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| {{about|homology and cohomology ''of'' a group|homology or cohomology groups of a space or other object|Homology (mathematics)}}
| | Nothing to write about myself I think.<br>Enjoying to be a part of wmflabs.org.<br>I just wish Im useful at all<br><br>Here is my blog - Hostgator Coupon; [https://cyclingboutique.zendesk.com/entries/47733040-Hostgator-Status-Twitter https://cyclingboutique.zendesk.com/entries/47733040-Hostgator-Status-Twitter], |
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| In [[abstract algebra]], [[homological algebra]], [[algebraic topology]] and [[algebraic number theory]], as well as in applications to [[group theory]] proper, '''group cohomology''' is a way to study [[group (mathematics)|groups]] using a sequence of [[functor]]s ''H<sup>n</sup>''. The study of fixed points of groups acting on [[module (mathematics)|modules]] and [[quotient module]]s is a motivation, but the cohomology can be defined using various constructions. There is a dual theory, group homology, and a generalization to non-abelian coefficients.
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| These algebraic ideas are closely related to topological ideas. The group cohomology of a group ''G'' can be thought of as, and is motivated by, the [[singular cohomology]] of a suitable space having ''G'' as its [[fundamental group]], namely the corresponding [[Eilenberg–MacLane space]]. Thus, the group cohomology of '''Z''' can be thought of as the singular cohomology of the circle '''S'''<sup>1</sup>, and similarly for '''Z'''/2'''Z''' and '''P'''<sup>∞</sup>('''R''').
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| A great deal is known about the cohomology of groups, including interpretations of low dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.
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| == Motivation ==
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| A general paradigm in [[group theory]] is that a [[group (mathematics)|group]] ''G'' should be studied via its [[group representation]]s. A slight generalization of those representations are the [[G-module|''G''-modules]]: a ''G''-module is an [[abelian group]] ''M'' together with a [[group action]] of ''G'' on ''M'', with every element of ''G'' acting as an [[automorphism]] of ''M''. In the sequel we will write ''G'' multiplicatively and ''M'' additively.
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| Given such a ''G''-module ''M'', it is natural to consider the submodule of [[G-invariant|''G''-invariant]] elements:
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| :<math> M^{G} = \lbrace x \in M \ | \ \forall g \in G : \ gx=x \rbrace. </math>
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| Now, if ''N'' is a submodule of ''M'' (i.e. a subgroup of ''M'' mapped to itself by the action of ''G''), it isn't in general true that the invariants in ''M/N'' are found as the quotient of the invariants in ''M'' by those in ''N'': being invariant 'modulo ''N'' ' is broader. The first group cohomology ''H''<sup>1</sup>(''G'',''N'') precisely measures the difference. The group cohomology functors ''H*'' in general measure the extent to which taking invariants doesn't respect [[exact sequence]]s. This is expressed by a [[long exact sequence]].
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| == Formal constructions ==
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| In this article, ''G'' is a finite group. The collection of all ''G''-modules is a [[category theory|category]] (the morphisms are group homomorphisms ''f'' with the property ''f''(''gx'') = ''g''(''f''(''x'')) for all ''g'' in ''G'' and ''x'' in ''M''). This category of ''G''-modules is an abelian category with enough injectives (since it is isomorphic to the category of all [[module (mathematics)|modules]] over the [[group ring]] '''Z'''[''G'']).
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| Sending each module ''M'' to the group of invariants ''M<sup>G</sup>'' yields a [[functor]] from this category to the category '''Ab''' of abelian groups. This functor is [[left exact functor|left exact]] but not necessarily right exact. We may therefore form its right [[derived functor]]s; their values are abelian groups and they are denoted by ''H<sup>n</sup>''(''G'', ''M''), "the ''n''-th cohomology group of ''G'' with coefficients in ''M''". ''H''<sup>0</sup>(''G'', ''M'') is identified with ''M<sup>G</sup>''.
