nLab
group cohomology

Idea

As described at cohomology, a notion of cohomology exists for every (infinity,1)-topos $H$: for $X$ and $A$ two objects of $H$,

• an $A$-valued cocycle on $X$ is an object in the ∞-groupoid $H(X,A)$;

• a coboundary between two such cocycles is a morphism in $H(X,A)$

• the cohomology classes are the equivalence classes of $H(X,A)$, so that the cohomology set of $A$-valued cohomology on $X$ is

  $$H(X,A):={\Pi }_{0}H(X,A)={\mathrm{Ho}}_H(X,A),$$H(X,A) := \Pi_0 \mathbf{H}(X,A) = Ho_{\mathbf{H}}(X,A) \,,

  where $H$ is the homotopy category of the (∞,1)-category $H$.

Now, ordinary groupoid nonabelian cohomology is the cohomology obtained for $H=$ ∞Grpd $\simeq$ Top: cohomology on ∞-groupoids (or topological spaces) with coefficients in $\mathrm{\infty }$-groupoids.

The various notions of group cohomology are special cases of this:

• Group cohomology with coefficients in a trivial module is the cohomology in $H=$ ∞Grpd for the case that

  • the domain $X=BG$ is a one-object groupoid being the delooping of a group $G$;

  • the coefficient object is $A=B^{n}K$, the strict ω-groupoid coming from the crossed complex that is concentrated in degree $n$, where it is the abelian group $K$ (see Eilenberg-MacLane object for more details).

  For $n\in \mathbb{N}$ One writes

  $$H_{\mathrm{Grp}}^n(G,K):=H(BG,B^{n}K)={\mathrm{Ho}}_{\mathrm{\infty }\mathrm{Grpd}}(BG,B^{n}K)$$H_{Grp}^n(G,K) := H(\mathbf{B}G, \mathbf{B}^n K) = Ho_{\infty Grpd}(\mathbf{B}G , \mathbf{B}^n K)

  for the degree-$n$ group cohomology of $G$ with values in $K$.

• Nonabelian group cohomology is obtained from this by allowing the coefficient object to be of the form

  • $A=B{K}_{(n)}$, for $K_{(n)}$ an arbitrary n-group

  For instance for $K=\mathrm{AUT}(H)$ the automorphism 2-group of a possibly nonabelian group $H$, nonabelian group cohomology classified $H$-extensions of $G$ (see also gerbe (general idea)).

  Details on this case are at nonabelian group cohomology

• Group cohomology with coefficients in a nontrivial module is in turn twisted cohomology version of nonabelian group cohomology:

  • let $A:=B_{\rho }^{n}K$ be a strict ω-groupoid coming from a crossed complex of the form

    $$[B_{\rho }^{n}K]:=(\cdots \to *\to *\to K\to \cdots \to *\to *\to G\overrightarrow{\to }*)$$[\mathbf{B}^n_\rho K] := ( \cdots \to {*} \to {*} \to K \to \cdots \to {*} \to {}* \to G \stackrel{\to}{\to}{*})

    with the abelian group $K$ in degree $n$ and for

    $$\rho :G\to \mathrm{Aut}(K)$$\rho : G \to Aut(K)

    the action of $G$ on $K$ required by the structure of a crossed complex;

  The $n$th group cohomology of $G$ with coefficients in the module $(K,\rho )$ is the connected components of the $\mathrm{\infty }$-groupoid of sections $\sigma$

  $$\begin{array}{ccc}& & B_{\rho }^{n}K\\ & {}^{\sigma }\nearrow & \downarrow \\ BG& \to & BG\end{array}.$$\array{ && \mathbf{B}^n_\rho K \\ & {}^{\sigma}\nearrow & \downarrow \\ \mathbf{B}G &\to& \mathbf{B}G } \,.

  This is an example of twisted cohomology, as explained there.

Examples

We spell out in detail how the above reproduces the ordinary definition of group cohomology.

For the case of ordinary abelian group cohomology, the context of strict omega-groupoids is in principle fully sufficient, since the domain object $BG$ in that case is a 1-groupoid, clearly a strict infinity-groupoid, as are the abelian coefficient $n$-groupoids $B^{n}K$, manifestly so as images of crossed complexes under the equivalence of crossed complexes with strict omega-groupoids.

