group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The group cohomology of a group $G$ is the cohomology of its delooping $\mathbf{B}G$. This cohomology classifies group extensions of $G$.
More generally, the group cohomology of an ∞-group $G$ is the cohomology of its delooping $\mathbf{B}G$ and it classifies ∞-group extensions of $G$ or equivalently principal ∞-bundles over $\mathbf{B}G$ (for coefficients with trivial ∞-action) or associated ∞-bundles (for coefficients with nontrivial ∞-action).
More in detail, if $A$ is any abelian group then a cocycle in $G$-group cohomology with coefficients in $A$ regarded as equipped with the trivial action is a morphism
and the cohomology group is the homotopy equivalence classes of this
More generally, $A$ here may be equipped with a $G$-action $\rho \colon A \times G \to A$. There is the the corresponding action groupoid or associated ∞-bundle $\mathbf{B}^n A\sslash G \to \mathbf{B}G$ and now a cocycle is a morphism $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^n A\sslash G$ fitting into a diagram
Equivalently this means that the group cohomology of $G$ with coefficients in an abelian group $A$ with $G$-action $\rho$ is the twisted cohomology of the delooping $\mathbf{B}G$ with respect to the local coefficient ∞-bundle $\mathbf{B}^n A \sslash G$.
All this generalizes to $G$ itself any ∞-group and $\mathbf{B}^n A$ replaced by any $G$-∞-action $\rho \colon V \times G \to G$ in which case a group cocycle is now a morphism
hence a cocycle in the twisted cohomology of $\mathbf{B}G$ with coefficients in the local coefficient ∞-bundle given by the universal $\rho$-associated $V$-fiber ∞-bundle.
In other words, the general notion of group cohomology of $G$ is just the most general notion of cohomology of $\mathbf{B}G$.
This general definition we discuss below in
The special case where $V = \mathbf{B}^n A$ is the $n$-fold delooping of an abelian group is important for applications and also because in this case powerful tools of homological algebra can be applied and group cohomology of ordinary groups may be computed in tersm of of Ext-functors. This we discuss in
Finally one can break this further down into components In
we give some standard formulas for group cohomology in low degree.
We give the general abstract definition in the language of (∞,1)-topos theory / homotopy type theory.
Let $\mathbf{H}$ be an (∞,1)-topos. Let $G \in Grp(\mathbf{H})$ be a group object, an ∞-group, in $\mathbf{H}$. Write $\mathbf{B}G \in \mathbf{H}$ for its delooping.
An ∞-action $\rho : V \times G \to V$ of $G$ on a $V \in \mathbf{H}$ is equivalently, as discussed there, exhibited by a fiber sequence
Regarded as an object in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}G}$ this is the categorical semantics of what in the syntax of homotopy type theory this is the dependent type
Also, $\bar \rho$ is the local coefficient bundle for $G$-group cohomology with coefficients in $V$ equipped with this $G$-∞-action. this means that the group cohomology of $G$ with coefficients in $V$ is the hom in the slice (∞,1)-topos over $\mathbf{H}$ as base (∞,1)-topos
where we denote on the right by $\mathbf{B}G$ the terminal object in the slice $\mathbf{H}_{/\mathbf{B}G}$. Notice that in $\mathbf{H}$ this is the trivial fiber sequence
This is the categorical semantics of what in the syntax of homotopy type theory is
By the discussion at ∞-action, this expresses the ∞-invariants of the conjugation action of $G$ on the morphisms $* \to V$ of the underlying objects. Since the action on the point is trivial, these are just the ∞-invariants of $V$.
In the special case that the $G$-∞-action on $V$ is trivial, the group cohomology is equivalently just the set of connected components of the hom space
In particular if $V = \mathbf{B}^n A$ for $A$ an abelian group, this is
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type | ∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |
Let $G$ be an ordinary group, specifically a group object in a topos $\mathcal{T}$ such that the abelian category $Ab(\mathcal{T})$ has projective object. If $G$ is an ordinary discrete group then this means that in the ambient set theory we assume the axiom of choice or ar least the presentation axiom.
Write then
for the group algebra of $G$ over the integers. Write
for the category $\mathbb{Z}[G]$Mod of modules over $\mathbb{Z}G$.