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| ===Long exact sequence of cohomology===
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| In practice, one often computes the cohomology groups using the following fact: if
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| :<math> 0 \to L \to M \to N \to 0 </math>
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| is a [[short exact sequence]] of ''G''-modules, then a long exact sequence
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| :<math>0\to L^G\to M^G\to N^G\overset{\delta^0}{\to} H^1(G,L) \to H^1(G,M) \to H^1(G,N)\overset{\delta^1}{\to} H^2(G,L)\to \cdots</math>
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| is induced. The maps δ<sup>''n''</sup> are called the "[[connecting homomorphism]]s" and can be obtained from the [[snake lemma]].<ref>Section VII.2 of [[#Reference-Se1979|Serre 1979]]</ref>
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| ===Cochain complexes===
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| Rather than using the machinery of derived functors, the cohomology groups can also be defined more concretely, as follows.<ref>Page 62 of [[#Reference-Mil2008|Milne 2008]] or section VII.3 of [[#Reference-Se1979|Serre 1979]]</ref> For ''n'' ≥ 0, let ''C<sup>n</sup>''(''G'', ''M'') be the group of all [[function (mathematics)|function]]s from ''G<sup>n</sup>'' to ''M''. This is an abelian group; its elements are called the (inhomogeneous) ''' ''n''-cochains'''. The '''coboundary homomorphisms'''
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| :<math>d^n : C^n (G,M) \rightarrow C^{n+1}(G,M) </math>
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| are defined as
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| :<math> \left(d^n\varphi\right)(g_1,\dots,g_{n+1}) = g_1\cdot \varphi(g_2,\dots,g_{n+1}) </math>
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| ::<math> {} + \sum_{i=1}^n (-1)^{i} \varphi(g_1,\dots,g_{i-1},g_i g_{i+1},g_{i+2},\dots,g_{n+1})</math>
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| ::<math> {} + (-1)^{n+1} \varphi(g_1,\dots,g_n) </math>
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| The crucial thing to check here is
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| :<math> d^{n+1} \circ d^n = 0 </math>
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| thus we have a [[cochain complex]] and we can compute cohomology. For ''n'' ≥ 0, define the group of ''' ''n''-cocycles''' as:
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| :<math>Z^n(G,M) = \operatorname{ker}(d^n) </math>
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| and the group of '''''n''-coboundaries''' as
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| :<math> \begin{cases} B^0(G,M) = {0} \\ B^n(G,M)= \operatorname{im}(d^{n-1}), \ n \geq 1 \end{cases} </math>
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| and
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| :<math>H^n(G,M) = Z^n(G,M)/B^n(G,M).\ </math>
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| ===The functors Ext<sup>''n''</sup> and formal definition of group cohomology===
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| Yet another approach is to treat ''G''-modules as modules over the [[group ring]] '''Z'''[''G''], which allows one to define group cohomology via [[Ext functor]]s:
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| :<math>H^{n}(G,M) = \operatorname{Ext}^{n}_{\mathbf{Z}[G]}(\mathbf{Z},M),</math>
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| where ''M'' is a '''Z'''[''G'']-module.
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| Here '''Z''' is treated as the trivial ''G''-module: every element of ''G'' acts as the identity. These Ext groups can also be computed via a projective resolution of '''Z''', the advantage being that such a resolution only depends on ''G'' and not on ''M''. We recall the definition of Ext more explicitly for this context. Let ''F'' be a [[projective resolution|projective '''Z'''[''G'']-resolution]] (e.g. a [[free resolution| free '''Z'''[''G'']-resolution]]) of the trivial '''Z'''[''G'']-module '''Z''': | |
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| :<math> \dots \to F_n\to F_{n-1} \to\dots \to F_0\to \mathbf Z.</math>
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| e.g., one may always take the resolution of group rings, ''F<sub>n</sub>'' = '''Z'''[''G''<sup>''n''+1</sup>], with morphisms
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| :<math> f_n : \mathbf Z[G^{n+1}] \to \mathbf Z[G^n], \quad (g_0, g_1, \dots, g_n) \mapsto \sum_{i=0}^{n} (-1)^i(g_0, \dots, \widehat{g_i}, \dots, g_n). </math>
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| Recall that for '''Z'''[''G'']-modules ''N'' and ''M'', Hom<sub>''G''</sub>(''N'', ''M'') is an [[abelian group]] consisting of '''Z'''[''G'']-homomorphisms from ''N'' to ''M''. Since Hom<sub>''G''</sub>(–, ''M'') is a [[contravariant functor]] and reverses the arrows, applying Hom<sub>''G''</sub>(–, ''M'') to ''F'' termwise produces a [[cochain complex]] Hom<sub>''G''</sub>(''F'', ''M''):
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| :<math> \cdots \leftarrow \operatorname{Hom}_G(F_n,M)\leftarrow \operatorname{Hom}_G(F_{n-1},M)\leftarrow \dots \leftarrow \operatorname{Hom}_G(F_0,M)\leftarrow \operatorname{Hom}_G(\mathbf Z,M).</math>
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| The cohomology groups ''H*''(''G'', ''M'') of ''G'' with coefficients in the module ''M'' are defined as the cohomology of the above cochain complex:
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| :<math> H^n(G,M)=H^n({\rm Hom}_{G}(F,M))</math>
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| for ''n'' ≥ 0.