So one possibility is to model ${\mathrm{Ho}}_{\mathrm{\infty }\mathrm{Grpd}}$ in this case as the homotopy category induced by the model structure on strict omega-groupoids.

This is, more or less implicitly, the route taken in chapter 12 of

• Ronnie Brown, Phil Higgins, R. Sivera, Nonabelian algebraic topology (pdf, web)

Since every $\mathrm{\infty }$-groupoid is fibrant, this model category category of strict $\mathrm{\infty }$-groupoids is in fact a category of fibrant objects and hence the hom-sets in its homotopy category may be computed as colimits over $\mathrm{\infty }$-anafunctors, namely

$$H(BG,B^{n}K)={\mathrm{colim}}_{Y\stackrel{\sim }{\twoheadrightarrow }BG}\mathrm{Hom}(Y,B^{n}K)$$H(\mathbf{B}G, \mathbf{B}^n K) = colim_{Y \stackrel{\sim}{\twoheadrightarrow} \mathbf{B}G} Hom(Y,\mathbf{B}^n K)

where the colimit is over all strict $\omega$-groupoids $Y$ with $Y\stackrel{\sim }{\twoheadrightarrow }BG$ an acyclic fibration, which here is a k-surjective functor for all $k$.

On the other hand, since also the full model structure is around, this colimit localizes on the cofibrant replacement $Y=F(N(BG))$ of $BG$. But this is nothing but the free strict $\omega$-groupoid on the nerve of $BG$, which is the usual bar resolution of $G$ (see the discusson at nerve):

$$N(BG)=(\cdots \to G\times G\times G\overrightarrow{\overrightarrow{\to }}G\times G\overrightarrow{\to }G\to *)$$N (\mathbf{B}G) = \left( \cdots \to G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \times G \stackrel{\to}{\to} G \to {*} \right)

This is of course nothing but the incarnation of $BG$ as an object in the category of weak infinity-groupoids modeled as Kan complexes.

For instance the 2-cells in $N(BG)$ are of the form

$$N(BG{)}_2=\{\begin{array}{ccc}& & *\\ & {}^{g_1}\nearrow & & {\searrow }^{g_2}\\ *& & \stackrel{g_1{g}_2}{\to }& & *\end{array}\mid g_1,g_2\in G\},$$N(\mathbf{B}G)_2 = \left\{ \left. \array{ && {*} \\ & {}^{g_1}\nearrow && \searrow^{g_2} \\ {*} &&\stackrel{g_1 g_2}{\to}&& {*} } \right| g_1, g_2 \in G \right\} \,,

where the diagram indicates what the face maps on $N(BG)=G\times G$ are.

Accordingly, the 3-cells look like

$$N(BG{)}_3=\{\begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {}^{g_1{g}_2}\nearrow & & {\downarrow }^{g_3}\\ *& & \stackrel{g_1{g}_2{g}_3}{\to }& & *\end{array}\Rightarrow \begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {\searrow }^{g_2{g}_3}& & {\downarrow }^{g_3}\\ *& & \stackrel{g_1{g}_2{g}_3}{\to }& & *\end{array}\mid g_1,g_2,g_3\in G\}.$$N(\mathbf{B}G)_3 = \left\{ \left. \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&{}^{g_1 g_2}\nearrow&& \downarrow^{g_3} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& {*} } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&\searrow^{g_2 g_3}&& \downarrow^{g_3} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& {*} } \right| g_1, g_2, g_3 \in G \right\} \,.

The free strict $\omega$-groupoid on $N(BG)$ has as $n$-morphisms the free $n$-groupoids generated from one $n$-oriental per such $n$-simplex in $N(BG)$.

In chapter 12 of Brown-Higgins-Sivera group cocycles are computed as morphisms out of this cofibrant replacement $F(N(BG))$ of the ordinary 1-groupoid $BG$ in the category of strict omega-groupoids. (Or rather, there the equivalent crossed complexes) are used.

