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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
+
{{TEX|done}}
  
<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/n06690012.png" /></td> </tr></table>
+
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  $  C ^{0} $,
 +
$  C ^{1} $
 +
be groups, let  $  C ^{2} $
 +
be a set with a distinguished point  $  e $,
 +
let  $  \mathop{\rm Aff}\nolimits \  C ^{1} $
 +
be the holomorph of  $  C ^{1} $(
 +
i.e. the semi-direct product of  $  C ^{1} $
 +
and  $  \mathop{\rm Aut}\nolimits ( C ^{1} ) $;
 +
cf. also [[Holomorph of a group|Holomorph of a group]]), and let  $  \mathop{\rm Aut}\nolimits \  C ^{2} $
 +
be the group of permutations of  $  C ^{2} $
 +
that leave  $  e $
 +
fixed. Then a non-Abelian cochain complex is a collection $$
 +
C ^{*}  =   (C ^{0} ,\  C ^{1} ,\  C ^{2} ,\  \rho ,\
 +
\sigma ,\  \delta ),
 +
$$
 +
where  $  \rho : \  C ^{0} \rightarrow  \mathop{\rm Aff}\nolimits \  C ^{1} $,
 +
$  \sigma : \  C ^{0} \rightarrow  \mathop{\rm Aut}\nolimits \  C ^{2} $
 +
are homomorphisms and  $  \delta : \  C ^{1} \rightarrow C ^{2} $
 +
is a mapping such that $$
 +
\delta (e)  =   e 
 +
\textrm{ and } 
 +
\delta ( \rho (a) b)  =   \sigma (a) \delta (b), 
 +
a \in C ^{0} ,  b \in C ^{1} .
 +
$$
 +
Define the  $  0 $-
 +
dimensional cohomology group by $$
 +
H ^{0} (C ^{*} )  =   \rho ^{-1} (  \mathop{\rm Aut}\nolimits \  C ^{1} ),
 +
$$
 +
and the  $  1 $-
 +
dimensional cohomology set (with distinguished point) by $$
 +
H ^{1} (C ^{*} )  =  Z ^{1} / \rho ,
 +
$$
 +
where  $  Z ^{1} = \delta ^{-1} (e) \subseteq C ^{1} $
 +
and the factorization is modulo the action  $  \rho $
 +
of the group  $  C ^{0} $.
  
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
 
 
<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/n06690016.png" /></td> </tr></table>
 
 
Define the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690017.png" />-dimensional cohomology group by
 
 
<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/n06690018.png" /></td> </tr></table>
 
 
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
 
 
<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/n06690020.png" /></td> </tr></table>
 
 
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.===
 
===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
+
1) Let $  X $
 
+
be a topological space with a sheaf of groups $  {\mathcal F} $,  
<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/n06690028.png" /></td> </tr></table>
+
and let $  \mathfrak U $
 
+
be a covering of $  X $;  
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]]),
+
one then has the Čech complex $$
 
+
C ^{*} ( \mathfrak U ,\  {\mathcal F} )  =   (C ^{0} ( \mathfrak U ,\
<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/n06690030.png" /></td> </tr></table>
+
{\mathcal F} ),
 
+
C ^{1} ( \mathfrak U ,\  {\mathcal F} ),  C ^{2} ( \mathfrak U ,\
<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/n06690031.png" /></td> </tr></table>
+
{\mathcal F} )),
 
+
$$
<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/n06690032.png" /></td> </tr></table>
+
where $  C ^{i} ( \mathfrak U ,\  {\mathcal F} ) $
 
+
are defined as in the Abelian case (see [[Cohomology|Cohomology]]), $$
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]].
+
( \sigma (a) (c)) _{ijk}  =   a _{i} c _{ijk} a _{i} ^{-1} ,
 
+
$$
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690046.png" /> be a group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690047.png" /> be a (not necessarily Abelian) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690048.png" />-group, i.e. an operator group with group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690049.png" />. Denote the action of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690050.png" /> on an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690051.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690052.png" />. Define a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690053.png" /> by the formulas
+
$$
 
+
( \delta b) _{ijk}  =   b _{ij} b _{jk} b _{ik} ^{-1} ,
<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/n06690054.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/n06690055.png" /></td> </tr></table>
+
a  \in  C ^{0} ,  b  \in  C ^{1} ,  c  \in  C ^{2} .
 