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| This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of Hom<sub>''G''</sub>(''F'', ''M'') i.e functions φ<sub>''n''</sub>: ''G<sup>n</sup>'' → ''M'' that obey
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| :<math> g\phi_n(g_1,g_2,\ldots, g_n)= \phi_n(gg_1,gg_2,\ldots, gg_n).</math>
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| The coboundary operator δ: ''C<sup>n</sup>'' → ''C''<sup>''n''+1</sup> is now naturally defined by, for example,
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| :<math> \delta \phi_2(g_1, g_2,g_3)= \phi_2(g_2,g_3)-\phi_2(g_1,g_3)+ \phi_2(g_1,g_2).</math>
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| The relation to the coboundary operator ''d'' that was defined in the previous section, and which acts on the "inhomogeneous" cochains <math> \varphi</math>, is given by reparameterizing so that
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| :<math>\begin{align}
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| \varphi_2(g_1,g_2) &= \phi_3(1, g_1,g_1g_2) \\
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| \varphi_3(g_1,g_2,g_3) &= \phi_4(1, g_1,g_1g_2, g_1g_2g_3),
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| \end{align}</math>
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| and so on. Thus
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| :<math>\begin{align}
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| d \varphi_2(g_1,g_2,g_3) &= \delta \phi_3(1,g_1, g_1g_2,g_1g_2g_3)\\
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| & = \phi_3(g_1, g_1g_2,g_1g_2g_3) - \phi_3(1, g_1g_2, g_1g_2g_3) +\phi_3(1,g_1, g_1g_2g_3) - \phi_3(1,g_1,g_1g_2) \\
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| & = g_1\phi_3(1, g_2,g_2g_3) - \phi_3(1, g_1g_2, g_1g_2g_3) +\phi_3(1,g_1, g_1g_2g_3) - \phi_3(1,g_1,g_1g_2) \\
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| & = g_1\varphi_2(g_2,g_3) -\varphi_2(g_1g_2,g_3)+\varphi_2(g_1,g_2g_3) -\varphi_2(g_1,g_2),
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| \end{align}</math>
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| as in the preceding section.
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| ===Group homology ===
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| Dually to the construction of group cohomology there is the following definition of '''group homology''': given a [[G-module|''G''-module]] ''M'', set ''DM'' to be the [[submodule]] [[generating set of a group|generated]] by elements of the form ''g''·''m'' − ''m'', ''g'' ∈ ''G'', ''m'' ∈ ''M''. Assigning to ''M'' its so-called ''[[coinvariant]]s'', the [[quotient group|quotient]]
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| :<math>M_G:=M/DM, \, </math>
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| is a [[right exact functor]]. Its [[left derived functor]]s are by definition the group homology
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| :<math>H_n\left(G,M\right)</math>.
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| Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted "co-" switches:
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| * superscripts correspond to cohomology ''H*'' and invariants ''X<sup>G</sup>'' while
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| * subscripts correspond to homology ''H''<sub>∗</sub> and coinvariants ''X<sub>G</sub>'' := ''X''/''G''.