Alternatively, we can pass along the inclusion

$$\mathrm{Str}\omega \mathrm{Grpd}\hookrightarrow \mathrm{\infty }\mathrm{Grpd}\hookrightarrow \mathrm{SSet}$$Str \omega Grpd \hookrightarrow \infty Grpd \hookrightarrow SSet

of strict $\mathrm{\infty }$-groupoids into all $\mathrm{\infty }$-groupoids modeled as Kan complexes and compute the homotopy classes of morphisms there. Every Kan complex is already cofibrant (while of course still also being fibrant), so once the situation is interpreted in SSet we can compute group cohomology in terms of ordinary morphisms $N(BG)\to N(B^{n}K)$ without having to resolve further, without having to resort to anafunctors etc. Of course it is the nerve operation involved both in forming the cofibrant replacement in $\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$ as well as in passing to $\mathrm{SSet}$ that accomplishes the required resolutions in either case.

The upshot of all this is just that the following illustrative pictures may be interpreted either in $\mathrm{Strct}\mathrm{\infty }\mathrm{Grpod}$ or in $\mathrm{SSet}$:

degree-$1$ group cohomology

A degree-one group cocycle $c$, $[c]\in H_{\mathrm{Grp}}^1(G,K)$ is just a functor $c:BG\to BK$. This is a group homomorphism $G\to K$.

Degree-$2$ group cohomology

A degree-2 group cocycle $c$, $[c]\in H_{\mathrm{Grp}}^2(G,K)$ is on 2-cells a map

$$c_2:\left(\begin{array}{ccc}& & *\\ & {}^{g_1}\nearrow & & {\searrow }^{g_2}\\ *& & \stackrel{g_2{g}_1}{\to }& & *\end{array}\right)\mapsto \left(\begin{array}{ccc}& & *\\ & {}^{*}\nearrow & {\Downarrow }^{c(g_1,g_2)}& {\searrow }^{*}\\ *& & \stackrel{*}{\to }& & *\end{array}\right)$$c_2 \;\; : \;\; \left( \array{ && {*} \\ & {}^{g_1}\nearrow && \searrow^{g_2} \\ {*} &&\stackrel{g_2 g_1}{\to}&& {*} } \right) \;\;\; \mapsto \;\;\; \left( \array{ && {*} \\ & {}^{{*}}\nearrow &\Downarrow^{c(g_1,g_2)}& \searrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&& {*} } \right)

i.e. a map $c:G\times G\to K$ such that it extends to a morphism on 3-cells:

$$\begin{array}{rl}{c}_3& :(\begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {}^{g_2{g}_1}\nearrow & & {\downarrow }^{g_3}\\ *& & \stackrel{g_3{g}_2{g}_1}{\to }& & *\end{array}\Rightarrow \begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {\searrow }^{g_3{g}_2}& & {\downarrow }^{g_3}\\ *& & \stackrel{g_3{g}_2{g}_1}{\to }& & *\end{array})\\ & \mapsto \left(\begin{array}{ccccc}*& & \stackrel{*}{\to }& & *\\ {\uparrow }^{*}& {\Downarrow }^{c(g_1,g_2)}& {}^{*}\nearrow & {\Downarrow }^{c(g_2,g_3)}& {\downarrow }^{*}\\ *& & \stackrel{*}{\to }& & *\end{array}\stackrel{\mathrm{Id}}{\Rightarrow }\begin{array}{ccccc}*& & \stackrel{*}{\to }& & *\\ {\uparrow }^{*}& {\Downarrow }^{c(g_1,g_2{g}_3)}& {\searrow }^{*}& {\Downarrow }^{c(g_2,g_3)}& {\downarrow }^{*}\\ *& & \stackrel{*}{\to }& & *\end{array}\right)\end{array}.$$\begin{aligned} c_3 \;\;\; &: \;\;\; \left( \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&{}^{g_2 g_1}\nearrow&& \downarrow^{g_3} \\ {*} &&\stackrel{g_3 g_2 g_1}{\to}&&{*} } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&\searrow^{g_3 g_2}&& \downarrow^{g_3} \\ {*} &&\stackrel{g_3 g_2 g_1}{\to}&&{*} } \right) \\ & \mapsto \left( \array{ {*} &&\stackrel{{*}}{\to}&& {*} \\ \uparrow^{{*}} &\Downarrow^{c(g_1,g_2)} &{}^{{*}}\nearrow&\Downarrow^{c(g_2,g_3)}& \downarrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&&{*} } \;\;\;\; \stackrel{Id}{\Rightarrow} \;\;\;\; \array{ {*} &&\stackrel{{*}}{\to}&& {*} \\ \uparrow^{{*}} &\Downarrow^{c(g_1,g_2 g_3)} &\searrow^{{*}}&\Downarrow^{c(g_2, g_3)}& \downarrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&&{*} } \right) \end{aligned} \,.

