Questions tagged [classifying-spaces]

The classifying space BG of a group G classifies principal G-bundles, in that homotopy classes of maps [X, BG] are naturally identified with isomorphism classes of principal G-bundles P ⭢ X.

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6
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1answer
217 views

Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity

Suppose $G$ is a Lie group, with $\pi_0(G)$ not necessarily finite, but might as well assume $G_0$, the connected component of the identity, is compact. In the case that $\pi_0(G)$ is finite, then we ...
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2answers
221 views

Low dimensional integral cohomology of $BPSO(4n)$

Toda has calculated the $\mathbb{Z}/2$‐cohomology ring of $BPSO(4n+2)$, and also gave the simple exceptional calculation of the $\mathbb{Z}/2$‐cohomology of $BPSO(4)$, in Hiroshi Toda, Cohomology of ...
3
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1answer
168 views

Universal bundles over algebraic stacks

$\DeclareMathOperator\el{ell}$In the topological case, given a group $G$, we can define the classifying space $BG$ for principal $G$-bundles and there is a universal $G$-principal bundle $EG \to BG$ ...
5
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1answer
265 views

$1$-cocycle associated to universal $G$-bundle $EG \to BG$

Let $G$ be a (topological) group whose identity element $e_G$ is a nondegenerated basepoint (e.g. if $G$ is a Lie group). Then that's a known fact that there is for every 'nice' enough topological ...
2
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1answer
138 views

Group homology and singular homology

It is well-known that the singular homology of the classifying space of a group $G$ is isomorphic to the group homology of $G$ with coefficients in the trivial $G$-module $\mathbb{Z}$, i.e. $H_*(BG,\...
2
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0answers
34 views

Bounded and Lipschitz De Rham cohomology representatives for pull-backs from classifying spaces

Let $(M,g)$ be a Riemannian manifold with $\pi_1(M)=\Gamma$. Let $\tilde{M}$ be its universal cover, and let $f\colon M\to B\Gamma$ be a classifying map. Given any smooth differential form $\omega$ on ...
8
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225 views

Milnor's universal bundle as a colimit?

I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
2
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0answers
83 views

Classifying space induces a equivalence of categories between $\operatorname{PBun}_G(M)$ and $\Pi(M,BG)$ for finite groups $G$

Let $G$ be a finite group, $BG$ its classifying space and M a manifold. Then it is mentioned in Schweigert and Woike - Orbifold construction for topological field theories (Remark 2.3 d) that there is ...
6
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2answers
207 views

CW-presentation of configurations of points in plane and space

I know from the the theory of Artin groups that, as the $K(\pi,1)$ conjecture is known for Braids group, that using Salvetti complexes we have a fairly explicit finite CW-complex presentation of the ...
3
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1answer
150 views

$E_\infty$-space structure of $B\mathrm{GL}(\mathbb S_{(p)})$

In Geometric Topology - Localization, Periodicity, and Galois Symmetry by Dennis Sullivan, we can read that there is a decomposition $$B\mathrm{SL}(\mathbb S_{(p)})\times K((\mathbf Z_{(p)})^\times)\...
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162 views

Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
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53 views

Definition of universal bundle in category of smooth manifolds

Let $(P, M, G)$ a smooth principal bundle over smooth manifold $M$ with Lie group $G$. We denote by $k_{G}(M)$ the set of all classes of principal G-bundle isomorphisms. If i want to define universal ...
5
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165 views

Applications of Chow rings of classifying spaces in algebraic geometry

For an algebraic group $G$, the Chow ring of its classifying space $BG$, in the sense of Totaro, The Chow ring of a classifying space has been computed in many cases. Is there any interesting ...
4
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1answer
299 views

Delooping of a group object as a one object groupoid

According to https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid we can think of delooping of a group as the one object groupoid $BG$ consisting of a single object and whose ...
9
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2answers
258 views

$p$-completeness of the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$

Let $S$ be a finite $p$-group and $K$ a compact Lie group, in the paper A Segal conjecture for $p$-completed classifying spaces, it is said that the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\...
3
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1answer
145 views

$(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ induced by inclusion is an equivalence

Let $\mathbb{Z}/p^{\infty}$ denote the Prufer group. By $p$-completion properties, it follows that $(B\mathbb Z/p^{\infty})^{\wedge}_p\simeq K(\mathbb{Z}^{\wedge}_p,2)\simeq(BS^1)^{\wedge}_p$. But, ...
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0answers
107 views

Classifying spaces of amalgamated topological monoids

Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
7
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1answer
149 views

Groupoid completion of a topological category vs its homotopy category?