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| The [[covariant functor]] which assigns ''M<sub>G</sub>'' to ''M'' is isomorphic to the functor which sends ''M'' to '''Z''' ⊗<sub>'''Z'''[''G'']</sub> ''M'', where '''Z''' is endowed with the trivial ''G''-action. Hence one also gets an expression for group homology in terms of the [[Tor functor]]s,
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| :<math>H_n(G,M) = \operatorname{Tor}_n^{\mathbf{Z}[G]}(\mathbf{Z},M)</math>
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| Recall that the tensor product ''N'' ⊗<sub>'''Z'''[''G'']</sub> ''M'' is defined whenever ''N'' is a right '''Z'''[''G'']-module and ''M'' is a left '''Z'''[''G'']-module. If ''N'' is a left '''Z'''[''G'']-module, we turn it into a right '''Z'''[''G'']-module by setting ''a'' ''g'' = ''g''<sup>−1</sup> ''a'' for every ''g'' ∈ ''G'' and every ''a'' ∈ ''N''. This convention allows to define the tensor product ''N'' ⊗<sub>'''Z'''[''G'']</sub> ''M'' in the case where both ''M'' and ''N'' are left '''Z'''[''G'']-modules.
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| Specifically, the homology groups ''H''<sub>''n''</sub>(''G'', ''M'') can be computed as follows. Start with a [[projective resolution]] ''F'' of the trivial '''Z'''[''G'']-module '''Z''', as in the previous section. Apply the covariant functor ⋅ ⊗<sub>'''Z'''[''G'']</sub> ''M'' to ''F'' termwise to get a [[chain complex]] ''F'' ⊗<sub>'''Z'''[''G'']</sub> ''M'':
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| :<math> \dots \to F_n\otimes_{\mathbf{Z}[G]}M\to F_{n-1}\otimes_{\mathbf{Z}[G]}M \to\dots \to F_0\otimes_{\mathbf{Z}[G]}M\to \mathbf Z\otimes_{\mathbf{Z}[G]}M.</math>
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| Then ''H''<sub>''n''</sub>(''G'', ''M'') are the homology groups of this chain complex, <math>H_n(G,M)=H_n(F\otimes_{\mathbf{Z}[G]}M)</math> for ''n'' ≥ 0.
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| Group homology and cohomology can be treated uniformly for some groups, especially [[finite group]]s, in terms of complete resolutions and the [[Tate cohomology group]]s.
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| ==Functorial maps in terms of cochains==
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| ===Connecting homomorphisms===
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| For a short exact sequence 0 → ''L'' → ''M'' → ''N'' → 0, the connecting homomorphisms δ<sup>n</sup> : ''H''<sup>n</sup>(''G'', ''N'') → ''H''<sup>n+1</sup>(''G'', ''L'') can be described in terms of inhomogeneous cochains as follows.<ref>Remark II.1.21 of [[#Reference-Mil2008|Milne 2008]]</ref> If ''c'' is an element of ''H''<sup>n</sup>(''G'', ''N'') represented by an ''n''-cocycle φ : ''G''<sup>n</sup> → N, then δ<sup>n</sup>(''c'') is represented by ''d''<sup>n</sup>(ψ), where ψ is an ''n''-cochain ''G''<sup>n</sup> → M "lifting" φ (i.e. such that φ is the composition of ψ with the surjective map ''M'' → ''N'').
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| ==Non-abelian group cohomology==
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| {{see also|Nonabelian cohomology}}
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| Using the ''G''-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group ''G'' with coefficients in a non-abelian group. Specifically, a ''G''-group is a (not necessarily abelian) group ''A'' together with an action by ''G''.
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| The ''zeroth cohomology of G with coefficients in A'' is defined to be the subgroup
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| :<math>H^{0}(G,A)=A^{G},\,</math>
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| of elements of ''A'' fixed by ''G''.
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| The ''first cohomology of G with coefficients in A'' is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map φ to be a 1-cocycle is that φ(''gh'') = φ(''g'')[''g''φ(''h'')] and <math>\ \varphi\sim \varphi'</math> if there is an ''a'' in ''A'' such that <math>\ a\varphi'(g)=\varphi(g)\cdot(ga)</math>. In general, ''H''<sup>1</sup>(''G'', ''A'') is not a group when ''A'' is non-abelian. It instead has the structure of a [[pointed set]] – exactly the same situation arises in the 0th [[homotopy group]], <math>\ \pi_0(X;x)</math> which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point.
