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 $G$.

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20
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4answers
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 ...
19
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1answer
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 ...
15
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1answer
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 $G$ ...
13
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3answers
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 $G \to H$, ...
10
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3answers
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 ...
9
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0answers
280 views

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\...
8
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1answer
288 views

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 $\dim X<m$ homotopy classes of mappings $X \to \mathbb{...
8
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1answer
1k views

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 $\Sigma$ ...
8
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1answer
257 views

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?
8
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0answers
196 views

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)...
7
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1answer
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...
7
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1answer
239 views

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) \...
7
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0answers
115 views

Different groups with the same classifying space.

Let $G$ be a topological group and $BG$ its classifying space. From the LES of the universal bundle, we get $\pi_i(BG)\cong\pi_{i-1}(G)$, so given the classifying space, we know all homotopy groups of ...
6
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1answer
660 views

Classifying space $B$SU(n) [closed]

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 ...
6
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1answer
351 views

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 ...
6
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1answer
1k views

The oriented Grassmannian $\widetilde{\text{Gr}}(k,\mathbb{R}^n)$ is simply connected for $n>2$

I saw this result mentioned a lot in many references, but it is always stated as a fact or an exercise. My approach would be to see the oriented grassmannian as the quotient $$\frac{SO(n)}{(SO(k)\...
6
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2answers
441 views

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.
6
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1answer
912 views

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 ...
6
votes
2answers
420 views

Is there a classifying space for covering maps?

It is often said that a sheaf on a topological space $X$ is a "continuously-varying set" over $X$, but the usual definition does not reflect this because a sheaf is not a continuous map from $X$ to ...
6
votes
2answers
225 views

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 ...
6
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1answer
83 views

Weyl group action on maximal torus

This is a classical result in the theory of Lie groups, but I am stuck on trying to understand the action of the Weyl group on the classifying space of the torus. Namely, Let $G$ be a compact ...
6
votes
1answer
331 views

Classifying map of tensor product of two line bundles

We know that $\mathbb{C}P^{\infty}$ is the classifying space of line bundles. Also we know that $\mathbb{C}P^{\infty}$ is an H space that is we have $$\mu: \mathbb{C}P^{\infty} \times \mathbb{C}P^{\...
6
votes
1answer
460 views

How to correct a wrong proof about the Birman exact sequence?

I've given a proof of the exactness of the Birman exact sequence of groups: $$1\to\pi_1(S_{g,r}^s)\to MCG(S_{g,r}^{s+1})\overset{\lambda}{\to} MCG(S_{g,r}^s)\to 1$$ making use of classifying spaces ...
6
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1answer
834 views

Calculation of the first Chern class of the canonical line bundle over $\mathbb{CP}^n$

There are different ways of defining and thereafter calculating the Chern classes. Right now I'm studying from the lecture notes which introduce the first Chern class through the classifying spaces as ...
5
votes
1answer
877 views

Proof that classifying spaces for discrete groups are the Eilenberg-MacLane spaces

A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ by a free action of $G$. The claim is that if $G$ is a discrete group then $EG/G$ is an ...
5
votes
2answers
2k views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
5
votes
2answers
118 views

Relation/Difference between moduli spaces and classifying spaces.

From what I have read so far, a classifying space is a representing object of some (co)representable functor. For example, the $n^\text{th}$ Eilenberg–MacLane space is the classifying space for the $...
5
votes
1answer
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 ...
5
votes
1answer
47 views

Which integral Steifel-Whitney classes are universally $0$?

Let $BO(n)$ denote the classifying space of the orthogonal group $O(n)$. Then there is the well-known ring isomorphism $$H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\dots,w_n] $$ where $w_i \in H^...
5
votes
1answer
140 views

The classifying space of open covers of a manifold

Let $M$ be a manifold of dimension $d$ and let $\mathsf{Disk}_{/M}$ be the category of open subsets of $M$ that are diffeomorphic to $\mathbb{R}^d$ with morphisms given by inclusions. Let $\mathrm{B} ...
5
votes
1answer
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 ...
5
votes
1answer
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 ...
5
votes
1answer
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 $X$...
5
votes
0answers
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 ...
5
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0answers
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 ...
5
votes
0answers
127 views

Proof of the isomorphism $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2).$

I want to prove the following: $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2)$ via the map $[f] \mapsto f^* w_1(\gamma^1)$ where $\gamma^1$ denotes the tautological line bundle over $\mathbb{...
4
votes
3answers
193 views

What is the classifying space, $K(G,1)$ for $\mathbb Z[1/2]$?

I'm wondering about the classifying space for the diadic fractions $\mathbb Z[1/2]$? I have no idea how to begin answering the question, so my apologies for showing a lack of effort. More generally, ...
4
votes
1answer
436 views

Group (co)homology and classyfing spaces

I would like to ask where I can find in the literature the proof of the following fact: the group cohomology of the group $G$ is naturally isomorphic with the ordinary (say singular) cohomology of the ...
4
votes
1answer
241 views

Classification of line bundles by group homomorphisms from the fundamental group to $\mathbb{Z}_2$

Let $X$ be a path-connected manifold. Then by the classification of vector bundles, the collection of all (real) line bundles on $X$ is $$ \text{Vect}^1(X)\cong [X,BO(1)]=[X,\mathbb{R}P^\infty]=[X,B\...
4
votes
1answer
910 views

what is the classifying space of a monoid

In the paper Homology Fibrations and the "Group-Completion". Theorem. McDuff, D.; Segal, G., 1976, the first line: A topological monoid $M$ has a classifying space $BM$. I do not understand this ...
4
votes
1answer
76 views

homotopy equivalent structure groups have h.e. classifying spaces (if $G\simeq H$ then $BG\simeq BH$)

I believe there is a theorem like if $G$ and $H$ are homotopy equivalent topological groups, then $G$-bundles can have their structure group reduced to $H$ or equivalently (?) the classifying space $...
4
votes
1answer
116 views

Cohomology of the Eilenberg-Maclane space $K(\mathbb{R},1)$

Let $\mathbb{R}$ be the reals as an abelian group. A connected topological space $X$ is called an Eilenberg–MacLane space of homotopy type $K(\mathbb{R},1)$, if it has fundamental group isomorphic to $...
4
votes
1answer
149 views

Classification of circle bundles over $\mathbb{RP}^2$

I am trying to understand isomorphism classes of bundles $$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$ These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$. ATTEMPT 1. $...
4
votes
1answer
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 ...
4
votes
1answer
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 ...
4
votes
1answer
145 views

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 $\varepsilon^1$ ...
4
votes
2answers
697 views

A group-like topological monoid is a loop space

I am looking for an elementary reference for the following fact. Let $M$ be a topological monoid and suppose moreover that it is group-like, ie. $\pi_{0}(M)$ is a group. Then the canonical map $M \...
4
votes
1answer
139 views

Which $BG$s are also $K(\pi,n)$s?

As a motivation for the question, note that $\mathbb{C}P^\infty$ is at the same time a $BU(1)$ and a $K(\mathbb{Z},2)$; therefore, $H^2(X,\mathbb{Z})$ classifies complex line bundles on a space $X$. ...
4
votes
1answer
115 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
4
votes
1answer
279 views

Classifying space of the reals

What's the classifying space $B\mathbb{R}$ of the additive group of real numbers provided with the Euclidean topology ? By the extension $\mathbb{Z} \hookrightarrow \mathbb{R} \twoheadrightarrow S^1$...