+
$$
<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/n06690056.png" /></td> </tr></table>
+
Taking limits with respect to coverings, one obtains from the cohomology sets $  H ^{i} (C ^{*} ( \mathfrak U ,\  {\mathcal F} )) $,
 
+
$  i = 0,\  1 $,  
<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/n06690057.png" /></td> </tr></table>
+
the cohomology $  H ^{i} (X,\  {\mathcal F} ) $,
 
+
$  i = 0,\  1 $,  
<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/n06690058.png" /></td> </tr></table>
+
of the space $  X $
 
+
with coefficients in $  {\mathcal F} $.  
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.
+
Under these conditions, $  H ^{0} (X,\  {\mathcal F} ) = {\mathcal F} (X) $.  
 
+
If $  {\mathcal F} $
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" />;
+
is the sheaf of germs of continuous mappings with values in a topological group $  G $,  
 
+
then $  H ^{1} (X,\  {\mathcal F} ) $
<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/n06690078.png" /></td> </tr></table>
+
can be interpreted as the set of isomorphism classes of topological principal bundles over $  X $
 
+
with structure group $  G $.  
<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/n06690079.png" /></td> </tr></table>
+
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 $  G $-
 
+
object]].
<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/n06690080.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/n06690081.png" /></td> </tr></table>
 
 
 
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>
 
 
 
<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/n066900111.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/n066900117.png" /></td> </tr></table>
 
  
and
+
2) Let  $  G $
 +
be a group and let  $  A $
 +
be a (not necessarily Abelian)  $  G $-
 +
group, i.e. an operator group with group of operators  $  G $.
 +
Denote the action of an operator  $  g \in G $
 +
on an element  $  a \in A $
 +
by  $  a ^{g} $.
 +
Define a complex  $  C ^{*} (G,\  A) $
 +
by the formulas $$
 +
C ^{k}  =    \mathop{\rm Map}\nolimits (G ^{k} ,\  A), 
 +
k = 0,\  1,\  2,
 +
$$
 +
$$
 +
( \rho (a) (b)) (g)  =  ab (g) (a ^{g} ) ^{-1} ,
 +
$$
 +
$$
 +
( \sigma (a) (c)) (g,\  h)  =  a ^{g} c (g,\  h) (a ^{g} ) ^{-1} ,
 +
$$
 +
$$
 +
\delta (b) (g,\  h)  =  b (g) ^{-1} b (gh) (b (h) ^{g} ) ^{-1} ,
 +
$$
 +
$$
 +
a  \in  C ^{0} ,  b  \in  C ^{1} ,  c  \in  C ^{2} ,  g  \in  G.
 +
$$
 +
The group  $  H ^{0} (G,\  A) = H ^{0} (C ^{*} (G,\  A)) $
 +
is the subgroup  $  A ^{G} $
 +
of  $  G $-
 +
fixed points in  $  A $,
 +
while  $  H ^{1} (G,\  A) = H ^{1} ( C ^{*} ( G ,\  A ) ) $
 +
is the set of equivalence classes of crossed homomorphisms  $  G \rightarrow A $,
 +
interpreted as the set of isomorphism classes of principal homogeneous spaces (cf. [[Principal homogeneous space|Principal homogeneous space]]) over  $  A $.
 +
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.
  
