Cohomology of groups

Historically, the earliest theory of a cohomology of algebras.

With every pair $ ( G, A) $, where $ G $ is a group and $ A $ a left $ G $- module (that is, a module over the integral group ring $ \mathbf Z G $), there is associated a sequence of Abelian groups $ H ^ { n } ( G, A) $, called the cohomology groups of $ G $ with coefficients in $ A $. The number $ n $, which runs over the non-negative integers, is called the dimension of $ H ^ { n } ( G, A) $. The cohomology groups of groups are important invariants containing information both on the group $ G $ and on the module $ A $.

By definition, $ H ^ {0} ( G, A) $ is $ \mathop{\rm Hom} _ {G} ( \mathbf Z , A) \simeq A ^ {G} $, where $ A ^ {G} $ is the submodule of $ G $- invariant elements in $ A $. The groups $ H ^ { n } ( G, A) $, $ n > 1 $, are defined as the values of the $ n $- th derived functor of the functor $ A \mapsto H ^ {0} ( G, A) $. Let

$$ \dots \rightarrow ^ { {d _ n} } \ P _ {n} \rightarrow ^ { {d _ {n} - 1 } } \ P _ {n - 1 } \rightarrow \dots \rightarrow \ P _ {0} \rightarrow \mathbf Z \rightarrow 0 $$

be some projective resolution of the trivial $ G $- module $ \mathbf Z $ in the category of $ G $- modules, that is, an exact sequence in which every $ P _ {i} $ is a projective $ \mathbf Z G $- module. Then $ H ^ { n } ( G, A) $ is the $ n $- th cohomology group of the complex

$$ 0 \rightarrow \mathop{\rm Hom} _ {G} ( P _ {0} , A) \rightarrow ^ { {d _ 0} ^ \prime } \ \mathop{\rm Hom} _ {G} ( P _ {1} , A) \rightarrow \dots , $$

where $ d _ {n} ^ { \prime } $ is induced by $ d _ {n} $, that is, $ H ^ { n } ( G, A) = \mathop{\rm Ker} d _ {n} ^ { \prime } / \mathop{\rm Im} d _ {n - 1 } ^ { \prime } $.

The homology groups of a group are defined using the dual construction, in which $ \mathop{\rm Hom} _ {G} $ is replaced everywhere by $ \otimes _ {G} $.

The set of functors $ A \mapsto H ^ { n } ( G, A) $, $ n = 0, 1 \dots $ is a cohomological functor (see Homology functor; Cohomology functor) on the category of left $ G $- modules.

A module of the form $ B = \mathop{\rm Hom} ( \mathbf Z [ G], X) $, where $ X $ is an Abelian group and $ G $ acts on $ B $ according to the formula

$$ ( g \phi ) ( t) = \ \phi ( tg),\ \ \phi \in B,\ \ t \in \mathbf Z G, $$

is said to be co-induced. If $ A $ is injective or co-induced, then $ H ^ { n } ( G, A) = 0 $ for $ n \geq 1 $. Every module $ A $ is isomorphic to a submodule of a co-induced module $ B $. The exact homology sequence for the sequence

$$ 0 \rightarrow A \rightarrow B \rightarrow B/A \rightarrow 0 $$

then defines isomorphisms $ H ^ { n } ( G, B/A) \simeq H ^ { n + 1 } ( G, A) $, $ n \geq 1 $, and an exact sequence

$$ B ^ {G} \rightarrow \ ( B/A) ^ {G} \rightarrow \ H ^ {1} ( G, A) \rightarrow 0. $$

Therefore, the computation of the $ ( n + 1) $- dimensional cohomology group of $ A $ reduces to calculating the $ n $- dimensional cohomology group of $ B/A $. This device is called dimension shifting.

Dimension shifting enables one to give an axiomatic definition of cohomology groups, namely, they can be defined as a sequence of functors $ A \mapsto H ^ { n } ( G, A) $ from the category of $ G $- modules into the category of Abelian groups forming a cohomological functor and satisfying the condition that $ H ^ { n } ( G, B) = 0 $, $ n \geq 1 $, for every co-induced module $ B $.

The groups $ H ^ { n } ( G, A) $ can also be defined as equivalence classes of exact sequences of $ G $- modules of the form

$$ 0 \rightarrow A \rightarrow M _ {1} \rightarrow \dots \rightarrow M _ {n} \rightarrow \mathbf Z \rightarrow 0 $$

with respect to a suitably defined equivalence relation (see [1], Chapt. 3, 4).

