# 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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### "Whenever we have some interesting invariant of spaces, we try to cook up a space that represents this invariant"

In his essay Classifying Spaces Made Easy
Baez writes:
We've seen this trick a couple of times lately, and it's actually a
big theme in homotopy theory: whenever we have some interesting
invariant of ...

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### Is there a classifying space for transitive Lie algebroids? If so, what is it?

Let $M$ be a manifold. The data of a Lie groupoid over $M$ is equivalent to the data of a singular foliation $M=\sqcup\mathcal{F}_i$ and, for each $i$, a map (mod homotopy) $f_i:F_i\to BG_i$ (where $...

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### What are the obstacles for a complex to be a space of loops?

It is known that any space of loops is an H-space. So my question has two parts:
What are the obstacles for a complex to be an H-space? Is there any hope to somehow reasonably classify/characterize ...

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### Chain-level representability of simplicial cohomology

There's a simple construction for a classifying space $K(G,1)$ of a finite group $G$ as an infinite simplicial complex consisting of one $i$-simplex for each possible simplicial $1$-cocycle restricted ...

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### Do classifying spaces determine categories of principal bundles?

If $X$ is a topological space, $G$ a topological group and $E G \to BG$ a universal bundle, isomorphism classes of numerable principal $G$-bundles over $X$ are in one-to-one correspondence with ...

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### What characteristic classes are there?

Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(...

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### Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

I am stuck at a critical step in my master's thesis. If someome can help me out here, I will give appropriate credit.
We know that the data of a 2-group $G$ can be given by a group $\tau_0(G)$ and a 3-...

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### If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?

We know that if $G$ is a topological group that contains a torsion element and $G$ satisfies additional conditions such as $G$ discrete or $G$ finite-dimensional, then the classifying space $BG$ is ...

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### Classifying space $\text{BU}(n)$ from the differential-geometric point of view?

The classifying space of a topological group $G$ is the quotient of $EG$ (a topological space with vanishing homotopy groups) by a proper free action of $G$. The standard notation for the classifying ...

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### Cohomology ring $H^*(BG,\mathbb{Z}_2)$ for $G=\mathbb{Z}_2\ltimes B^2\mathbb{Z}^2$

$$
\newcommand{\Z}{\mathbb{Z}}
$$
Consider the group $G=\Z_2\ltimes B^2\Z^2$ where $\Z_2$ acts on $\Z^2$ by interchanging the two factors of $\Z^2$, hence $\Z_2$ also acts on the Eilenberg-MacLane ...

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### Classifying space of bundles over bundles

Consider a sufficiently nice topological space $X$ as well as topological groups $G$ and $H$. Consider the functor $F$ that associates to $X$ the set of all isomorphism classes of all principal $H$-...

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### Geometric models for the classifying spaces of the spin and string covers of the orthogonal, symplectic, and symmetric groups

$\newcommand{\oUConf}{\widehat{\mathrm{UConf}}}\newcommand{\UConf}{\mathrm{UConf}}\newcommand{\oGr}{\widehat{\mathrm{Gr}}}\newcommand{\Gr}{\mathrm{Gr}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\Spin}{\...

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### Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms

In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...

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### construction of open subsets in classifying space $BG$

Let $G$ be an arbitrary group and we
construct the classifying space $BG$ as quotient
of $EG$ where the latter one is considered in
this discussion to be constructed in natural way as $\Delta$-complex ...

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### Classifying spaces of monoidal categories and deloopings

$\newcommand{\abs}[1]{|#1|}$The classifying space $\abs{\mathcal{C}}$ of a category $\mathcal{C}$ is the geometric realisation $\abs{\mathrm{N}_{\bullet}(\mathcal{C})}$ of its nerve $\mathrm{N}_\...

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### A step in Toda's computation of a Cotor

I am trying to understand a proof from Toda's paper Cohomology of classifying spaces. The step I am stuck on is at page 96. Here is the setup.
We work with cohomology with $\mathbb{F}_2$ coefficients. ...

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### 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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### 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 ...

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### 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 $EG \to BG$ ...

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### $1$-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 ...

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### 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,\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### $E_\infty$-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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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### $p$-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^{\...

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### $(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ 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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### 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 ...

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### 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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### 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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### 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 ...

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### 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 $\...

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### 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 $LBG \sim BLG$?)...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 (...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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) ...

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### 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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### 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{...