<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/n066900118.png" /></td> </tr></table>
+
3) Let  $  X $
 +
be a smooth manifold,  $  G $
 +
a Lie group and  $  \mathfrak g $
 +
the Lie algebra of  $  G $.
 +
The non-Abelian de Rham complex  $  R _{G} ^{*} (X) $
 +
is defined as follows: $  R _{G} ^{0} (X) $
 +
is the group of all smooth functions  $  X \rightarrow G $;  
 +
$  R _{G} ^{k} (X) $,
 +
$  k = 1,\  2 $,
 +
is the space of exterior  $  k $-
 +
forms on  $  X $
 +
with values in  $  \mathfrak g $;
 +
$$
 +
\rho (f \  ) ( \alpha )  =   df \cdot f ^ {\  -1} + (  \mathop{\rm Ad}\nolimits \  f \  ) \alpha ;
 +
$$
 +
$$
 +
\sigma (f \  ) ( \beta )  =   (  \mathop{\rm Ad}\nolimits \  f \  ) \beta ,
 +
$$
 +
$$
 +
\delta \alpha  =   d \alpha - {
 +
\frac{1}{2}
 +
} [ \alpha ,\  \alpha ],
 +
$$
 +
$$
 +
f  \in  R _{G} ^{0} ,  a  \in  R _{G} ^{1} ,  \beta  \in  R _{G} ^{2} .
 +
$$
 +
The set  $  H ^{1} (R _{G} (X)) $
 +
is the set of classes of totally-integrable equations of the form  $  df \cdot f ^ {\  -1} = \alpha $,
 +
$  \alpha \in R _{G} ^{1} $,
 +
modulo gauge transformations. An analogue of the de Rham theorem provides an interpretation of this set as a subset of the set  $  H ^{1} ( \pi _{1} (M),\  G) $
 +
of conjugacy classes of homomorphisms  $  \pi _{1} (M) \rightarrow G $.  
 +
In the case of a complex manifold  $  M $
 +
and a complex Lie group  $  G $,
 +
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.
  
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]]).
+
For each subcomplex of a non-Abelian cochain complex there is an associated exact cohomology sequence. For example, for the complex  $  C ^{*} (G,\  A) $
 +
of Example 2 and its subcomplex  $  C ^{*} (G,\  B) $,
 +
where  $  B $
 +
is a  $  G $-
 +
invariant subgroup of $  A $,
 +
this sequence is $$
 +
e  \rightarrow  H ^{0} (G,\  B)  \rightarrow  H ^{0} (G,\  A)  \rightarrow 
 +
(A/B) ^{G } \rightarrow
 +
$$
 +
$$
 +
\rightarrow 
 +
H ^{1} (G,\  B)  \rightarrow  H ^{1} (G,\  A).
 +
$$
 +
If  $  B $
 +
is a normal subgroup of  $  A $,
 +
the sequence can be continued up to the term  $  H ^{1} (G,\  A/B) $,
 +
and if  $  B $
 +
is in the centre it can be continued to  $  H ^{2} (G,\  B) $.  
 +
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.
  
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
+
Apart from the 0- and  $  1 $-
 +
dimensional non-Abelian cohomology groups just described, there are also  $  2 $-
 +
dimensional examples. A classical example is the  $  2 $-
 +
dimensional cohomology of a group  $  G $
 +
with coefficients in a group  $  A $;
 +
the definition is as follows. Let  $  {\mathcal Z} ^{2} (G,\  A) $
 +
denote the set of all pairs  $  (m,\  \phi ) $,
 +
where  $  m: \  G \times G \rightarrow A $,
 +
$  \phi : \  G \rightarrow  \mathop{\rm Aut}\nolimits \  A $
 +
are mappings such that $$
 +
\phi (g _{1} ) \phi (g _{2} ) \phi (g _{1} g _{2} ) ^{-1}
 +
  =    \mathop{\rm Int}\nolimits \  m (g _{1} ,\  g _{2} ),
 +
$$
 +
$$
 +
m (g _{1} ,\  g _{2} ) m (g _{1} g _{2} ,\  g
 +
_{3} )  =  \phi (g _{1} ) (m (g _{2} ,\  g _{3} )) m (g _{1} ,\  g _{2} \  g _{3} );
 +
$$
 +
here  $  \mathop{\rm Int}\nolimits \  a $
 +
is the inner automorphism generated by the element  $  a \in A $.  
 +
Define an equivalence relation in  $  {\mathcal Z} ^{2} (G,\  A) $
 +
by putting  $  (m,\  \phi ) \sim (m ^ \prime  ,\  \phi ^ \prime  ) $
 +
if there is a mapping $  h: \  G \rightarrow A $
 +
such that $$
 +
\phi ^ \prime  (g)  =   (  \mathop{\rm Int}\nolimits \  h (g)) \phi (g)
 +
$$
 +
and $$
 +
m ^ \prime  (g _{1} ,\  g _{2} )  =   h (g _{1} ) ( \phi (g
 +
_{1} ) (h (g _{2} ))) m (g _{1} ,\  g _{2} )
 +
h (g _{1} ,\  g _{2} ) ^{-1} .
 +
$$
 +
The equivalence classes thus obtained are the elements of the cohomology set $  {\mathcal H} ^{2} (G,\  A) $.  
 +
They are in one-to-one correspondence with the equivalence classes of extensions of  $  A $
 +
by  $  G $(
 +
see [[Extension of a group|Extension of a group]]).
  
