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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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Ring structure of $\Bbb CP^n$ and Chern class.

In this notes Prop 1.71 in nlab, the author aims to compute $H^*(\Bbb C P^n, \Bbb Z)$. I have two confusions. What makes it justified to use $c_1$ as the generator of $H^2(\Bbb CP^n, \Bbb Z)$? There ...
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A description of the map between Grassmanians $G_1^k \rightarrow G_k$,

We know that $G_k:=co\lim G_k(\Bbb C^n)$ is the classifying space for $k$ dimensional complex vector bundles. With total space $E_k = \{(x,v) \, :|, x \in G_k, v \in \Bbb C ^\infty \}$. So we may ...
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Classifying maps and transition functions

Suppose a space $X'$ is obtained from $X$ by attaching an $i$-cell, i.e., $$X' = X \cup_\phi e^i$$ where $\phi: \partial e^i \to X$ is the attaching map. Let $G$ be a structure group, and $P$ be a ...
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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 ...
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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. ...
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Cohomological dimension of a topological group with torsion

I'm interested in a proof (or counter-example) of the following: Let $G$ be a topological group. If $G$ contains torsion then $H^n(BG)\neq 0$ for infinitely many $n$. I know this is true for ...
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Classifying Space of a Category Contractible [duplicate]

My question refers to a statement in Laures' and Szymik's "Grundkurs Topologie" (page 233). Sorry, there exist only a German version. Here the relevant excerpt: My question is why a category having a ...
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Different universal bundle definitions

When reading about classifying spaces and universal (principal) bundles, multiple definitions seem to occur. One has the following possibilities (as far as I can see): The universal $G$-bundle $EG\...
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Loop Space of Classifying Space [duplicate]

Let $\operatorname{U}(n)$ be the group of unitary $n$-matrices. My question is why we have following isomorphism $$\Omega BU(n) \cong U(n)$$ where $BU(n)= E/U(n)$ is the classifying space and $\...
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Milnor construction and deloopings

To construct a classifying space (and universal bundle) of a topological group $G$ one can use the well-known Milnor construction based on the infinite join of $G$. On the other hand one can (at ...
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Group cohomology and singular cohomology of classifying space

Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ ...
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Deducing the statement $I(G) \rightarrow H^*(BG; \Bbb R)$, Chern-weil theory

So I know: Let $G$ be a Lie group Given a smooth principal $G$ bundle $P \rightarrow M$, we may define an algebra homomorphism $$I(G) \rightarrow H^{ev}(M; \Bbb R)$$ where $I(G)$ the graded ...
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Homotopy invariance of pullbacks of principal bundles

This is the proof of lemma 7.2 in a notes by Stephen Mitchell, on classifying spaces. Essentially one step of the proof claims that: Let $p:Y \rightarrow B \times I$ be a principal bundle. If $B$...
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$\mathrm{Οut}(G) = [BG, BG]$?

Let $G$ be a finite group and $BG$ its classifying space. Let $[BG, BG]$ denote the set of self-maps of $BG$ up to homotopy equivalence. Automorphisms of $G$ give such self-maps, and inner ...
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Does the center of $\pi_1(Y)$ act trivial on $[X,Y]_\star$?

Let $X$ and $Y$ be based (and well-pointed) and connected. We have an action of $\pi_1(Y)$ on the set $[X,Y]_\star$ of based homotopy classes of based maps. The quotient is just the set $[X,Y]$ of ...
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Classifying map for the hyperplane bundle

Let $G$ be a topological group and $X$ be a CW complex. Then principal $G$-bundles on $X$ is classified by the classifying space $BG$ in the sense that, given a principal $G$-bundle $P \to X$, there ...
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Classifying space of a monoid with 5 elements

In this paper https://arxiv.org/pdf/math/0202260.pdf, the author proves that $BP$ is homotopic to $S^2$, using homology (In the first lemma of the paper). Well, I tried to use the definition of ...
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Serre spectral sequence of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2=\mathbb{R}P^\infty$

As the title shows, we have a fibration of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2\sim\mathbb{R}P^\infty$. I am trying to check my understanding of Serre spectral sequence with ...
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A 2-group $\mathbb{G}$, so that always exists $0 \to BG_b \to \mathbb{G} \to G_a \to 0?$

If $\mathbb{G}$ is a 2-group, does there always exists a short exact sequence for this $\mathbb{G}$, such that $$ 0 \to BG_b \to \mathbb{G} \to G_a \to 0? $$ where both $G_a$ and $G_b$ are nontrivial ...
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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 ...
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$\mathbf{B}A$ as a 2-group in a long fiber sequence

I am trying to digest the following statement about 2-group: From nlab Observation 4.2: "Let $A \to \hat G$ be the inclusion of a subgroup, exhibiting a central extension $A \to \hat G \to G$ ...
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What is the total space of the universal bundle over $B\mathbb{Q}$?

