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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4
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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 \...
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1answer
762 views

Universal quotient bundles of $G(2,4)$ and $\mathbb{G}(1,\mathbb{P}^3)$

Let $V$ be an $n$-dimentional complex vector space, $G=G(k,V)$ the Grassmannian of $k$-planes in $V$, and let $\mathcal{V}:=V \otimes \mathcal{O}_G$ the rank-$n$ trivial vector bundle on $G$. We ...
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1answer
318 views

Creating a lift chart for a classification tree

This is likely a simple question but I'm new to data mining techniques and am trying to compare two different predictive models. I've created a logistic regression and a classification tree and would ...
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) \...
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1answer
217 views

Group Extension and Classifying Space

If $$ 0 \to H \to G \to G/H \to 0\ $$ is a group extension, under what conditions do we have a fibration of the form $$ BH \to BG \to B(G/H), $$ where $BG$ is a classifying space of $G$? Suppose ...
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0answers
125 views

What are the non-degenerate faces of $N\mathbb{Z}_2$

I don't understand the nerve construction. For $\mathbb{Z}_2$, Wikipedia says $\bullet \overset{1}\longrightarrow \bullet \overset{1}\longrightarrow \bullet$ should produce a nondegenerate 2-simplex, ...
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1answer
343 views

Classifying space for finite-dimensional torus

Note that $K({\bf Z},1)=S^1$ but $BS^1 = {\bf CP}^\infty$. For finite groups $H$, $G$, $$K(G\times H,1) = K(G,1)\times K(H,1)$$ Does it works for classifying spaces of continuous groups ? As far ...
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 ...
3
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1answer
256 views

Can one apply the classifying space functor $B$ more than once?

For a topological monoid $M$, the classifying space $BM$ is at least a pointed topological space as far as I know. From where to where is the construction $B$ a functor actually? Can I plug in an $...
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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)...
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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 $\mathbb{Z} \hookrightarrow \mathbb{R} \twoheadrightarrow S^1$...
4
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1answer
248 views

Cohomology of $\Bbb CP^{\infty}=BU_1, BU_2,\dots$ : A reference request

Where can I find the calculation of the cohomology rings of the classifying spaces $BU_n,~BO_n$ and $BO,~BU$? I took a class where extensive use was made of these cohomology rings, but I missed the ...
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1answer
866 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 ...
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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 $\Sigma$ ...
5
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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 ...
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0answers
231 views

VC dimension of an oriented hyperplane

What is VC dimension (Vapnik-Chervonenkis dimension) of an oriented hyperplane? I know that VC dimension of set of oriented hyperplanes is $n+1$. Is it the same? I came across this question recently......
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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 ...
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1answer
411 views

Is $K(G,1) = BG$?

Is an Eilenberg-MacLane space $K(G,1)$ the same as the classifying space $BG$ for a group $G$ ?
5
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1answer
365 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
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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 ...
7
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1answer
756 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...
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0answers
149 views

Question involving the Chern character from the book “Fibre Bundles”

On page 311 of Dale Husemöller's book Fibre Bundles in Theorem 11.6 he has the following commutative diagram $$\begin{array} & & K(BG)\\ &\nearrow &\downarrow\\ R(G)&\...
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1answer
461 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 ...
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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 ...
15
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1answer
567 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 $G$ ...
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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 ...

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