<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>
+
The correspondence  $  (m,\  \phi ) \rightarrow \phi $
 +
gives a mapping  $  \theta $
 +
of the set  $  {\mathcal H} ^{2} (G,\  A) $
 +
into the set of all homomorphisms $$
 +
G  \rightarrow    \mathop{\rm Out}\nolimits \  A  =   \mathop{\rm Aut}\nolimits \  A/  \mathop{\rm Int}\nolimits \  A;
 +
$$
 +
let  $  H _ \alpha  ^{2} (G,\  A) = \theta ^{-1} ( \alpha ) $
 +
for  $  \alpha \in  \mathop{\rm Out}\nolimits \  A $.
 +
If one fixes  $  \alpha \in  \mathop{\rm Out}\nolimits \  A $,
 +
the centre  $  Z (A) $
 +
of  $  A $
 +
takes on the structure of a  $  G $-
 +
module and so the cohomology groups  $  H ^{k} (G,\  Z (A)) $
 +
are defined. It turns out that  $  H _ \alpha  ^{2} (G,\  A) $
 +
is non-empty if and only if a certain class in  $  H ^{3} (G,\  Z (A)) $
 +
is trivial. Moreover, under this condition the group  $  H ^{2} (G,\  Z (A)) $
 +
acts simplely transitively on the set  $  H _ \alpha  ^{2} (G,\  A) $.
  
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.
 
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.

Latest revision as of 10:30, 20 December 2019


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 $ C ^{0} $, $ C ^{1} $ be groups, let $ C ^{2} $ be a set with a distinguished point $ e $, let $ \mathop{\rm Aff}\nolimits \ C ^{1} $ be the holomorph of $ C ^{1} $( i.e. the semi-direct product of $ C ^{1} $ and $ \mathop{\rm Aut}\nolimits ( C ^{1} ) $; cf. also Holomorph of a group), and let $ \mathop{\rm Aut}\nolimits \ C ^{2} $ be the group of permutations of $ C ^{2} $ that leave $ e $ fixed. Then a non-Abelian cochain complex is a collection $$ C ^{*} = (C ^{0} ,\ C ^{1} ,\ C ^{2} ,\ \rho ,\ \sigma ,\ \delta ), $$ where $ \rho : \ C ^{0} \rightarrow \mathop{\rm Aff}\nolimits \ C ^{1} $, $ \sigma : \ C ^{0} \rightarrow \mathop{\rm Aut}\nolimits \ C ^{2} $ are homomorphisms and $ \delta : \ C ^{1} \rightarrow C ^{2} $ is a mapping such that $$ \delta (e) = e \textrm{ and } \delta ( \rho (a) b) = \sigma (a) \delta (b), a \in C ^{0} , b \in C ^{1} . $$ Define the $ 0 $- dimensional cohomology group by $$ H ^{0} (C ^{*} ) = \rho ^{-1} ( \mathop{\rm Aut}\nolimits \ C ^{1} ), $$ and the $ 1 $- dimensional cohomology set (with distinguished point) by $$ H ^{1} (C ^{*} ) = Z ^{1} / \rho , $$ where $ Z ^{1} = \delta ^{-1} (e) \subseteq C ^{1} $ and the factorization is modulo the action $ \rho $ of the group $ C ^{0} $.


Examples.