What is the total space of the universal bundle over $B\mathbb{Q}$, i.e. what is $E\mathbb{Q}$ for $B\mathbb{Q}=E\mathbb{Q}/\mathbb{Q}$ where $B\mathbb{Q}$ is the classifying space? Thoughts/Attempt: ...
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A direct proof of the Chern-Weil isomorphism

Given a principal $G$-bundle $P \to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism $$S^*(g)^G \to H_{DR}^*(M)$$ associated to any invariant polynomial on $...
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1answer
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BO(-) example in Weiss Calculus

I'nm reading Orthogonal Calculus by Michael Weiss, and trying to understand example 2.7, concerning the derivatives of the functor $BO$, which sends a (finite dimensional) inner product space to the ...
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1answer
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“2”-group cohomology

In order to define the cohomology of a topological group G, we first have to introduce the concept of a classifying space. A classifying space BG is the base space of a principal G bundle EG. The EG ...
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Two notions of classifying space

Why is the categorical classifying space for a group G, i.e., geometric realization of the nerve of G(as a category of One object), the same as the topological classifying space for principle G ...
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1answer
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If $H\leq G$, then $BH \to BG$ is a fiber bundle with fiber $G/H$

Suppose $G$ is a topological group and $H\leq G$ is a closed subgroup. The inclusion $H\to G$ induces a map on classifying spaces $BH\to BG$. I've seen in some sources that $BH\to BG$ is actually a ...
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3answers
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Eilenberg–MacLane space $K(\mathbb{Z}_2,n)$

We know that the generalized classifying space / Eilenberg–MacLane space $$ B\mathbb{Z}_2=\mathbb{RP}^{\infty} $$ $$ BU(1)=\mathbb{CP}^{\infty} $$ How do one construct/derive the (infinite ...
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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 ...
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“Eilenberg-MacLane property” for the classifying space of a groupoid

Given a groupoid $G$, its classifying space is defined as the standard geometric realisation of the nerve. My question is: since the classifying space of a group is the only space up to homotopy that ...
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Classifying spaces [related to Eilenberg–MacLane] for explicit group examples

I am interested in knowing the generalized classifying spaces (related to Eilenberg–MacLane space $K(G,n)$ when $G$ is discrete) for explicit group examples ($G=$ the entries given at the top row) ...
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Eilenberg–MacLane space for explicit group examples

I am interested in knowing the explicit answers of Eilenberg–MacLane space $K(G,n)$ for explicit group examples ($G=$ the entries given at the top row) given below. Can someone fill in the Table? $$\...
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How to compute sheaf cohomology of a classifying space?

David Wigner showed that the group cohomology invented by Calvin Moore can be related to sheaf cohomology when the coefficient is discrete by constructing a locally constant sheaf on the classifying ...
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1answer
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Is the classifying space of a non-compact Lie group equal the classifying space of the maximal compact subgroup?

I am currently reading the review article by Stasheff on continuous group cohomology. On page 552, it is said that $BK\simeq BG$, where $K$ is the maximal compact subgroup of the Lie group $G$. I am ...
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1answer
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Classifying space of a category / classifying space of a group

Consider the classifying space of a category as defined in this post. Consider the case of a group $G$. I can't see something pretty simple, that is that this coincides with the 'standard' classifying ...
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1answer
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explicit map of classifying spaces

The general construction of classifying spaces says that for any group homomorphism $H \rightarrow G$ there is a associated map between the classifying spaces $BH \rightarrow BG$. I am trying to see ...
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1answer
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References for classifying spaces and cohomology of classifying spaces

I am interested in understanding classifying spaces and cohomology groups of clasifying spaces to understand characteristic classes as in How do one Introduce characteristic classes question. Any ...
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Cohomology of Classifying Space/Simplicial Manifold

Given a simplicial manifold $\,X^{\mathbf{\cdot}}$ (say a classifying spae $BG$ of a Lie group $G$) we have a differential given by $d_n^*=\sum_i (-1)^id^*_{n,i}\,,$ acting on functions $f_n:X^n\to A\,...
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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, ...
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Higher stacks and BG

I am wondering how one should think of the higher stacks $B^n(G)$? Here is what I mean: The stack $BG$ can be thought of as the quotient of a point by the group $G$ ($BG = */G$) in the proper ...
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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 ...
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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 ...
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1answer
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Atiyah–Jänich for $K_1$

Atiyah–Jänich's theorem says that $$ \left[X\to\mathcal{F}\left(\mathcal{H}\right)\right] = K_0\left(X\right) $$where $\mathcal{H}$ is any separable complex Hilbert space, $\mathcal{F}\left(\mathcal{H}...
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Classifying Space for What is the Infinite Unitary Group?

There is the well known result that $$ \left[X\to Gr_n\left(\mathbb{C}^{\infty}\right)\right] = Vect_n(X)$$ That is, homotopy classes of maps from a topological space $X$ into the $n$-Grassmannian are ...
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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 $X$. ...
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Classifying space of Sporadic groups?

Which are the classifying spaces of the $26$ Sporadic groups? Is there something special in those classifying spaces $BG$ more than the fact that $\pi_1(BG)=G$?
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What is the classifying space of the Abelianization of the fundamental group?

Let $G$ be the free group generated by two elements. Then, it is not hard to see that the classifying space of $G$ is given by gluing two circles at a point. A CW-decomposition of this space consists ...
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Group cohomology homomorphism action under natural embedding $O(n) \times O(k) \to O(n+k)$

Let we have embedding $O(n) \times O(k) \to O(n+k)$ that induces homomorphism on cohomology $H^\cdot(BO(n+k),\mathbb{Z}_2) \to H^\cdot(BO(n),\mathbb{Z}_2) \otimes H^\cdot(BO(k),\mathbb{Z}_2)$ or, in ...
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1answer
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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^{\...
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1answer
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Cohomology morphism $H^*(B\mathbb{Z}_2^n,\mathbb{Z}_2) \to H^*(BS\mathbb{Z}_2^n,\mathbb{Z}_2)$

Let $S\mathbb{Z}_2^n := (\epsilon_1,...,\epsilon_n) : \sum_i \epsilon_i = 0 \mod 2$. So we have embedding $S\mathbb{Z}_2^n \to \mathbb{Z}_2^n$ and, hence, embedding $BS\mathbb{Z}_2^n \to B\mathbb{Z}_2^...