To compute the cohomology groups, the standard resolution of the trivial $ G $- module $ \mathbf Z $ is generally used, in which $ P _ {n} = \mathbf Z [ G ^ {n + 1 } ] $ and, for $ ( g _ {0} \dots g _ {n} ) \in G ^ {n + 1 } $,

$$ d _ {n} ( g _ {0} \dots g _ {n} ) = \ \sum _ {i = 0 } ^ { n } (- 1) ^ {i} ( g _ {0} \dots \widehat{g} _ {i} \dots g _ {n} ), $$

where the symbol $ \widehat{ {}} $ over $ g _ {i} $ means that the term $ g _ {i} $ is omitted. The cochains in $ \mathop{\rm Hom} _ {G} ( P _ {n} , A) $ are the functions $ f ( g _ {0} \dots g _ {n} ) $ for which $ gf ( g _ {0} \dots g _ {n} ) = f ( gg _ {0} \dots gg _ {n} ) $. Changing variables according to the rules $ g _ {0} = 1 $, $ g _ {1} = h _ {1} $, $ g _ {2} = h _ {1} h _ {2} \dots g _ {n} = h _ {1} \dots h _ {n} $, one can go over to inhomogeneous cochains $ f ( h _ {1} \dots h _ {n} ) $. The coboundary operation then acts as follows:

$$ d ^ \prime f ( h _ {1} \dots h _ {n + 1 } ) = \ h _ {1} f ( h _ {2} \dots h _ {n + 1 } ) + $$

$$ + \sum _ {i = 1 } ^ { n } (- 1) ^ {i} f ( h _ {1} \dots h _ {i} h _ {i + 1 } \dots h _ {n + 1 } ) + $$

$$ + (- 1) ^ {n + 1 } f ( h _ {1} \dots h _ {n} ). $$

For example, a one-dimensional cocycle is a function $ f: G \rightarrow A $ for which $ f ( g _ {1} g _ {2} ) = g _ {1} f ( g _ {2} ) + f ( g _ {1} ) $ for all $ g _ {1} , g _ {2} \in G $, and a coboundary is a function of the form $ f ( g) = ga - a $ for some $ a \in A $. A one-dimensional cocycle is also said to be a crossed homomorphism and a one-dimensional coboundary a trivial crossed homomorphism. When $ G $ acts trivially on $ A $, crossed homomorphisms are just ordinary homomorphisms and all the trivial crossed homomorphisms are 0, that is, $ H ^ {1} ( G, A) = \mathop{\rm Hom} ( G, A) $ in this case.

The elements of $ H ^ {1} ( G, A) $ can be interpreted as the $ A $- conjugacy classes of sections $ G \rightarrow F $ in the exact sequence $ 1 \rightarrow A \rightarrow F \rightarrow G \rightarrow 1 $, where $ F $ is the semi-direct product of $ G $ and $ A $. The elements of $ H ^ {2} ( G, A) $ can be interpreted as classes of extensions of $ A $ by $ G $. Finally, $ H ^ {3} ( G, A) $ can be interpreted as obstructions to extensions of non-Abelian groups $ H $ with centre $ A $ by $ G $( see [1]). For $ n > 3 $, there are no analogous interpretations known (1978) for the groups $ H ^ { n } ( G, A) $.

If $ H $ is a subgroup of $ G $, then restriction of cocycles from $ G $ to $ H $ defines functorial restriction homomorphisms for all $ n $:

$$ \mathop{\rm res} : \ H ^ { n } ( G, A) \rightarrow \ H ^ { n } ( H, A). $$

For $ n = 0 $, $ \mathop{\rm res} $ is just the imbedding $ A ^ {G} \subset A ^ {H} $. If $ G/H $ is some quotient group of $ G $, then lifting cocycles from $ G/H $ to $ G $ induces the functorial inflation homomorphism