Given a category $\mathcal{C}$ enriched in spaces, we can take the nerve (a simplicial space) and then geometric realization to get a space $B\mathcal{C}$. If we view spaces as $\infty$-groupoids, ...
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235 views

Is every space a classifying space?

Despite a pretty thorough look (I think) I can’t find the answer to the following question: Is every (reasonable?) path connected space weakly equivalent to the classifying space of some topological ...
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251 views

Roadmap to homotopical group theory

I have been lurking here for a long time just enjoying the scenery from my beginner's viewpoint. I have a math.SE account but I think this question is appropriate here based on the nature of the ...
9
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1answer
464 views

About the cohomology of $BG^\delta$. Making a Lie group discrete

Let $G$ be a connected Lie group. Recall that the topological group $G^\delta$ is $G$ endowed with the discrete topology. The inclusion $G^\delta \to G$ induces a map between the classifying spaces $\...
8
votes
1answer
621 views

For which G is BLG weak homotopy equivalent to LBG?

Let $G$ be a (Edit: path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?)...
4
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1answer
434 views

Classifying space BG and contractable space EG

This question is probably not research level that's why I asked it previously on MSE a week ago. Unfortunately it doesn't get much attention there and I thought I would try it here. Choose a ...
12
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1answer
531 views

Classifying space for Thompson's group F?

Let $\mathcal C$ be the free monoidal category generated by an object $X$, and a morphism $X \otimes X \to X$. This category contains exactly two connected components: that of the monoidal unit $1\in ...
16
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350 views

Can the intermediate Chern classes be expressed as Euler classes?

General question: We know that the top Chern class $c_n(\xi)$ of an $n$-dimensional complex vector bundle $\xi$ is its Euler class, while the first Chern class, $c_1(\xi)$, is the Euler class of its ...
7
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234 views

Positive instances of the Eilenberg-Ganea conjecture with families

The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\...
3
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0answers
85 views

Displaying displayed categories

Displayed categories provide a natural categorification from classifying functions to the world of functors. The spirit of the idea is to encode a functor $ F: D \to C $ using a suitable 2-functor (...
6
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2answers
541 views

1-dimensional pure gauge theory

I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie, and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard even in the first section ($n = 1$), which was "trivial but ...
6
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1answer
361 views

Map from a classifying space to a stack

Let $G$ be an algebraic group over a field $k$, and let $BG$ be its classifying space. Let $X$ be a stack over $k$ (e.g. $X$ could be the Picard stack $Pic(S)$, for some scheme $S$). I'm trying to ...
4
votes
1answer
159 views

Group completion of $E_k$-algebras

Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is ...
6
votes
1answer
183 views

A sufficient condition for $\pi_1(\operatorname B\Gamma) = 0$

I am currently reading Ghys and Sergiescu's paper Sur un groupe remarquable de difféomorphisms du cercle (French only I'm afraid), but part of their proof of Corollary 3.4 (page 20 of the pdf) is ...
8
votes
1answer
242 views

When does $BG \to BA$ loop to a homomorphism?

