png" />. Under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690040.png" /> is the sheaf of germs of continuous mappings with values in a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690042.png" /> can be interpreted as the set of isomorphism classes of topological principal bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690043.png" /> with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690044.png" />. 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 G-object|Principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690045.png" />-object]].

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690046.png" /> be a group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690047.png" /> be a (not necessarily Abelian) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690048.png" />-group, i.e. an operator group with group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690049.png" />. Denote the action of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690050.png" /> on an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690051.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690052.png" />. Define a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690053.png" /> by the formulas

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The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690059.png" /> is the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690060.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690061.png" />-fixed points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690062.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690063.png" /> is the set of equivalence classes of crossed homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690064.png" />, interpreted as the set of isomorphism classes of principal homogeneous spaces (cf. [[Principal homogeneous space|Principal homogeneous space]]) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690065.png" />. For applications and actual computations of non-Abelian cohomology groups see [[Galois cohomology|Galois cohomology]]. Analogous definitions yield the non-Abelian cohomology of categories and semi-groups.

3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690066.png" /> be a smooth manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690067.png" /> a Lie group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690068.png" /> the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690069.png" />. The non-Abelian de Rham complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690070.png" /> is defined as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690071.png" /> is the group of all smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690072.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690074.png" />, is the space of exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690075.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690076.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690077.png" />;

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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690082.png" /> is the set of classes of totally-integrable equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690084.png" />, modulo gauge transformations. An analogue of the de Rham theorem provides an interpretation of this set as a subset of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690085.png" /> of conjugacy classes of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690086.png" />. In the case of a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690087.png" /> and a complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690088.png" />, 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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690089.png" /> of Example 2 and its subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690091.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690092.png" />-invariant subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690093.png" />, this sequence is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690094.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690095.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690096.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690097.png" />, the sequence can be continued up to the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690098.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690099.png" /> is in the centre it can be continued to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900100.png" />. 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 [[#References|[1]]], [[#References|[6]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900101.png" />-dimensional non-Abelian cohomology groups just described, there are also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900102.png" />-dimensional examples. A classical example is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900103.png" />-dimensional cohomology of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900104.png" /> with coefficients in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900105.png" />; the definition is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900106.png" /> denote the set of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900109.png" /> are mappings such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900110.png" /></td> </tr></table>

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here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900112.png" /> is the inner automorphism generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900113.png" />. Define an equivalence relation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900114.png" /> by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900115.png" /> if there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900116.png" /> such that

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and

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The equivalence classes thus obtained are the elements of the cohomology set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900119.png" />. They are in one-to-one correspondence with the equivalence classes of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900120.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900121.png" /> (see [[Extension of a group|Extension of a group]]).

The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900122.png" /> gives a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900123.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900124.png" /> into the set of all homomorphisms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900125.png" /></td> </tr></table>

let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900126.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900127.png" />. If one fixes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900128.png" />, the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900129.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900130.png" /> takes on the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900131.png" />-module and so the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900132.png" /> are defined. It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900133.png" /> is non-empty if and only if a certain class in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900134.png" /> is trivial. Moreover, under this condition the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900135.png" /> acts simplely transitively on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900136.png" />.

This definition of a two-dimensional cohomology can be generalized, carrying it over to sites (see [[#References|[2]]], where the applications of this concept are also presented). A general algebraic scheme that yields a two-dimensional cohomology is outlined in [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Giraud, "Cohomologie non abélienne" , Springer (1971) {{MR|0344253}} {{ZBL|0226.14011}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.K. Tolpygo, "Two-dimensional cohomologies and the spectral sequence in the nonabelian theory" ''Selecta Math. Sov.'' , '''6''' (1987) pp. 177–197 {{MR|0548342}} {{ZBL|0619.18006}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P. Dedecker, "Three-dimensional nonabelian cohomology for groups" , ''Category theory, homology theory and their applications (Battelle Inst. Conf.)'' , '''2''' , Springer (1968) pp. 32–64 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Frenkel, "Cohomology non abélienne et espaces fibrés" ''Bull. Soc. Math. France'' , '''85''' : 2 (1957) pp. 135–220 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H. Goldschmidt, "The integrability problem for Lie equations" ''Bull. Amer. Math. Soc.'' , '''84''' : 4 (1978) pp. 531–546 {{MR|0517116}} {{ZBL|0439.58025}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> T.A. Springer, "Nonabelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900137.png" /> 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 {{MR|209297}} {{ZBL|0193.48902}} </TD></TR></table>