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]

213 questions

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### 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

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### 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

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### 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

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### 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

**10**

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### 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

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### 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

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### 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

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### 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

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### 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

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### 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

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### 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

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### Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?

Shiquan

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### 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

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### Why is this space aspherical?

Let $X = Y \cup Z$ be a connected, path-connected Hausdorff space. Suppose that $Y$, $Z$, and $Y\cap Z$ are all connected, path-connected, and aspherical, and that the homomorphism $\pi_1(Y\cap Z) \to \pi_1(Y)$ induced by inclusion is…

Magritte

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### Relation between two notions of $BG$

The following is something that's always niggled me a little bit. I usually think about stacks over schemes, so I'm a bit out of my element—I apologize if I say anything silly below.
Let $G$ be a sufficiently topological group (e.g. you can assume a…

Alex Youcis

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