1) Let $ X $ be a topological space with a sheaf of groups $ {\mathcal F} $, and let $ \mathfrak U $ be a covering of $ X $; one then has the Čech complex $$ C ^{*} ( \mathfrak U ,\ {\mathcal F} ) = (C ^{0} ( \mathfrak U ,\ {\mathcal F} ), C ^{1} ( \mathfrak U ,\ {\mathcal F} ), C ^{2} ( \mathfrak U ,\ {\mathcal F} )), $$ where $ C ^{i} ( \mathfrak U ,\ {\mathcal F} ) $ are defined as in the Abelian case (see Cohomology), $$ ( \sigma (a) (c)) _{ijk} = a _{i} c _{ijk} a _{i} ^{-1} , $$ $$ ( \delta b) _{ijk} = b _{ij} b _{jk} b _{ik} ^{-1} , $$ $$ a \in C ^{0} , b \in C ^{1} , c \in C ^{2} . $$ Taking limits with respect to coverings, one obtains from the cohomology sets $ H ^{i} (C ^{*} ( \mathfrak U ,\ {\mathcal F} )) $, $ i = 0,\ 1 $, the cohomology $ H ^{i} (X,\ {\mathcal F} ) $, $ i = 0,\ 1 $, of the space $ X $ with coefficients in $ {\mathcal F} $. Under these conditions, $ H ^{0} (X,\ {\mathcal F} ) = {\mathcal F} (X) $. If $ {\mathcal F} $ is the sheaf of germs of continuous mappings with values in a topological group $ G $, then $ H ^{1} (X,\ {\mathcal F} ) $ can be interpreted as the set of isomorphism classes of topological principal bundles over $ X $ with structure group $ G $. 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.

2) Let $ G $ be a group and let $ A $ be a (not necessarily Abelian) $ G $- group, i.e. an operator group with group of operators $ G $. Denote the action of an operator $ g \in G $ on an element $ a \in A $ by $ a ^{g} $. Define a complex $ C ^{*} (G,\ A) $ by the formulas $$ C ^{k} = \mathop{\rm Map}\nolimits (G ^{k} ,\ A), k = 0,\ 1,\ 2, $$ $$ ( \rho (a) (b)) (g) = ab (g) (a ^{g} ) ^{-1} , $$ $$ ( \sigma (a) (c)) (g,\ h) = a ^{g} c (g,\ h) (a ^{g} ) ^{-1} , $$ $$ \delta (b) (g,\ h) = b (g) ^{-1} b (gh) (b (h) ^{g} ) ^{-1} , $$ $$ a \in C ^{0} , b \in C ^{1} , c \in C ^{2} , g \in G. $$ The group $ H ^{0} (G,\ A) = H ^{0} (C ^{*} (G,\ A)) $ is the subgroup $ A ^{G} $ of $ G $- fixed points in $ A $, while $ H ^{1} (G,\ A) = H ^{1} ( C ^{*} ( G ,\ A ) ) $ is the set of equivalence classes of crossed homomorphisms $ G \rightarrow A $, interpreted as the set of isomorphism classes of principal homogeneous spaces (cf. Principal homogeneous space) over $ A $. 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 $ X $ be a smooth manifold, $ G $ a Lie group and $ \mathfrak g $ the Lie algebra of $ G $. The non-Abelian de Rham complex $ R _{G} ^{*} (X) $ is defined as follows: $ R _{G} ^{0} (X) $ is the group of all smooth functions $ X \rightarrow G $; $ R _{G} ^{k} (X) $, $ k = 1,\ 2 $, is the space of exterior $ k $- forms on $ X $ with values in $ \mathfrak g $; $$ \rho (f \ ) ( \alpha ) = df \cdot f ^ {\ -1} + ( \mathop{\rm Ad}\nolimits \ f \ ) \alpha ; $$ $$ \sigma (f \ ) ( \beta ) = ( \mathop{\rm Ad}\nolimits \ f \ ) \beta , $$ $$ \delta \alpha = d \alpha - { \frac{1}{2} } [ \alpha ,\ \alpha ], $$ $$ f \in R _{G} ^{0} , a \in R _{G} ^{1} , \beta \in R _{G} ^{2} . $$ The set $ H ^{1} (R _{G} (X)) $ is the set of classes of totally-integrable equations of the form $ df \cdot f ^ {\ -1} = \alpha $, $ \alpha \in R _{G} ^{1} $, modulo gauge transformations. An analogue of the de Rham theorem provides an interpretation of this set as a subset of the set $ H ^{1} ( \pi _{1} (M),\ G) $ of conjugacy classes of homomorphisms $ \pi _{1} (M) \rightarrow G $. In the case of a complex manifold $ M $ and a complex Lie group $ G $, 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 $ C ^{*} (G,\ A) $ of Example 2 and its subcomplex $ C ^{*} (G,\ B) $, where $ B $ is a $ G $- invariant subgroup of $ A $, this sequence is $$ e \rightarrow H ^{0} (G,\ B) \rightarrow H ^{0} (G,\ A) \rightarrow (A/B) ^{G } \rightarrow $$ $$ \rightarrow H ^{1} (G,\ B) \rightarrow H ^{1} (G,\ A). $$ If $ B $ is a normal subgroup of $ A $, the sequence can be continued up to the term $ H ^{1} (G,\ A/B) $, and if $ B $ is in the centre it can be continued to $ H ^{2} (G,\ B) $. 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 $ 1 $- dimensional non-Abelian cohomology groups just described, there are also $ 2 $- dimensional examples. A classical example is the $ 2 $- dimensional cohomology of a group $ G $ with coefficients in a group $ A $; the definition is as follows. Let $ {\mathcal Z} ^{2} (G,\ A) $ denote the set of all pairs $ (m,\ \phi ) $, where $ m: \ G \times G \rightarrow A $, $ \phi : \ G \rightarrow \mathop{\rm Aut}\nolimits \ A $ are mappings such that $$ \phi (g _{1} ) \phi (g _{2} ) \phi (g _{1} g _{2} ) ^{-1} = \mathop{\rm Int}\nolimits \ m (g _{1} ,\ g _{2} ), $$ $$ m (g _{1} ,\ g _{2} ) m (g _{1} g _{2} ,\ g _{3} ) = \phi (g _{1} ) (m (g _{2} ,\ g _{3} )) m (g _{1} ,\ g _{2} \ g _{3} ); $$ here $ \mathop{\rm Int}\nolimits \ a $ is the inner automorphism generated by the element $ a \in A $. Define an equivalence relation in $ {\mathcal Z} ^{2} (G,\ A) $ by putting $ (m,\ \phi ) \sim (m ^ \prime ,\ \phi ^ \prime ) $ if there is a mapping $ h: \ G \rightarrow A $ such that $$ \phi ^ \prime (g) = ( \mathop{\rm Int}\nolimits \ h (g)) \phi (g) $$ and $$ m ^ \prime (g _{1} ,\ g _{2} ) = h (g _{1} ) ( \phi (g _{1} ) (h (g _{2} ))) m (g _{1} ,\ g _{2} ) h (g _{1} ,\ g _{2} ) ^{-1} . $$ The equivalence classes thus obtained are the elements of the cohomology set $ {\mathcal H} ^{2} (G,\ A) $. They are in one-to-one correspondence with the equivalence classes of extensions of $ A $ by $ G $( see Extension of a group).