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| Using explicit calculations, one still obtains a ''truncated'' long exact sequence in cohomology. Specifically, let
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| :<math>1\to A\to B\to C\to 1\,</math>
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| be a short exact sequence of ''G''-groups, then there is an exact sequence of pointed sets
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| :<math>1\to A^G\to B^G\to C^G\to H^1(G,A) \to H^1(G,B) \to H^1(G,C).\,</math>
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| ==Connections with topological cohomology theories==
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| <!-- {{refimprovesect|date=September 2007}} -->
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| Group cohomology can be related to topological cohomology theories: to the topological group ''G'' there is an associated [[classifying space]] ''BG''. (If ''G'' has no topology about which we care, then we assign the [[discrete topology]] to ''G''. In this case, ''BG'' is an [[Eilenberg-MacLane space]] K(''G'',1), whose [[fundamental group]] is ''G'' and whose higher [[homotopy group]]s vanish). The ''n''-th cohomology of ''BG'', with coefficients in ''M'' (in the topological sense), is the same as the group cohomology of ''G'' with coefficients in ''M''. This will involve a [[local system|local coefficient system]] unless ''M'' is a trivial ''G''-module. The connection holds because the total space ''EG'' is contractible, so its chain complex forms a projective resolution of ''M''. These connections are explained in {{harv|Adem-Milgram|2004}}, Chapter II.
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| When ''M'' is a ring with trivial ''G''-action, we inherit good properties which are familiar from the topological context: in particular, there is a [[cup product]] under which
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| :<math>H^*(G;M)=\bigoplus_n H^n(G;M)\,</math>
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| is a [[graded module]], and a [[Künneth formula]] applies.
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| If, furthermore, ''M'' = ''k'' is a field, then ''H*''(''G''; ''k'') is a graded ''k''-algebra. In this case, the Künneth formula yields
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| :<math>H^*(G_1\times G_2;k)\cong H^*(G_1;k)\otimes H^*(G_2;k).\,</math>
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| For example, let ''G'' be the group with two elements, under the discrete topology. The real [[projective space]] '''P'''<sup>∞</sup>('''R''') is a classifying space for ''G''. Let ''k'' = '''F'''<sub>2</sub>, the [[field (mathematics)|field]] of two elements. Then
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| :<math>H^*(G;k)\cong k[x],\,</math>
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| a polynomial ''k''-algebra on a single generator, since this is the [[singular cohomology|cellular cohomology]] ring of '''P'''<sup>∞</sup>('''R''').
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| Hence, as a second example, if ''G'' is an [[elementary abelian group|elementary abelian 2-group]] of rank ''r'', and ''k'' = '''F'''<sub>2</sub>, then the Künneth formula gives
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| :<math>H^*(G;k)\cong k[x_1, \ldots, x_r]</math>,
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| a polynomial ''k''-algebra generated by ''r'' classes in ''H''<sup>1</sup>(''G''; ''k'').
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| == Properties ==
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| In the following, let ''M'' be a ''G''-module.
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| ===Functoriality===
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| Group cohomology depends contravariantly on the group ''G'', in the following sense: if ''f'' : ''H'' → ''G'' is a [[group homomorphism]], then we have a naturally induced morphism ''H<sup>n</sup>''(''G'',''M'') → ''H<sup>n</sup>''(''H'',''M'') (where in the latter, ''M'' is treated as an ''H''-module via ''f'').
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| Given a morphism of ''G''-modules ''M''→''N'', one gets a morphism of cohomology groups in the ''H<sup>n</sup>''(''G'',''M'') → ''H<sup>n</sup>''(''G'',''N'').
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| ===''H''<sup>1</sup>===
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| The first cohomology group is the quotient of the so-called ''crossed homomorphisms'', i.e. maps (of sets) ''f'' : ''G'' → ''M'' satisfying ''f''(''ab'') = ''f''(''a'') + ''af''(''b'') for all ''a'', ''b'' in ''G'', modulo the so-called ''principal crossed homomorphisms'', i.e. maps ''f'' : ''G'' → ''M'' given by ''f''(''a'') = ''am''−''m'' for some fixed ''m'' ∈ ''M''. This follows from the definition of cochains above.
