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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Lie group cohomology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory} [[!include group theory - contents]] \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{structured_group_cohomology}{Structured group cohomology}\dotfill \pageref*{structured_group_cohomology} \linebreak \noindent\hyperlink{TopologicalGroupCohomology}{Topological group cohomology}\dotfill \pageref*{TopologicalGroupCohomology} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{SequenceDiffCohomology}{Long sequence in differential cohomology}\dotfill \pageref*{SequenceDiffCohomology} \linebreak \noindent\hyperlink{relation_to_intrinsic_cohomology_of_smooth_groupoids}{Relation to intrinsic cohomology of smooth $\infty$-groupoids}\dotfill \pageref*{relation_to_intrinsic_cohomology_of_smooth_groupoids} \linebreak \noindent\hyperlink{relation_to_lie_algebra_cohomology}{Relation to Lie algebra cohomology}\dotfill \pageref*{relation_to_lie_algebra_cohomology} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \emph{Lie group cohomology} generalizes the notion of [[group cohomology]] from [[discrete group]]s to [[Lie group]]s. From the [[nPOV]] on [[cohomology]], a natural definition is that for $G$ a Lie group, its cohomology is the intrinsic cohomology of its [[delooping]] [[Lie groupoid]] $\mathbf{B}G$ in the [[(∞,1)-topos]] $\mathbf{H} =$ [[?LieGrpd]]. In the literature one finds a sequence of definitions that approach this intrisic topos-theoretic definition. This is discussed below. For a detailed discussion of the relation of this to the intrinsic topos-theoretic definition see the section at [[∞-Lie groupoid]]. \hypertarget{structured_group_cohomology}{}\subsubsection*{{Structured group cohomology}}\label{structured_group_cohomology} If the groups in question are not [[group object]]s internal to [[Set]] but groups with extra structure, such as [[topological group]]s or [[Lie group]]s, then their cohomology has to be understood in the corresponding natural context. In parts of the literature cohomology of structured groups $G$ is defined in direct generalization of the formulas above as homotopy classes of morphisms from the simplicial object \begin{displaymath} \left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \right) \end{displaymath} to a simplicial object $N (\mathbf{B}^n A)$. This is what is described above. But this does \textbf{not} in general give the right answer for structured groups: namely [[cohomology]] is really about homotopy classes of maps in the suitable ambient [[(∞,1)-topos]]. For plain groups as in the above entry, we are working in the $(\infty,1)$-topos [[∞Grpd]]. That may be modeled by the standard [[model structure on simplicial sets]]. In that model structure, all objects a cofibrant and [[Kan complex]]es are fibrant. That means all objects we are dealing with here are both cofibrant and fibrant, and hence the simplicial set of maps between them is the correct [[derived hom-space]] between these objects. But this changes as we consider groups with extra structure. For a [[Lie group]] $G$, the object \begin{displaymath} \left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \right) \end{displaymath} has to be considered as an [[Lie ∞-groupoid]]: an object in the [[model structure on simplicial presheaves]] over a [[site]] such as [[Diff]] or [[CartSp]]. As such it is in general \textbf{not} both cofibrant and fibrant. To that extent plain morphisms out of this object do \textbf{not} compute the correct [[derived hom-space]]s. Instead, the right definition of structured group cohomology uses the correct fibrant and cofibrant replacements. \hypertarget{TopologicalGroupCohomology}{}\subsubsection*{{Topological group cohomology}}\label{TopologicalGroupCohomology} In \begin{itemize}% \item [[Jim Stasheff]], \emph{Continuous cohomology of groups and classifying spaces} Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 (\href{http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540920}{web}) \end{itemize} $n$-cocycles on a topological group $G$ with valzues in a topological abelian group $A$ are considered as [[continuous functions]] $G^{\times n}\to A$ (p. 3 ). (``[[continuous cohomology]]'') A definition in terms of [[derived functor|Ext-functor]]s and comparison with the naive definition is in \begin{itemize}% \item David Wigner, \emph{Algebraic cohomology of topological groups} Transactions of the American Mathematical Society, volume 178 (1973)(\href{http://egg.epfl.ch/~nmonod/bonn/Wigner_1973.pdf}{pdf}) \end{itemize} A classical reference that considers the cohomology of Lie groups as topological spaces is \begin{itemize}% \item [[Armand Borel]], \emph{Homology and cohomology of compact connected Lie groups} (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063923/pdf/pnas01596-0040.pdf}{pdf}) \end{itemize} A corrected definition of topological group cohomology has been given by Segal \begin{itemize}% \item [[Graeme Segal]], \emph{Cohomology of topological groups} In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377\{387. Academic Press, London, (1970). \item [[Graeme Segal]], \emph{A classifying space of a topological group in the sense of Gel'fand-Fuks. Funkcional. Anal. i Prilozen., 9(2):48\{50, (1975).} \end{itemize} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} For $G$ a [[Lie group]] and $A$ an abelian Lie group, write \begin{displaymath} H_{naive}^n(G,A) = \{smooth G^{\times n } \to A\}/_\sim \end{displaymath} for the naive notion of cohomology on $G$. A refined definition of Lie group cohomology, denoted $H^n_{diff}(G,A)$, was given in (\hyperlink{Brylinski}{Brylinski}) following (\hyperlink{Blanc}{Blanc}) and effectively rediscovers Segal's definition. See section 4 of (\hyperlink{Shommer-Pries}{Schommer-Pries}) for a review and applications. \begin{udefn} \textbf{(Brylinski)} Let $G$ be a [[Lie group]] ([[paracompact space|paracompact]]) and $A$ an abelian Lie group. For eack $k \in \mathbb{N}$ we can pick a [[good open cover]] $\{U^{k}_{i} \to G^{\times_k}| i \in I_k\}$ such that \begin{itemize}% \item the index sets arrange themselves into a [[simplicial set]] $I : [k] \mapsto I_k$; \item and for $d_j(U^k_i)$ and $s_j(U^k_i)$ the images of the face and degeneracy maps of $G^{\times\bullet}$ we have \begin{displaymath} d_j(U^k_i) \subset U^{k-1}_{d_j(i)} \end{displaymath} and \begin{displaymath} s_j(U^k_i) \subset U^{k+1}_{s_j(i)} \,. \end{displaymath} \end{itemize} Then the \textbf{differentiable group cohomology} of $G$ with coefficients in $A$ is the cohomology of the total complex of the [[Cech cohomology|Cech]] double complex $C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)$ whose differentials are the alternating sums of the face maps of $G^{\times_\bullet}$ and of the [[Cech nerve]]s, respectively: \begin{displaymath} H^n_{diff}(G,A) := H^n Tot C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A) \end{displaymath} \end{udefn} This is definition 1.1 in (\hyperlink{Brylinski}{Brylinski}) As discussed there, this is equivalent to other definitions, notably to a definition given earlier by [[Graeme Segal]]. There is an evident morphism \begin{displaymath} H^n_{naive}(G,A) \to H^n_{diff}(G,A) \end{displaymath} obtained by pulling back a globally defined smooth cocycle to a cover. At [[∞-Lie groupoid]] it is discussed that there is a further refinement \begin{displaymath} H^n_{diff}(G,A) \to H^n(\mathbf{B}G,A) \,, \end{displaymath} where on the right we have the [[cohomology|intrinsic cohomology]] of [[∞-Lie groupoid]]s. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{general}{}\subsubsection*{{General}}\label{general} From now on, for definiteness by a \emph{Lie group} $G$ we mean (following \href{http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=3}{Bry, page3}) a [[paracompact space|paracompact]] [[Frechet manifold]] equipped with a group structure such that the product and the inverse maps are smooth, and there is an everywhere defined exponential map $exp : \mathfrak{g} \to G$ where $\mathfrak{g}$ is the [[Lie algebra]] of $G$. \begin{uprop} If the coefficient Lie group $A$ is a [[topological vector space]], then the naive group cohomology $H^n(G,A) = \{smooth G^{\times n} \to A\}/_\sim$ coincides with the correct Lie group cohomology \begin{displaymath} (A top.\;vect.\;space) \Rightarrow (H^n_{naive}(G,A) \stackrel{\simeq}{\to} H^n_{diff}(G,A) ) \,. \end{displaymath} If the coefficient Lie group $A$ is discrete, then Lie group cohomology coindices with the topological cohomology of the [[classifying space]] $\mathcal{B}G$ \begin{displaymath} (A discrete) \Rightarrow (H^n(G,A) \simeq H^n(\mathcal{B}G), A) \,. \end{displaymath} \end{uprop} \begin{proof} This is \href{http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=4}{Bry, prop. 1.3} and \href{http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=5}{Bry, lemma 1.5}. \end{proof} \begin{uprop} $H^2_{diff}(G,A)$ classifies central [[group extension|extensions of Lie groups]] \begin{displaymath} A \to \hat G \stackrel{\pi}{\to} G \end{displaymath} such that $\pi : \hat G \to G$ is a locally trivial smooth [[principal bundle|principal]] $A$-fibration. The image of $H^2_{naive}(G,A) \to H^2_{diff}(G,A)$ consists of those central extensions for which is bundle is trivial. \end{uprop} \begin{proof} This is \href{http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=4}{Bry, prop. 1.6} and \href{http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=5}{Bry, lemma 1.5}. \end{proof} \hypertarget{SequenceDiffCohomology}{}\subsubsection*{{Long sequence in differential cohomology}}\label{SequenceDiffCohomology} For the purpose of this section we specifically conceive Lie group cohomology inside the [[(∞,1)-topos]] [[?LieGrpd]] of [[∞-Lie groupoid]]s, as described there. This is a [[local (∞,1)-topos]], hence in particular an [[∞-connected (∞,1)-topos]] and therefore it admits [[schreiber:differential cohomology in an (∞,1)-topos]]. By the theorem about the we have for $G$ a [[Lie group]], $\mathbf{B}G$ its [[delooping]], $\mathbf{B}^{n} U(1)$ the [[∞-Lie groupoid|circle n+1-group]] a long sequence in cohomology \begin{displaymath} \cdots \to H^n_{diff}(G, (1))\to H^n_G(G,U(1)) \to H_{dR, G}^{n+1}(G) \to \cdots \,, \end{displaymath} where $H_G(G,-)$ denotes $G$-equivariant cohomology, in that \begin{displaymath} H^n_G(G,A) := \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A) \end{displaymath} and so on. The point to note is that we may identify [[Lie algebra cohomology]] inside $H^n_{dR,G}(G)$ and may therefore regard the map \begin{displaymath} H^n_G(G,U(1)) \to H_{dR, G}^{n+1}(G) \end{displaymath} as the differentiation map that take a smooth group cocycle to a Lie algebra cocycle. This morphism operates by putting constructing a [[circle n-bundle with connection]] over $\mathbf{B}G$ and then computing its [[curvature]] forms. \textbf{Example} For $\mathfrak{g}$ a [[semisimple Lie algebra]],$G$ its simply connected Lie group, let $\mathbf{B}G \to \mathbf{B}^3 U(1)$ be the group cocycle that classifies the [[string 2-group]]. Its image in $H^3_{dR,G}(G)$ is the curvature of the [[Chern-Simons circle 3-bundle]] over $\mathbf{B}G$. This is represented by a simplicial differential form consisting of two pieces \begin{itemize}% \item on $G$ the form $\langle \theta\wedge \theta \wedge \theta \rangle$ obtained by feeding the [[Maurer-Cartan form]] on $G$ into the canoical