Since there are no non-identity 3-morphisms in $B^{2}K$ (non-degenerate 3-cells in $N(B^{2}K)$) the 3-cell on the right is required to be the identity. Since the composition of the 2-cells on the right is their addition (group multiplication in the abelian group $K$) this says that the assignment $c_2:G\times G\to G$ has to be such that

$$c(g_1,g_2)-c(g_1,g_2\cdot g_3)+c(g_1\cdot g_2,g_3)-c(g_2,g_3)=0$$c(g_1, g_2) - c(g_1, g_2 \cdot g_3) + c(g_1 \cdot g_2, g_3) - c(g_2, g_3) = 0

This expresses the commutativity of the above tetrahedra. And it is indeed the ordinary formula for a cocycle in degree-2 group cohomology.

Degree-$3$ group cohomology

similarly…

Structured group cohomology

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

$$(\cdots G\times G\overrightarrow{\overrightarrow{\to }}G\overrightarrow{\to }*)$$\left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \right)

to a simplicial object $N(B^{n}A)$.

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 $(\mathrm{\infty },1)$-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

$$(\cdots G\times G\overrightarrow{\overrightarrow{\to }}G\overrightarrow{\to }*)$$\left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \right)

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.

Topological group cohomology

In

• Jim Stasheff, Continuous cohomology of groups and classifying spaces Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 (web)

$n$-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

• David Wigner, Algebraic cohomology of topological groups Transactions of the American Mathematical Society, volume 178 (1973)(pdf)

A classical reference that considers the cohomology of Lie groups as topological spaces is

• Armand Borel, Homology and cohomology of compact connected Lie groups (pdf)

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).

See also Lie group cohomology.

Lie group cohomology

The content of this subsection is at Lie group cohomology.

Nonabelian group cohomology

If the coefficient group $K$ is nonabelian, its higher deloopings $B^{n}K$ to not exist. But n-groupoids approximating this non-existant delooping do exists. Cohomology of $BG$ with coefficients in these is called nonabelian group cohomology or Schreier theory. See there for more details.

Examples

Cohomology of $U(n)$, $O(n)$, etc.

We consider for $G$ a topological group such as

• the unitary group $U(n)$;

• the special unitary group $\mathrm{SU}(n)$;

• the symplectic group $\mathrm{Sp}(n)$

the corresponding group cohomology in terms of the cohomology of the classifying space/delooping $BG$.

For all $n\in \mathbb{N}$ we have

$$\begin{array}{rl}{H}^{•}(\mathrm{BU}(n);\mathbb{Z})& =\mathbb{Z}[c_1,\cdots ,c_n]\\ H^{•}(\mathrm{BSU}(n);\mathbb{Z})& =\mathbb{Z}[c_2,\cdots ,c_n]\\ H^{•}(\mathrm{BSp}(n);\mathbb{Z})& =\mathbb{Z}[p_1,\cdots ,p_n]\end{array}$$\begin{aligned} H^\bullet(BU(n);\mathbb{Z}) &= \mathbb{Z}[c_1,\cdots,c_n] \\ H^\bullet(BSU(n);\mathbb{Z}) &= \mathbb{Z}[c_2,\cdots,c_n] \\ H^\bullet(BSp(n);\mathbb{Z}) &= \mathbb{Z}[p_1,\cdots,p_n] \end{aligned}

where $c_i\in H^{2i}$ and $p_i\in H^{4i}$.

Heisenberg cocycle

The additive group on the Cartesian space ${ℝ}^2$ with group operation

$$(a,b)+(a\prime ,b\prime )=(a+a\prime ,b+b\prime )$$(a,b) + (a',b') = (a + a' , b + b')

carries a degree-2 group cocycle $\omega$ with values in $ℝ$ given by

$$\omega :((a_1,b_1),(a_2,b_2))\mapsto a_1\cdot b_2.$$\omega : ((a_1,b_1), (a_2,b_2)) \mapsto a_1 \cdot b_2 \,.