If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...
4
votes
1answer
131 views

Injectivity on rational homtopy implies surjectivity on rational cohomology for classifiying spaces BO and BTOP

Take $BO=\bigcup_{n\geq 1}BO(n)$ and $BTOP=\bigcup_{n\geq 1}BTOP(n)$ where $TOP(n)$ is the set of homeomorphisms of $\mathbb{R}^n$ which send $0$ to $0$ and let $\phi: BO \rightarrow BTOP$ be the map ...
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96 views

Classifying map of a simple circle bundle

Let $\mathbb{K}_0 \subset \mathbb{K}$ be two tori (subtori of $(S^1)^n$). We suppose that $\mathbb{K}_0$ is obtained from $\mathbb{K}$ by the following procedure: consider, on the lie algebra $\text{...
0
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0answers
108 views

Topological groups with homeomorphic underlying spaces, isomorphic abstract groups and homotopy equivalent classifying spaces

Define the classifying space $BG$ of a well-pointed topological group $G$ as the fat realization of the nerve of $G$. Let $G$, $H$ be well-pointed topological groups. Assume that there is a ...
6
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1answer
355 views

Approximation of classifying space BG of compact Lie group G by finite CW complexes

The classifying space of a topological group $G$ is usually constructed as follows: one constructs a sequence of spaces $E_1G$, $E_2G$, $E_3G$, … with a free $G$-action such that the (homotopy) ...
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0answers
99 views

Associated bundle construction and classifying space

Let $\theta:G\rightarrow H$ be a morphism of Lie groups. Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\...
4
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0answers
124 views

Different definitions of a structure on principal bundles

Let $P\to B$ be a principal $G$-bundle and $\psi:H\to G$ a homomorphism of topological groups. A $\psi$-structure for $P$ can be defined in two different ways. I am trying to prove their equivalence. ...
10
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1answer
404 views

What does the classifying space of a topological monoid classify?

The classifying space $BG$ of a topological group $G$ classifies principal $G$ bundles. I have come to appreciate this. I hope the following question is appropriate for MathOverflow: What does the ...
4
votes
1answer
246 views

Trivialization of Pontryagin square on oriented $4$-manifolds

I'm sorry for not clearly stating my question, thanks to Robert Bruner for answering my original question, let me restate it. Let $\mathcal{P}:H^2(-,\mathbb{Z}/2)\to H^4(-,\mathbb{Z}/4)$ be the ...
6
votes
1answer
399 views

Homotopy type of a specific discrete monoid

Consider the discrete monoid $M$ of nondecreasing continuous maps from $[0,1]$ to itself preserving the extremities. Note that the monoid is right-cancellative ($x.z=y.z$ implies $x=y$, since $z$ is ...
5
votes
1answer
266 views

Interesting properties in $…\to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to …$

Let $K(G,n)$ be the Eilenberg Maclane space. Consider the map from $$ K(\mathbb{Z}_2,1) \to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to \dots, $$ It ...
5
votes
1answer
382 views

Classification of Principal $G$ bundles and vector bundles in smooth sense

Suppose $G$ is a Topological group then classification theorem of Principal $G$ bundles says that there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a ...
3
votes
1answer
301 views

Motivation for classifying vector bundles

The statement I am familiar with regarding classification of vector bundles is : Given a paracompact space $X$. The set of isomorphism classes of rank $n$ vector bundles over $X$ is in bijective ...
3
votes
0answers
109 views

Cobordant of 5d manifolds, and the generalization of bordisms

Some of the 5-dimensional manifolds are (co)bordant via oriented cobordism. For example, if I understand correctly, 5-dimensional Dold manifold and Wu manifold are manifolds which are cobordant to ...
4
votes
1answer
304 views

$\mathbb{P}^1$-bundle over compact base

1) Is it possible to construct a $\mathbb{P}^1$-bundle $P\to B$ where $B$ is a proper variety and $P$ is not $\mathbb{P}(V)$ for a rank 2 vector bundle $V\to B$? If we drop the properness assumption ...
3
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0answers
126 views

Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$

Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...
5
votes
1answer
303 views

Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$

I am interested in knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(3)$ or $BO(3)$. Here we can take either: $B^...
5
votes
0answers
129 views

Filling points to a simplex in models for EG

I have a question which is related to higher Dehn functions of groups. I also have a group $G$ with a finite $K(G,1)$. Let us denote by $EG$ the universal cover of this complex. We choose a path-...
8
votes
1answer
435 views

fibrations of classifying spaces - Leray Hirsch Theorem converse

Let $G$ be a topological group and let $H$ be a closed subgroup. Assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces $$G/H \rightarrow BH \rightarrow ...