$$ \inf : \ H ^ { n } ( G/H,\ A ^ {H} ) \rightarrow \ H ^ { n } ( G, A). $$

Let $ \phi : G ^ \prime \rightarrow G $ be a homomorphism. Then every $ G $- module $ A $ can be regarded as a $ G ^ \prime $- module by setting $ g ^ \prime a = \phi ( g ^ \prime ) a $ for $ g ^ \prime \in G ^ \prime $. Combining the mappings $ \mathop{\rm res} $ and $ \inf $ gives mappings $ H ^ { n } ( G ^ \prime , A) \rightarrow H ^ { n } ( G, A) $. In this sense $ H ^ {*} ( G, A) $ is a contravariant functor of $ G $. If $ \Pi $ is a group of automorphisms of $ G $, then $ H ^ { n } ( G, A) $ can be given the structure of a $ \Pi $- module. For example, if $ H $ is a normal subgroup of $ G $, the groups $ H ^ { n } ( H, A) $ can be equipped with a natural $ G/H $- module structure. This is possible thanks to the fact that inner automorphisms of $ G $ induce the identity mapping on the $ H ^ { n } ( G, A) $. In particular, for a normal subgroup $ H $ in $ G $, $ \mathop{\rm Im} \mathop{\rm res} \subset H ^ { n } ( H, A) ^ {G/H} $.

Let $ H $ be a subgroup of finite index in the group $ G $. Using the norm map $ N _ {G/H} : A ^ {H} \rightarrow A ^ {G} $, one can use dimension shifting to define the functorial co-restriction mappings for all $ n $:

$$ \mathop{\rm cores} : \ H ^ { n } ( H, A) \rightarrow \ H ^ { n } ( G, A). $$

These satisfy $ \mathop{\rm cores} \cdot \mathop{\rm res} = ( G: H) $.

If $ H $ is a normal subgroup of $ G $ then there exists the Lyndon spectral sequence with second term $ E _ {2} ^ {p,q} = H ^ { p } ( G/H, H ^ { q } ( H, A)) $ converging to the cohomology $ H ^ { n } ( G, A) $( see [1], Chapt. 11). In small dimensions it leads to the exact sequence

$$ 0 \rightarrow H ^ {1} ( G/H, A ^ {H} ) \mathop \rightarrow \limits ^ { \inf } \ H ^ {1} ( G, A) \mathop \rightarrow \limits ^ { { \mathop{\rm res}} } \ H ^ {1} ( H, A) ^ {G/H} \mathop \rightarrow \limits ^ { { \mathop{\rm tr}} } $$

$$ \mathop \rightarrow \limits ^ { { \mathop{\rm tr}} } H ^ {2} ( G/H, A ^ {H} ) \mathop \rightarrow \limits ^ { \inf } H ^ {2} ( G, A), $$

where $ \mathop{\rm tr} $ is the transgression mapping.

For a finite group $ G $, the norm map $ N _ {G} : A \rightarrow A $ induces the mapping $ \widehat{N} _ {G} : H _ {0} ( G, A) \rightarrow H ^ {0} ( G, A) $, where $ H _ {0} ( G, A) = A/J _ {G} A $ and $ J _ {G} $ is the ideal of $ \mathbf Z G $ generated by the elements of the form $ g - 1 $, $ g \in G $. The mapping $ N _ {G} $ can be used to unite the exact cohomology and homology sequences. More exactly, one can define modified cohomology groups (also called Tate cohomology groups) $ \widehat{H} {} ^ {n } ( G, A) $ for all $ n $. Here

$$ \widehat{H} {} ^ {n } ( G, A) = H ^ { n } ( G, A) \ \ \textrm{ for } n \geq 1, $$

$$ \widehat{H} {} ^ {n } ( G, A) = H _ {- n - 1 } ( G, A) \ \textrm{ for } n \leq - 1, $$

$$ \widehat{H} {} ^ {-} 1 ( G, A) = \mathop{\rm Ker} \widehat{N} _ {G} \ \textrm{ and } \ \widehat{H} _ {0} ( G, A) = \mathop{\rm Coker} \widehat{N} _ {G} . $$

For these cohomology groups there exists an exact cohomology sequence that is infinite in both directions. A $ G $- module $ A $ is said to be cohomologically trivial if $ \widehat{H} {} ^ {n } ( H, A) = 0 $ for all $ n $ and all subgroups $ H \subseteq G $. A module $ A $ is cohomologically trivial if and only if there is an $ i $ such that $ \widehat{H} {} ^ {i} ( H, A) = 0 $ and $ \widehat{H} {} ^ {i + 1 } ( H, A) = 0 $ for every subgroup $ H \subseteq G $. Every module $ A $ is a submodule or a quotient module of a cohomologically trivial module, and this allows one to use dimension shifting both to raise and to lower the dimension. In particular, dimension shifting enables one to define $ \mathop{\rm res} $ and $ \mathop{\rm cores} $( but not $ \inf $) for all integral $ n $. For a finitely-generated $ G $- module $ A $ the groups $ \widehat{H} {} ^ {n } ( G, A) $ are finite.

