# Questions tagged [classifying-spaces]

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 $G$.

175 questions
Filter by
Sorted by
Tagged with
2k views

### 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 ...
2k views

### 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 ...
566 views

### 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 $G$ ...
910 views

### 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 $G \to H$, ...
2k views

### 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 ...
280 views

755 views

### 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...
239 views

412 views

### When is $BG$ a topological group?

Let $G$ be a topological group, then it has a classifying space $BG$. When is $BG$ a topological group? My motivation for asking this question is that I was thinking about the $B$-analogue of my ...
47 views

376 views

### Homotopy-Fibre Sequence of Classifying Spaces

Let $G$ be a topological group and $H$ be a normal subgroup of $G$ (I think $H$ is required to be admissible in the sense that the quotient map $G\to G/H$ is a principal $H$-bundle, am I right?). Then ...
865 views

### Which is the correct universal line bundle: the tautological bundle or its dual?

With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$ In ...
364 views

### Why is the group of covering transformations relative to the quotient map isomorphic to a subgroup of the Fundamental Group?

I'm trying to prove the classification theorem for covering spaces. I've got to the stage where I need to show the following: If $H$ a subgroup of $\Pi_1(X,x_0)$ then $\exists Y$ covering space of $X$...
90 views

### Expressing a term in $BH\to BG \to B(G/H)$ as a function of the other two. Classifying spaces and fibrations.

Let $G$ be a Lie compact group and $H<G$ be a normal subgroup. Then we have an $H$-principal bundle $$H\overset{i}{\to} G \overset{\pi}{\to}G/H.$$ The classifying space construction is functorial ...
60 views

### Source request for $H^*(B\mathrm{TOP},\mathbb{Q})\cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...
127 views

231 views

### map between classifying spaces

Assume that $H$ is a closed subgroup of a Lie Group $G$ and let $EG \rightarrow BG$ the universal bundle. It is well known the fact that there is an induced map $BH \rightarrow BG$ with homotopy ...
88 views

### Lifting the classyfing map and characteristic classes

This question was originally posted on mathoverfow: below the question there were some useful comments however no canonical answer was given. Normally I would offer a bounty for this question but it ...
### Proving that $\xi$ is a trivial vector bundle iff $\xi \oplus \varepsilon^1$ is trivial
I want to prove that $\xi$ is a rank $k$ vector bundle over an $n$ dimensional CW complex $X$ such that $k>n$. Then $\xi$ is trivial iff $\xi \oplus \varepsilon^1$ is trivial. Here $\varepsilon^1$ ...