The correspondence $ (m,\ \phi ) \rightarrow \phi $ gives a mapping $ \theta $ of the set $ {\mathcal H} ^{2} (G,\ A) $ into the set of all homomorphisms $$ G \rightarrow \mathop{\rm Out}\nolimits \ A = \mathop{\rm Aut}\nolimits \ A/ \mathop{\rm Int}\nolimits \ A; $$ let $ H _ \alpha ^{2} (G,\ A) = \theta ^{-1} ( \alpha ) $ for $ \alpha \in \mathop{\rm Out}\nolimits \ A $. If one fixes $ \alpha \in \mathop{\rm Out}\nolimits \ A $, the centre $ Z (A) $ of $ A $ takes on the structure of a $ G $- module and so the cohomology groups $ H ^{k} (G,\ Z (A)) $ are defined. It turns out that $ H _ \alpha ^{2} (G,\ A) $ is non-empty if and only if a certain class in $ H ^{3} (G,\ Z (A)) $ is trivial. Moreover, under this condition the group $ H ^{2} (G,\ Z (A)) $ acts simplely transitively on the set $ H _ \alpha ^{2} (G,\ A) $.


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
How to Cite This Entry:
Non-Abelian cohomology. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Non-Abelian_cohomology&oldid=44307
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. OnishchikA.K. Tolpygo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article