The cocycle condition for this is the identity

$$a_1\cdot (b_2+b_3)+a_2\cdot b_3=a_1\cdot b_2+(a_1+a_2)\cdot b_3$$a_1 \cdot (b_2 + b_3) + a_2 \cdot b_3 = a_1 \cdot b_2 + (a_1 + a_2) \cdot b_3

The group extension classified by this cocycle is the Heisenberg group.

Related concepts

• group cohomology

  • nonabelian group cohomology, groupoid cohomology

• group extension

• Lie group cohomology

  • ∞-Lie groupoid cohomology

References

General

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 cohmology of a group $G$ as the cohomology of the classifying space $BG$ is for instance in

• Joshua Roberts, Group cohomology: a classifying space perspective (pdf)

• Advanced course on classifying spaces and cohomology of groups (ps)

Aspects of the general point of view on group cohomology is described for instance in chaper 12 of

• R. Brown, P. Higgins, R. Sivera, Nonabelian algebraic topology (pdf, web)

Cohomology of simplicial groups is discussed for instance in

• Sebastian Thomas, On the second cohomology group of a simplicial group (pdf)

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 $\mathrm{AUT}(H)$ of some possibly nonabelian group $H$.

This is the topic of Schreier theory.

A random example for this use of terminology would be

• Roggenkamp, Scott, Automorphisms and nonabelian cohomology (pdf)

For a conceptual discussion of nonabelian group cohomology see

• John Baez, Mike Shulman, Lectures on n-Categories and Cohomology (arXiv)

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 $\mathrm{\infty }$-groups, this is discussed for instance in

• Johannes Huebschmann, Crossed $n$-fold extensions and group cohomology (web)

On various topological groups

Some references pertaining to the cohomology of the classifying space/delooping $BG$ for $G$ a topological group (characteristic classes).

Compact Lie groups

Cohomology of the classifying space $BG$ for $G$ the topological group underlying a compact Lie group.

• John Milnor, Jim Stasheff, Characteristic Classes , Princeton University Press and University of Tokyo Press, Princeton, New Jersey, (1974).

• 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)

• 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,

The unitary groups $U(n)$

For $U$ the unitary group, the integral cohomology of the classifying space $BU(n)$ consists of the Chern classes, one in every even degree.

The (special) orthogonal groups $O(n)$, $\mathrm{SO}(n)$

The cohomology of $BO(n)$ (orthogonal group) and $B\mathrm{SO}(n)$ (special orthogonal group) with coefficients in ${\mathbb{Z}}_2$ is discussed in (MilnorStasheff, 1974).

The cohomology of $BO(n)$ with coefficients in $\mathbb{Z}$ and ${\mathbb{Z}}_{2m}$ was found in

• Emery Thomas , On the cohomology of the real Grassman complexes and the characteristic classes of the $n$-plane bundle , Trans. Amer. Math. Soc. 96 (1960), 67–89.

The ring-structure on the cohomology with integer coefficients was given in

• E. Brown (Jr.), The cohomology of $B\mathrm{SO}(n)$ and $BO(n)$ with integer coefficients Proc. AMS Soc. 85 (1982)

• Mark Feshbach, The integral cohomology rings of the classifying spaces of $O(n)$ and $\mathrm{SO}(n)$, Indiana Univ. Math. J. 32 (1983), 511–516.

For local coefficients see

• Richard Lastovecki, Cohomology of $BO(n_1)\times \cdots \times BO(n_m)$ with local integer coefficients Comment.Math.Univ.Carolin. 46,1 (2005)21–32 (pdf)

For spin-group, string-group, …

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 $B\mathrm{Spin}$) 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 ${\mathrm{Spin}}^c(n)$ 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 $B\mathrm{String}$) is discussed in

• On the integral cohomology of the seven-connective cover of B O Bull. Austral. Math. Soc. Vol 38 (1988) (pdf)

The exceptional Lie groups

Cohomology of classifying spaces of exceptional Lie groups.

• Akira Kono, Mamoru Mimura, Cohomology mod 3 of the classifying space of the Lie group $E_6$ , Math. Scand. 46 (1980) (pdf)

Loop groups of compact Lie groups

• Daisuke Kishimoto, Akira Kono, Cohomology of the classifying spaces of loop groups (pdf)

Online references

Some of the above materiel is taken from discussion at

• MO, Group cohomology of compact Lie group with integer coeffient