The groups $ \widehat{H} {} ^ {n } ( G, A) $ are annihilated on multiplication by the order of $ G $, and the mapping $ \widehat{H} ( G, A) \rightarrow \oplus _ {p} \widehat{H} {} ^ {n } ( G _ {p} , A) $, induced by restrictions, is a monomorphism, where now $ G _ {p} $ is a Sylow $ p $- subgroup (cf. Sylow subgroup) of $ G $. A number of problems concerning the cohomology of finite groups can be reduced in this way to the consideration of the cohomology of $ p $- groups. The cohomology of cyclic groups has period 2, that is, $ \widehat{H} {} ^ {n } ( G, A) \simeq \widehat{H} {} ^ {n + 2 } ( G, A) $ for all $ n $.

For arbitrary integers $ m $ and $ n $ there is defined a mapping

$$ \widehat{H} {} ^ {n } ( G, A) \otimes \widehat{H} {} ^ {m} ( G, B) \rightarrow \ \widehat{H} {} ^ {n + m } ( G, A \otimes B), $$

(called $ \cup $- product, cup-product), where the tensor product of $ A $ and $ B $ is viewed as a $ G $- module. In the special case where $ A $ is a ring and the operations in $ G $ are automorphisms, the $ \cup $- product turns $ \oplus _ {n} \widehat{H} {} ^ {n } ( G, A) $ into a graded ring. The duality theorem for $ \cup $- products asserts that, for every divisible Abelian group $ C $ and every $ G $- module $ A $, the $ \cup $- product

$$ \widehat{H} {} ^ {n } ( G, A) \otimes \widehat{H} {} ^ {- n - 1 } ( G, \mathop{\rm Hom} ( A, C)) \rightarrow \ \widehat{H} {} ^ {-} 1 ( G, C) $$

defines a group isomorphism between $ \widehat{H} {} ^ {n } ( G, A) $ and $ \mathop{\rm Hom} ( \widehat{H} {} ^ {- n - 1 } ( G, \mathop{\rm Hom} ( A, C)) , \widehat{H} {} ^ {-} 1 ( G, C)) $( see [2]). The $ \cup $- product is also defined for infinite groups $ G $ provided that $ n, m > 0 $.

Many problems lead to the necessity of considering the cohomology of a topological group $ G $ acting continuously on a topological module $ A $. In particular, if $ G $ is a profinite group (the case nearest to that of finite groups) and $ A $ is a discrete Abelian group that is a continuous $ G $- module, one can consider the cohomology groups of $ G $ with coefficients in $ A $, computed in terms of continuous cochains [5]. These groups can also be defined as the limit $ \lim\limits _ \rightarrow H ^ { n } ( G/U, A ^ {U} ) $ with respect to the inflation mapping, where $ U $ runs over all open normal subgroups of $ G $. This cohomology has all the usual properties of the cohomology of finite groups. If $ G $ is a pro- $ p $- group, the dimension over $ \mathbf Z /p \mathbf Z $ of the first and second cohomology groups with coefficients in $ \mathbf Z /p \mathbf Z $ are interpreted as the minimum number of generators and relations (between these generators) of $ G $, respectively.

See [6] for different variants of continuous cohomology, and also for certain other types of cohomology groups. See Non-Abelian cohomology for cohomology with a non-Abelian coefficient group.

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The norm map $ N _ {G/H} : A ^ {H} \rightarrow A ^ {G} $ is defined as follows. Let $ g _ {1} \dots g _ {k} $ be a set of representatives of $ G/H $ in $ G $. Then $ N _ {G/H} ( a) = g _ {1} a + \dots + g _ {k} a $ in $ A ^ {G} $. For a definition of the transgression relation in general spectral sequences cf. Spectral sequence; for the particular case of group cohomology, where this gives a relation, sometimes called connection, between $ H ^ { n } ( G, A) $ and $ H ^ { n + 1 } ( G/H, A ^ {H} ) $ for all $ n > 0 $, cf. also [a1], Chapt. 11, Par. 9.

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