Notice that a module
is equivalently an abelian group equipped with a $G$-action. This or rather its $n$-fold suspension as a chain complex
is the kind of coefficient for the group cohomology of $G$ to which the following statement applies.
For $A$ a $G$-module, the invariants of $A$ are equivalently the $\mathbb{Z}[G]$-module homomorphisms from $\mathbb{Z}$ equipped with the trivial module structure
This equivalence is natural and hence the contravariant hom functor is equivalently the invariants-functor
By the fully general discussion above, group cohomology of $G$ with coefficients in some $A$ is the homotopy-version of the $G$-invariants? of $A$. In the context of homological algebra and in view of remark 1, this means that it is given by the derived functor of the hom functor out of the trivial $G$-module, hence by the Ext-functor:
For $A$ an abelian group equipped with a $G$-action, the degree-$n$ group cohomology of $G$ with coefficients in $A$ is the $n$th-Ext-group
where on the right $\mathbb{Z} \in \mathbb{Z}G Mod$ is regarded as equipped with the trivial $G$-action.
By the discussion at projective resolution this means more explicitly the following: let $F_\bullet \stackrel{\simeq_{qi}}{\to} \mathbb{Z}$ be a projective resolution $Ch_\bullet(\mathbb{Z}[G]Mod)$ of $\mathbb{Z}$ equipped with the trivial $G$-action, hence an exact sequence
of $\mathbb{Z}[G]$-modules. Let
be the corresponding cochain complex. Then the degree-$n$ group cohomology of $G$ with coefficient in $A$ is the degree-$n$ cochain cohomology of this complex
Give a normal subgroup $K \hookrightarrow G$ the invariants-functor may be decomposed as a composition of the functor that forms $K$-invariants with that which forms $(G/K)$-invariants for the quotient group. This decomposition gives rise to a Grothendieck spectral sequence for the group cohomology. This is called the Hochschild-Serre spectral sequence.
The fully general definition above subsumes various cases that are not always discussed on the same footing in traditional literature. For emphasis we highlight these special cases separately.
We unwind the general abstract definition of group cohomology above in terms of constructions on simplicial sets (for cohomology of discrete groups) and simplicial presheaves (for cohomology of general group objects).
$\,$
Let $G$ be a discrete group and $A$ an abelian discrete group, regarded as equipped with the trivial $G$-action. Let $n \in \mathbb{N}$.
Write $\overline{W}G = G^{\times^\bullet}\in$ sSet for the nerve of the groupoid $*\sslash G$ and write $DK(A[n]) \in$ sSet for the image under the Dold-Kan correspondence of the chain complex which is the $n$-fold suspension of a chain complex of $A$.
Then the degree-$n$ group cohomology of $G$ with coefficients in $A$ is the set
of homomorphisms of simplicial sets modulo simplicial homotopy.
By prop. 1 the group cohomology is
By assumption the relevant (∞,1)-topos here is $\mathbf{H} =$ ∞Grpd, which for emphasis we might write “Disc∞Grpd”. This is presented by the standard model structure on simplicial sets, $Disc\infty Grpd \simeq L_{whe} sSet$.
By the discussion at delooping and at ∞-group, a presentation in sSet, necessarily cofibrant, of the delooping $\mathbf{B}G \in \mathbf{H}$ is the standard bar construction
which is equivalently the nerve of the groupoid $*\sslash G$.
Moreover, by the discussion at Dold-Kan correspondence a presentation of the Eilenberg-MacLane object $\mathbf{B}^n A$ is $DK(A[n]) \in sSet$, and this is a Kan complex and hence a fibrant object in the model category structure.
Therefore by the discussion at derived hom-space we have that $sSet(\overline{W}G, DK(A[n]))$ is a Kan complex which presents the required hom-$\infty$-groupoid.
For low values of $n$ it is useful and easily possible to describe these simplicial maps explicitly. This we turn to now.
A degree-one group cocycle $c$, $[c] \in H^1_{Grp}(G,A)$ is just group homomorphism $G \to A$ – a character of $G$.
We discuss here in detail and in components the special case of degree-2 group cohomology of a discrete group $G$ with coefficients in $A$ an abelian discrete group and regarded as being equipped with the trivial $G$-action.