Lie algebra cocycle that is [[Chern-Simons element|in transtression]] with the [[Killing form]] [[invariant polynomial]]; \item on $G \times G$ something like $\langle \theta_1 \wedge \theta_2\rangle$. \end{itemize} (\ldots{}) \hypertarget{relation_to_intrinsic_cohomology_of_smooth_groupoids}{}\subsubsection*{{Relation to intrinsic cohomology of smooth $\infty$-groupoids}}\label{relation_to_intrinsic_cohomology_of_smooth_groupoids} We may naturally regard a Lie group as an [[∞-group]] in the [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s. As such, there is an intrinsic [[(∞,1)-topos]]-theoretic notion of its [[cohomology]]. \begin{uprop} For \begin{enumerate}% \item $G$ a [[Lie group]] and $A$ either a [[discrete group]] \item $G$ a [[compact topological space|compact]] [[Lie group]] and $A$ the additive Lie group of [[real numbers]] $\mathbb{R}$ or the [[circle group]] $\mathbb{R}/Z = U(1)$ \end{enumerate} the intrinsic cohomology of $G$ in [[Smooth∞Grpd]] coincides with the refined [[Lie group cohomology]] of (\hyperlink{Segal}{Segal})(\hyperlink{Brylinski}{Brylinski}) \begin{displaymath} H^n_{Smooth\infty Grpd}(\mathbf{B}G, A) \simeq H^n_{diffr}(G,A) \,. \end{displaymath} \end{uprop} This is discussed in detail at [[Smooth∞Grpd]] and proven at [[SynthDiff∞Grpd]]. \hypertarget{relation_to_lie_algebra_cohomology}{}\subsubsection*{{Relation to Lie algebra cohomology}}\label{relation_to_lie_algebra_cohomology} The content of a [[van Est isomorphism]] is that the canonical comparison map from Lie group cohomology to [[Lie algebra cohomology]] (by [[differentiation]]) is an [[isomorphism]] whenever the Lie group is sufficiently connected. \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[group cohomology]] \begin{itemize}% \item [[nonabelian group cohomology]], [[groupoid cohomology]] \item [[∞-group cohomology]] \end{itemize} \item [[group extension]] \item \textbf{Lie group cohomology} \begin{itemize}% \item [[∞-Lie groupoid cohomology]] \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} A textbook account of standard material is in chapter V in vol III of \begin{itemize}% \item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973) \end{itemize} The definition of the refined [[topological group]] cohomology in terms of degreewise [[abelian sheaf cohomology]] was given in \begin{itemize}% \item [[Graeme Segal]], \emph{Cohomology of topological groups} , Symposia Mathematica, Vol IV (1970) (1986?) p. 377 \end{itemize} It was later rediscovered for [[Lie group]]s in \begin{itemize}% \item [[Jean-Luc Brylinski]], \emph{Differentiable Cohomology of Gauge Groups} (\href{http://arxiv.org/abs/math/0011069}{arXiv}) \end{itemize} following \begin{itemize}% \item P. Blanc, \emph{Cohomologie diff\'e{}rentiable et changement de groupes, Ast\'e{}risque vol. 124-125 (1985), pp. 113-130} \end{itemize} Relevant background on the theory of abelian sheaf cohomology on simplicial spaces is at the beginning of \begin{itemize}% \item [[Pierre Deligne]], \emph{Th\'e{}orie de Hodge: III} Publication math\'e{}tematique de l'I.H.E.S, tome 44 (1974), p. 5-77 (\href{http://www.numdam.org/item?id=PMIHES_1974__44__5_0}{numdam}) \end{itemize} More discussion of Lie group cohomology along these lines is in \begin{itemize}% \item [[Chris Schommer-Pries]], \emph{A finite-dimensional String 2-group} (\href{http://arxiv.org/abs/0911.2483}{arXiv:0911.2483}) \end{itemize} A discussion of the relation between \emph{local} Lie group cohomology and [[Lie algebra cohomology]] is in \begin{itemize}% \item S. wierczkowski, \emph{Cohomology of group germs and Lie algebras} Pacific Journal of Mathematics, Volume 39, Number 2 (1971), 471-482. (\href{http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102969572}{pdf}) \end{itemize} [[!redirects differentiable Lie group cohomology]] \end{document}