group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Lie group cohomology generalizes the notion of group cohomology from discrete groups to Lie groups.
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 Cohomology of Lie groups at ∞-Lie groupoid.
If the groups in question are not group objects internal to Set but groups with extra structure, such as topological groups or Lie groups, 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
to a simplicial object $N (\mathbf{B}^n A)$.
This is what is described above. But this does 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 complexes 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 cofrrect derived hom-space between these objects.
But this changes as we consider groups with extra structure. For a Lie group $G$, the object
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 not both cofibrant and fibrant. To that extent plain morphisms out of this object do not compute the correct derived hom-spaces. Instead, the right definition of structured group cohomology uses the correct fibrant and cofibrant replacements.
In
$n$-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 Ext-functors and comparison with the naive definition is in
A classical reference that considers the cohomology of Lie groups as topological spaces is
A corrected definition of topological group cohomology has been given by Segal
Graeme Segal, Cohomology of topological groups In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377{387. Academic Press, London, (1970).
Graeme Segal, A classifying space of a topological group in the sense of Gel’fand-Fuks. Funkcional. Anal. i Prilozen., 9(2):48{50, (1975).
For $G$ a Lie group and $A$ an abelian Lie group, write
for the naive notion of cohomology on $G$.
A refined definition of Lie group cohomology, denoted $H^n_{diff}(G,A)$, was given in (Brylinski) following (Blanc) and effectively rediscovers Segal’s definition. See section 4 of (Schommer-Pries) for a review and applications.
(Brylinski)
Let $G$ be a Lie group (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
the index sets arrange themselves into a simplicial set $I : [k] \mapsto I_k$;
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
and
Then the differentiable group cohomology of $G$ with coefficients in $A$ is the cohomology of the total complex of the 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 nerves, respectively:
This is definition 1.1 in (Brylinski)
As discussed there, this is equivalent to other definitions, notably to a definition given earlier by Graeme Segal.
There is an evident morphism
obtained by pulling back a globally defined smooth cocycle to a cover.
At ∞-Lie groupoid it is discussed that there is a further refinement
where on the right we have the intrinsic cohomology of ∞-Lie groupoids.
From now on, for definiteness by a Lie group $G$ we mean (following Bry, page3) a 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 $G$.
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
If the coefficient Lie group $A$ is discrete, then Lie group cohomology coindices with the topological cohomology of the classifying space $\mathcal{B}G$
This is Bry, prop. 1.3 and Bry, lemma 1.5.
$H^2_{diff}(G,A)$ classifies central extensions of Lie groups
such that $\pi : \hat G \to G$ is a locally trivial smooth 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.
This is Bry, prop. 1.6 and Bry, lemma 1.5.
For the purpose of this section we specifically conceive Lie group cohomology inside the (∞,1)-topos ∞LieGrpd of ∞-Lie groupoids, as described there.
This is a local (∞,1)-topos, hence in particular an ∞-connected (∞,1)-topos and therefore it admits differential cohomology in an (∞,1)-topos. By the theorem about the differential fiber sequence we have for $G$ a Lie group, $\mathbf{B}G$ its delooping, $\mathbf{B}^{n} U(1)$ the circle n+1-group a long sequence in cohomology
where $H_G(G,-)$ denotes $G$-equivariant cohomology, in that
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
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.
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
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 in transtression with the Killing form invariant polynomial;
on $G \times G$ something like $\langle \theta_1 \wedge \theta_2\rangle$.
(…)
We may naturally regard a Lie group as an ∞-group in the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids. As such, there is an intrinsic (∞,1)-topos-theoretic notion of its cohomology.
For
$G$ a Lie group and $A$ either a discrete group
$G$ a compact Lie group and $A$ the additive Lie group of real numbers $\mathbb{R}$ or the circle group $\mathbb{R}/Z = U(1)$
the intrinsic cohomology of $G$ in Smooth∞Grpd coincides with the refined Lie group cohomology of (Segal)(Brylinski)
This is discussed in detail at Smooth∞Grpd and proven at SynthDiff∞Grpd.
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.
Lie group cohomology
A textbook account of standard material is in chapter V in vol III of
The definition of the refined topological group cohomology in terms of degreewise abelian sheaf cohomology was given in
It was later rediscovered for Lie groups in
following
Relevant background on the theory of abelian sheaf cohomology on simplicial spaces is at the beginning of
More discussion of Lie group cohomology along these lines is in
A discussion of the relation between local Lie group cohomology and Lie algebra cohomology is in