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| If the action of ''G'' on ''M'' is trivial, then the above boils down to ''H''<sup>1</sup>(''G'',''M'') = Hom(''G'', ''M''), the group of [[group homomorphism]]s ''G'' → ''M''.
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| ===''H''<sup>2</sup>===
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| If ''M'' is a trivial ''G''-module (i.e. the action of ''G'' on ''M'' is trivial), the second cohomology group ''H''<sup>2</sup>(''G'',''M'') is in one-to-one correspondence with the set of [[Group extension#Central extension|central extension]]s of ''G'' by ''M'' (up to a natural equivalence relation). More generally, if the action of ''G'' on ''M'' is nontrivial, ''H''<sup>2</sup>(''G'',''M'') classifies the isomorphism classes of all [[Group extension|extension]]s of ''G'' by ''M'' in which the induced action of ''G'' on ''M'' by [[inner automorphism]]s agrees with the given action.
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| ===Change of group===
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| The [[Hochschild–Serre spectral sequence]] relates the cohomology of a normal subgroup ''N'' of ''G'' and the quotient ''G/N'' to the cohomology of the group ''G'' (for (pro-)finite groups ''G''). From it, one gets the [[inflation-restriction exact sequence]].
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| ===Cohomology of finite groups is torsion ===
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| The cohomology groups of finite groups are all torsion. Indeed, by [[Maschke's theorem]] the category of representations of a finite group is semi-simple over any field of characteristic zero (or more generally, any field whose characteristic does not divide the order of the group), hence, viewing group cohomology as a derived functor in this abelian category, one obtains that it is zero. The other argument is that over a field of characteristic zero, the group algebra of a finite group is a direct sum of matrix algebras (possibly over division algebras which are extensions of the original field), while a matrix algebra is [[Morita equivalent]] to its base field and hence has trivial cohomology.
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| == History and relation to other fields ==
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| The low dimensional cohomology of a group was classically studied in other guises, long before the notion of group cohomology was formulated in 1943–45. The first theorem of the subject can be identified as [[Hilbert's Theorem 90]] in 1897; this was recast into ''[[Emmy Noether|Noether]]'s equations'' in [[Galois theory]] (an appearance of cocycles for ''H''<sup>1</sup>). The idea of ''[[factor set]]s'' for the [[extension problem]] for groups (connected with ''H''<sup>2</sup>) arose in the work of [[Otto Hölder|Hölder]] (1893), in [[Issai Schur]]'s 1904 study of projective representations, in [[Otto Schreier|Schreier]]'s 1926 treatment, and in [[Richard Brauer]]'s 1928 study of [[simple algebra]]s and the [[Brauer group]]. A fuller discussion of this history may be found in {{harv|Weibel|1999|pp=806–811}}.
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| In 1941, while studying ''H''<sub>2</sub>(''G'', '''Z''') (which plays a special role in groups), [[Heinz Hopf|Hopf]] discovered what is now called '''Hopf's integral homology formula''' {{harv|Hopf|1942}}, which is identical to Schur's formula for the [[Schur multiplier]] of a finite, finitely presented group:
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| :<math> H_2(G,\mathbf{Z}) \cong (R \cap [F, F])/[F, R]</math>,
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| where ''G'' ≅ ''F''/''R'' and ''F'' is a free group.
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| Hopf's result led to the independent discovery of group cohomology by several groups in 1943-45: [[Samuel Eilenberg|Eilenberg]] and [[Saunders Mac Lane|Mac Lane]] in the USA {{Harv|Rotman|1995|p=358}}; Hopf and [[Beno Eckmann|Eckmann]] in Switzerland; and [[Hans Freudenthal|Freudenthal]] in the Netherlands {{harv|Weibel|1999|p=807}}. The situation was chaotic because communication between these countries was difficult during World War II.
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| From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a model for the topological [[classifying space]] ''BG'' as discussed in [[#Connections with topological cohomology theories]] above. In practice, this meant using topology to produce the chain complexes used in formal algebraic definitions. From a module-theoretic point of view this was integrated into the [[Henri Cartan|Cartan]]–[[Samuel Eilenberg|Eilenberg]] theory of [[homological algebra]] in the early 1950s.
