User:Maximilian Janisch/latexlist/Algebraic Groups/Non-Abelian cohomology

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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) 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 $C ^ { 1 }$ be the holomorph of $C ^ { 1 }$ (i.e. the semi-direct product of $C ^ { 1 }$ and $( C ^ { 1 } )$; cf. also Holomorph of a group), and let $C ^ { 2 }$ be the group of permutations of $C ^ { 2 }$ that leave $E$ fixed. Then a non-Abelian cochain complex is a collection

\begin{equation} C ^ { * } = ( C ^ { 0 } , C ^ { 1 } , C ^ { 2 } , \rho , \sigma , \delta ) \end{equation}

where $\rho : C ^ { 0 } \rightarrow \text { Aff } C ^ { 1 }$, $\sigma : C ^ { 0 } \rightarrow \text { Aut } C ^ { 2 }$ are homomorphisms and $\delta : C ^ { 1 } \rightarrow C ^ { 2 }$ is a mapping such that

\begin{equation} \delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 } \end{equation}

Define the $0$-dimensional cohomology group by

\begin{equation} H ^ { 0 } ( C ^ { * } ) = \rho ^ { - 1 } ( \text { Aut } C ^ { 1 } ) \end{equation}

and the $1$-dimensional cohomology set (with distinguished point) by

\begin{equation} H ^ { 1 } ( C ^ { * } ) = Z ^ { 1 } / \rho \end{equation}

where $Z ^ { 1 } = \delta ^ { - 1 } ( e ) \subseteq C ^ { 1 }$ and the factorization is modulo the action $0$ of the group $C ^ { 0 }$.

Examples.

1) Let $x$ be a topological space with a sheaf of groups $F$, and let $2$ be a covering of $x$; one then has the Čech complex

\begin{equation} C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) ) \end{equation}

where $C ^ { i } ( \mathfrak { U } , F )$ are defined as in the Abelian case (see Cohomology),

\begin{equation} ( \sigma ( a ) ( c ) ) _ { i j k } = \alpha _ { i } c _ { i j k } a _ { i } ^ { - 1 } \end{equation}

\begin{equation} ( \delta b ) _ { i j k } = b _ { j } b _ { j k } b _ { i k } ^ { - 1 } \end{equation}

\begin{equation} \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 } , \quad c \in C ^ { 2 } \end{equation}

Taking limits with respect to coverings, one obtains from the cohomology sets $H ^ { i } ( C ^ { * } ( \mathfrak { U } , F ) )$, $i = 0,1$, the cohomology $H ^ { i } ( X , F )$, $i = 0,1$, of the space $x$ with coefficients in $F$. Under these conditions, $H ^ { 0 } ( X , F ) = F ( X )$. If $F$ is the sheaf of germs of continuous mappings with values in a topological group $k$, then $H ^ { 1 } ( X , F )$ can be interpreted as the set of isomorphism classes of topological principal bundles over $x$ with structure group $k$. 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 $k$-object.

2) Let $k$ be a group and let $4$ be a (not necessarily Abelian) $k$-group, i.e. an operator group with group of operators $k$. Denote the action of an operator $g \in G$ on an element $x \in A$ by $a ^ { g }$. Define a complex $C ^ { * } ( G , A )$ by the formulas

\begin{equation} C ^ { k } = \operatorname { Map } ( G ^ { k } , A ) , \quad k = 0,1,2 \end{equation}

\begin{equation} ( \rho ( \alpha ) ( b ) ) ( g ) = \alpha b ( g ) ( a ^ { g } ) ^ { - 1 } \end{equation}

\begin{equation} ( \sigma ( \alpha ) ( c ) ) ( g , h ) = \alpha ^ { g } c ( g , h ) ( \alpha ^ { g } ) ^ { - 1 } \end{equation}

\begin{equation} \delta ( b ) ( g , h ) = b ( g ) ^ { - 1 } b ( g h ) ( b ( h ) ^ { g } ) ^ { - 1 } \end{equation}

\begin{equation} \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 } , \quad c \in C ^ { 2 } , \quad g \in G \end{equation}

The group $H ^ { 0 } ( G , A ) = H ^ { 0 } ( C ^ { * } ( G , A ) )$ is the subgroup $A ^ { G }$ of $k$-fixed points in $4$, 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 $4$. 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, $k$ a Lie group and $8$ the Lie algebra of $k$. 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 $8$;

\begin{equation} \rho ( f ) ( \alpha ) = d f \cdot f ^ { - 1 } + ( \operatorname { Ad } f ) \alpha \end{equation}

\begin{equation} \sigma ( f ) ( \beta ) = ( \operatorname { Ad } f ) \beta \end{equation}

\begin{equation} \delta \alpha = d \alpha - \frac { 1 } { 2 } [ \alpha , \alpha ] \end{equation}

The set $H ^ { 1 } ( R _ { G } ( X ) )$ is the set of classes of totally-integrable equations of the form $d f . f ^ { - 1 } = \alpha$, $\alpha \in R _ { \overline { \zeta } } ^ { 1 }$, 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 $N$ and a complex Lie group $k$, 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 $k$-invariant subgroup of $4$, this sequence is

\begin{equation} e \rightarrow H ^ { 0 } ( G , B ) \rightarrow H ^ { 0 } ( G , A ) \rightarrow ( A / B ) ^ { G } \end{equation}

\begin{equation} \rightarrow H ^ { 1 } ( G , B ) \rightarrow H ^ { 1 } ( G , A ) \end{equation}

If $B$ is a normal subgroup of $4$, 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 $k$ with coefficients in a group $4$; the definition is as follows. Let $Z ^ { 2 } ( G , A )$ denote the set of all pairs $( m , \phi )$, where $m : G \times G \rightarrow A$, $\phi : G \rightarrow \text { Aut } A$ are mappings such that

\begin{equation} \phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } ) \end{equation}

\begin{equation} g ) = \phi ( g _ { 1 } ) ( m ( g _ { 2 } , g _ { 3 } ) \end{equation}

here $\operatorname { ln } t a$ is the inner automorphism generated by the element $x \in A$. Define an equivalence relation in $Z ^ { 2 } ( G , A )$ by putting $( m , \phi ) \sim ( m ^ { \prime } , \phi ^ { \prime } )$ if there is a mapping $h ; G \rightarrow A$ such that

\begin{equation} \phi ^ { \prime } ( g ) = ( \operatorname { Int } h ( g ) ) \phi ( g ) \end{equation}

and

\begin{equation} ( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 } \end{equation}

The equivalence classes thus obtained are the elements of the cohomology set $H ^ { 2 } ( G , A )$. They are in one-to-one correspondence with the equivalence classes of extensions of $4$ by $k$ (see Extension of a group).

The correspondence $( m , \phi ) \rightarrow \phi$ gives a mapping $6$ of the set $H ^ { 2 } ( G , A )$ into the set of all homomorphisms

\begin{equation} G \rightarrow \text { Out } A = \text { Aut } A / \operatorname { Int } A \end{equation}

let $H _ { \alpha } ^ { 2 } ( G , A ) = \theta ^ { - 1 } ( \alpha )$ for $x \in \text { Out } A$. If one fixes $x \in \text { Out } A$, the centre $Z ( A )$ of $4$ takes on the structure of a $k$-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 $H ^ { 2 }$ 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