Let $G$ be a discrete group and $A$ an abelian discrete group, regarded as being equipped with the trivial $G$-action.
Then a group 2-cocycle on $G$ with coefficients in $A$ is a function
such that for all $(g_1, g_2, g_3) \in G \times G \times G$ it satisfies the equation
(called the group 2-cocycle condition).
For $c, \tilde c$ two such cocycles, a coboundary $h \colon c \to \tilde c$ between them is a function
such that for all $(g_1,g_2) \in G \times G$ the equation
holds in $A$, where
is the group 2-coboundary encoded by $h$.
The degree-2 group cohomology is the set
of equivalence classes of group 2-cocycles modulo group 2-coboundaries. This is itself naturally an abelian group under pointwise addition of cocycles in $A$
where
This may be taken as the definition of degree-2 group cohomology (with coefficients in abelian groups and with trivial action). The following proof shows how this follows from the general simplicial presentation of prop. 2.
By prop. 2 we have $H^2_{Grp}(G,A) \simeq \pi_0 sSet(\overline{W}G, DK(A[2]))$.
Notice that fully explicitly the 2-simplices in $\overline{W}G$ are
and the 3-simplices are
Therefore a homomorphism of simplical sets $c \colon \overline{W}G \to DK(A[2])$ is in degree 2 a function
i.e. a map $c : G \times G \to K$. To be a simplicial homomorphism this has to extend to 3-simplices as:
Since there is a unique 3-cell in $DK(A[2])$ whenever the oriented su, of the $A$-labels of the boundary of the corresponding tetrahedron vanishes, the existence of the 3-cell on the right here is precisely the claimed cocycle condition.
A similar argument gives the coboundaries
We discuss now how in the computation of $H^2_{Grp}(G,A)$ one may concentrate on the normalized cocycles.
A group 2-cocycle $c \colon G \times G \to A$, def. 3 is called normalized if
For $c \colon G \times G \to A$ a group 2-cocycle, we have for all $g \in G$ that
The cocycle condition (1) evaluated on
says that
hence that
Similarly the 2-cocycle condition applied to
says that
hence that
Every group 2-cocycle $c \colon G \times G \to A$ is cohomologous to a normalized one, def. 2.
By lemma 1 it is sufficient to show that $c$ is cohomologous to a cocycle $\tilde c$ satisfying $\tilde c(e,e) = e$. Now given $c$, Let $h \colon G \to A$ be given by
Then $\tilde c \coloneqq c + d c$ has the desired property, with (2):
If the groups in question are not plain groups (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 for discrete groups. But this does not in general give the right answer for structured groups: while the simplicial set $\overline{W}G = G^{\times^\bullet}$ is cofibrant in the relevant model category presenting the ambient (∞,1)-topos Disc∞Grpd, for $G$ a structured group the simplicial object given by the same formula is not in general already cofibrant. It needs to be further resolved, instead.
Specifically, 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.
Doing requires more work. This is discussed at
See below at References - For structured groups for pointers to the literature.
If the coefficient group $K$ is nonabelian, its higher deloopings $\mathbf{B}^n K$ to not exist. But n-groupoids approximating this non-existant delooping do exists. Cohomology of $\mathbf{B}G$ with coefficients in these is called nonabelian group cohomology or Schreier theory. See there for more details.
For group cohomology of the group of order 2 $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$ see at Groupprops, Group cohomology of cyclic group Z2
We consider for $G$ a topological group such as
the unitary group $U(n)$;
the special unitary group $SU(n)$;
the symplectic group $Sp(n)$
the corresponding group cohomology in terms of the cohomology of the classifying space/delooping $B G$.
For all $n \in \mathbb{N}$ we have
where $c_i\in H^{2i}$ and $p_i\in H^{4i}$.
The additive group on the Cartesian space $\mathbb{R}^2$ with group operation
carries a degree-2 group cocycle $\omega$ with values in $\mathbb{R}$ given by
The cocycle condition for this is the identity
The group extension classified by this cocycle is the Heisenberg group.
The group cohomology of Galois groups is called Galois cohomology. See there for more details.