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| The application in [[algebraic number theory]] to [[class field theory]] provided theorems valid for general [[Galois extension]]s (not just [[abelian extension]]s). The cohomological part of class field theory was axiomatized as the theory of [[class formation]]s. In turn, this led to the notion of [[Galois cohomology]] and [[étale cohomology]] (which builds on it) {{harv|Weibel|1999|p=822}}. Some refinements in the theory post-1960 have been made, such as continuous cocycles and [[John Tate|Tate]]'s [[Tate cohomology group|redefinition]], but the basic outlines remain the same. This is a large field, and now basic in the theories of [[algebraic group]]s.
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| The analogous theory for [[Lie algebra]]s, called [[Lie algebra cohomology]], was first developed in the late 1940s, by [[Chevalley]]–Eilenberg, and [[Jean-Louis Koszul|Koszul]] {{harv|Weibel|1999|p=810}}. It is formally similar, using the corresponding definition of ''invariant'' for the action of a Lie algebra. It is much applied in [[representation theory]], and is closely connected with the [[BRST quantization]] of [[theoretical physics]].
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| Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of [[spontaneous symmetry breaking]] phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter -- short-range entangled states with symmetry. Short-range entangled states with symmetry are also known as [[symmetry protected topological order|symmetry protected topological states]].
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| ==Notes==
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| <references/>
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| ==References==
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| *{{Citation |last1= Adem |first1= Alejandro | first2=R. James | last2=Milgram |title= Cohomology of Finite Groups |publisher= [[Springer-Verlag]] |year= 2004 |isbn= 3-540-20283-8 |series=Grundlehren der Mathematischen Wissenschaften |mr=2035696 |volume=309 | zbl=1061.20044 | edition=2nd }}
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| * {{Citation | first=Kenneth S. | last1=Brown | title=Cohomology of Groups | authorlink=Kenneth Brown (mathematician) | publisher=[[Springer Verlag]] | year=1972 | isbn=0-387-90688-6 | series=[[Graduate Texts in Mathematics]] | mr=0672956 | volume=87 }}
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| * {{Citation | url=http://www.digizeitschriften.de/index.php?id=166&ID=132355&L=2 | first=Heinz | last=Hopf|title=Fundamentalgruppe und zweite Bettische Gruppe| journal=Comment. Math. Helv.|volume=14 | issue=1|year=1942|pages=257–309 | mr=6510 | doi=10.1007/BF02565622 | zbl=0027.09503 | jfm=68.0503.01 }}
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| * Chapter II of {{Citation | ref=Reference-Mil2008 | last1=Milne | first1=James | year=2007 | title=Class Field Theory | date=5/2/2008 | volume=v4.00 | url=http://www.jmilne.org/math| accessdate=8/9/2008}}
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| * {{Citation | last1=Rotman | first1=Joseph | year=1995 | title=An Introduction to the Theory of Groups | edition=4th | series=[[Graduate Texts in Mathematics]] | volume=148 | publisher=[[Springer-Verlag]] | isbn=978-0-387-94285-8 | mr = 1307623 }}
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| * Chapter VII of {{Citation | ref=Reference-Se1979 | last1=Serre | first1=Jean-Pierre | author1-link = Jean-Pierre Serre | title=Local fields | publisher=Springer-Verlag | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-90424-5 | zbl=0423.12016 | mr=554237 | year=1979 | volume=67 }}
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| * {{Citation | last1=Serre | first1=Jean-Pierre | title=Cohomologie galoisienne | edition=Fifth | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-58002-7 | mr = 1324577 | year=1994 | volume=5}}
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| * {{Citation | last1=Shatz | first1=Stephen S. | title=Profinite groups, arithmetic, and geometry | publisher=[[Princeton University Press]] | location=Princeton, NJ | isbn=978-0-691-08017-8 | mr = 0347778 | year=1972}}
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| * Chapter 6 of {{Weibel IHA}}
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| * {{Citation | last1=Weibel | first1=Charles A. | contribution=History of homological algebra | title=History of Topology |publisher=[[Cambridge University Press]] | isbn=0-444-82375-1 | mr = 1721123 | year=1999 | pages=797–836}}
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| [[Category:Algebraic number theory]]
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| [[Category:Cohomology theories]]
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| [[Category:Group theory]]
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| [[Category:Homological algebra]]
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