A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ by a free action of $G$. When $G$ is a discrete group $BG$ has homotopy type of $K(G,1)$ and (co)homology groups of $BG$ coincide with group cohomology of $G$.
Questions tagged [classifying-spaces]
278 questions
25
votes
1 answer
The loop space of the classifying space is the group: $\Omega(BG) \cong G$
Why does delooping the classifying space of a topological group $G$ return a space homotopy equivalent to $G$.
In symbols, why $\Omega(BG) \cong G$, where $G$ is a topological group and $BG$ its classifying space?
ArthurStuart
- 5,102
25
votes
4 answers
Why is the cohomology of a $K(G,1)$ group cohomology?
Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular cohomology of $K(G,1)$, and it is a theorem that…
Akhil Mathew
- 32,250
17
votes
1 answer
What functor does $K(G, 1)$ represent for nonabelian $G$?
For $G$ an abelian group, the Eilenberg-Maclane space $K(G, n)$ represents singular cohomology $H^n(-; G)$ with coefficients in $G$ on the homotopy category of CW-complexes. If $n > 1$, then $G$ must be abelian, but for $n = 1$ there are also…
Qiaochu Yuan
- 468,795
15
votes
3 answers
Is the classifying space a fully faithful functor?
Given a topological group $G$, we can form its classifying space $BG$; suppose we have chosen some specific construction, say the bar construction. $B$ is a functor - given any homomorphism $G \to H$, it induces a continuous map $BG \to BH$.
For…
user98602
12
votes
1 answer
Finite dimensional Eilenberg-Maclane spaces
Given a positive integer $n\geq 2$ and an abelian group $G$, is it possible to find a finite dimensional $K(G,n)$? In case it does, which are some examples?
Thanks...
MBL
- 1,022
12
votes
3 answers
Group structure on Eilenberg-MacLane spaces
How do we put a group structure on $K(G,n)$ that makes it a topological group?
I know that $\Omega K(G,n+1)=K(G,n)$ and since we have a product of loops this makes
$K(G,n)$ into a H-space. But what about being a topological group?
palio
- 11,466
12
votes
1 answer
Classifying space $B$SU(n)
We know that the classifying space
$$
BO(1)=B\mathbb{Z}_2=\mathbb{RP}^{\infty}
$$
$$
BU(1)=\mathbb{CP}^{\infty}
$$
$$
BSU(2)=\mathbb{HP}^{\infty}
$$
How do one construct/derive
$$
BSU(n)=?
$$
Can one explain $B\mathbb{Z}_2$, $BU(1)$, $BSU(n)$ in a…
wonderich
- 6,059
11
votes
0 answers
Characteristic classes of flat $G$-bundles, induced from $G$-invariant forms on $G/K$, where $G$ is a Lie group and $K$ its maximal compact subgroup
I'd like to solve or find a reference for Exercise 2 (a), Chapter 9 in the book Curvature and Characteristic Classes by Johan L. Dupont. https://mathscinet.ams.org/mathscinet/article?mr=500997
Statement of the exercise:
Exercise 2. Let G be a Lie…
Qing Lan
- 91
11
votes
1 answer
Is the classifying space $B^nG$ the Eilenberg-MacLane space $K(G, n)$?
Question: How should we interpret and understand the classifying space $B^nG$? Is that Eilenberg-MacLane space $K(G,n)$?
What one can learn about $BG$ follows the basic: A classifying space $BG$ of a topological group $G$ is the quotient of a…
wonderich
- 6,059
10
votes
1 answer
Homotopical classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$
Which are homotopy classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$ for $n < m$?
In real case, even for any cellular complex $X$ with $\dim X
evgeny
- 3,931
10
votes
1 answer
Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?
I'm looking for a reference for the following result:
If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ is trivial.
The proof supposedly uses homotopy…
Daan Michiels
- 3,811
9
votes
1 answer
A category whose classifying space has nontrivial higher homotopy groups
The classifying space of a category $\scr{C}$ is obtained by taking its nerve $N\scr{C}$, which is the simplicial set defined by
$$
N\mathscr{C}_n:= \mathrm{Fun}([n],\mathscr{C})
$$
and the classifying space is defined as
$$
B\mathscr{C}:=…
CWcx
- 1,391
9
votes
0 answers
Does a group homomorphism up to homotopy induce a map between classifying spaces?
Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces.
If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces $$Bf\colon BH\rightarrow BG.$$
Now suppose $f\colon…
Cuntero
- 91
- 2
8
votes
2 answers
Eilenberg-Maclane space $K(G\rtimes H, 1)$ for a semi-direct product.
We know that $K(G\times H, 1)=K(G,1)\times K(H,1)$. Do we know something like this for a semi-direct product, where $K(G,1)$ denotes the Eilenberg-Maclane space.
user114539
8
votes
0 answers
The classifying space of a gauge group
Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = g^{-1}f(p)g\ (\forall g \in G, p \in P) \rbrace.$$
I want to…
H. Shindoh
- 2,090