We may regard a Lie algebra as an infinitesimal group. Under this perspective Lie algebra cohomology and infinity-Lie algebra cohomology is a special case of (higher) group cohomology. See there for details.
group cohomology
Standard textbook references on group cohomology include
Adem, Milgram, Cohomology of finite groups
Alejandro Adem, Lectures on the cohomology of finite groups (pdf)
Kenneth Brown, Cohomology of Groups , Graduate Texts in Mathematics, 87 (1972)
An introduction to group cohomology of a group $G$ as the cohomology of the classifying space $B G$ is for instance in
Joshua Roberts, Group cohomology: a classifying space perspective (pdf)
Advanced course on classifying spaces and cohomology of groups (ps)
Discussion of the cohomology of discrete groups with abelian coefficients in terms of crossed modules instead of chain complexes (an intermediate step in the Dold-Kan correspondence) is in chapter 12 of
Cohomology of simplicial groups is discussed for instance in
Much of what is called “nonabelian cohomology” in the existing literature concerns the case of nonabelian group cohomology with coefficients in the automorphism 2-group $AUT(H)$ of some possibly nonabelian group $H$.
This is the topic of Schreier theory.
A random example for this use of terminology would be
For a conceptual discussion of nonabelian group cohomology see
Group cocycles classify group extensions. This is often discussed only for 2-cocycles and extensions by ordinary groups. Higher cocycles classify extensions by 2-groups and further by infinity-groups. In the context of crossed complexes, which are models for strict $\infty$-groups, this is discussed for instance in
In
$n$-cocycles on a topological group $G$ with values in a topological abelian group $A$ are considered as continuous maps $G^{\times n}\to A$ (p. 3 ).
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 reductive algebraic groups:
Some references pertaining to the cohomology of the classifying space/delooping $B G$ for $G$ a topological group (characteristic classes).
Cohomology of the classifying space $B G$ for $G$ the topological group underlying a compact Lie group.
Mark Feshbach, Some General Theorems on the Cohomology of Classifying Spaces of Compact Lie Groups Transactions of the American Mathematical Society Vol. 264, No. 1 (Mar., 1981), pp. 49-58 (JSTOR)
Donald Yau, Cohomology of unitary and symplectic groups (pdf)
D. Benson, John Greenlees, Commutative algebra for cohomology rings of classifying spaces of compact Lie groups (pdf)
Eric Friedlander, Guido Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups Commentarii Mathematici Helvetici Volume 59, Number 1, 347-361,
For $U$ the unitary group, the integral cohomology of the classifying space $B U(n)$ consists of the Chern classes, one in every even degree.
The cohomology of $B O(n)$ (orthogonal group) and $B SO(n)$ (special orthogonal group) with coefficients in $\mathbb{Z}_2$ is discussed in (MilnorStasheff, 1974).
The cohomology of $B O(n)$ with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_{2 m}$ was found in
The ring-structure on the cohomology with integer coefficients was given in
E. Brown (Jr.), The cohomology of $B SO(n)$ and $B O(n)$ with integer coefficients Proc. AMS Soc. 85 (1982)
Mark Feshbach, The integral cohomology rings of the classifying spaces of $O(n)$ and $SO(n)$, Indiana Univ. Math. J. 32 (1983), 511–516.
For local coefficients see
Martin Čadek, The cohomology of $B O(n)$ with twisted integer coefficients, J. Math. Kyoto Univ. 39 (1999), no. 2, 277–286 (Euclid)
Richard Lastovecki, Cohomology of $B O(n_1) \times \cdots \times B O(n_m)$ with local integer coefficients Comment.Math.Univ.Carolin. 46,1 (2005)21–32 (pdf)
The Whitehead tower of the orthogonal group starts out with
fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group
Group cohomology of the spin group (cohomology of the classifying space $B Spin$) is discussed in
Emery Thomas, On the cohomology groups of the classifying space for the stable spinor group , Bol. Soc. Mex. (1962), 57 - 69
Masana Harada, Akira Kono, Cohomology mod 2 of the classifying space of $Spin^c(n)$ Publications of the Research Institute for Mathematical Sciences archive Volume 22 Issue 3, Sept. (1986) (web)
Group cohomology of the string group (cohomology of the classifying space $B String$) is discussed in
Cohomology of classifying spaces of exceptional Lie groups.
Some of the above materiel is taken from discussion at