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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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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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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\...
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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. ...
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225 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 ...
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1answer
239 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 ...
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1answer
364 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 ...
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1answer
243 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 ...
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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 ...
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1answer
225 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 ...
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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 ...
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1answer
174 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 ...
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115 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 ...
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1answer
265 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^...
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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-...
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1answer
306 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 ...
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Why is any $G$-resolution a principal $G$-bundle?

In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a ...
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3answers
258 views

Classification of principal bundles

I'm trying to reconcile two results on the classification of principal bundles. First, we have $\mathrm{Prin}_G(X)$ (the equivalence classes of $G$-bundles on $X$) is isomorphic to $H^1(X;G)$ (the ...
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1answer
218 views

Calculating topological index

Consider the space $X=BSL(8,\mathbb{C})/(\mathbb{Z}/2)$. The topological Brauer group of $X$ is given by $Br_{top}(X)=Tor(H^{3}(X;\mathbb{Z}))=\mathbb{Z}/2$. I'm studying concepts of topological ...
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304 views

Cohomology of $BE_8$ and $BSU(2)$

What are the cohomology of the classifying space of $E_8$ group and $SU(2)$ group, $H^*(BE_8;\mathbb{Z})$ and $H^*(BSU(2);\mathbb{Z})$? In the paper http://homepages.math.uic.edu/~bshipley/...
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1answer
127 views

A group of type F that is an extension of type F-by-type F

Let us first recall that a group of type $F$ is a group admitting a compact classifying space. Let $K$ and $Q$ be groups of type $F$. Consider the family $\mathcal{G}(K, Q)$ consisting of groups $G$ ...
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2answers
447 views

Classifying space as the geometric realization of the nerve of $G$ viewed as a small category

Let $C$ be a small category: we define its nerve $(N(C)_k)_k$ as the following simplicial set: $N(C)_0=Ob(C)$ (the set of objects), $N(C)_1=Mor(C)$ (the set of all morphisms) and $N(C)_k$ to be a set ...
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1answer
520 views

Sullivan conjecture for compact Lie groups

Let $G$ be a topological group, and $M$ a connected compact smooth manifold. I'm studying $$ \pi_0 (map (BG,M)). $$ For $G$ a finite group, we know that this is just a point by the Sullivan ...
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1answer
111 views

non-simple local coefficient system on a fibration of classifying spaces

Long story short; I posted in MSE https://math.stackexchange.com/questions/2500745/local-system-of-coefficients-on-a-fibration-of-classyfing-spaces It is well known that if $G$ is a lie group ...
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139 views

Baum Connes conjecture and abstract isomorphism

Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...
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1answer
224 views

Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$

Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω_{n,O}(pt)$ and $Ω_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$? Here are some ...
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1answer
290 views

Bijection homotopy class of maps and homomorphisms of fundamental group

I was told there was a bijection between $[X;BG]$, the set of homotopy types of maps from a topological space $X$ to the classifying space $BG$, and the set of group homomorphisms $Hom(\pi_1 (X), G)$. ...
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151 views

Two definitions of central extensions of simplicial groups

This is a cross-post from MSE. Let $\overline W$ be a classifying space functor on $\mathrm{sGrp}$ with $G$ be a corresponding left adjoint (Kan's loop group). Def 1 : a sequence of maps $A\to E\...
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rational cohomology of classifying spaces of complex reductive Lie groups

I am looking for a reference or an ad-hoc proof of the following fact, which seems to be known to experts: Let $\mathbf{G}$ be a complex algebraic group with maximal (algebraic) torus $\mathbf{T}$ and ...
2
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1answer
121 views

Odd homotopy group of $BU(m)$

Let $U$ be the infinite unitary group $\lim_{n\to\infty}U(n)$. It is well known that, over the rationals, $BU$ is homotopy equivalent to $\prod_{n=1}^\infty K(\mathbb{Z}, 2n)$. Question: Is it true ...
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If $X\times Y$ is homotopy equivalent to a finite-dimensional CW Complex, are $X$ and $Y$ as well?

Is there a space $X$ that is not homotopy equivalent to a finite-dimensional CW complex for which there exists a space $Y$ such that the product space $X\times Y$ is homotopy equivalent to a finite-...
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1answer
884 views

Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space

Let $G$ be a compact, connected Lie group. There is an Atiyah–Hirzebruch spectral sequence $$H^*(BG;K^*) \implies K^*(BG)$$ connecting $H^*BG$, which generally contains torsion, with $K^*BG \cong \...
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2answers
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Group cohomology version of Deligne-Beilinson cohomology

I appreciate Deligne-Beilinson cohomology as a topological cohomology generalization of de Rham cohomology, which concerns the topological structure of manifolds. On the other hand, we know that ...
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1answer
134 views

Intuition for the construction of the space $M_G=EG\times _G M$

Reference: Atiyah & Bott, The moment map and Equivariant cohomology Question: What could be the motivation and the intuition behind the construction of the space $M_G=EG\times _G M$? When I am ...
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2answers
259 views

The mod p cohomologies of classifying spaces of compact Lie groups

I want to do some computation which need the mod p cohomologies of classifying spaces of connected compact Lie groups as input. I need the table for both the simply connected case and the central ...
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0answers
62 views

Classifying space for singular foliations

I apologize if this question is not appropriate for this site. I am aware of the n'th Haefliger groupoid, which acts as a classifying space for codimension-n foliations. Is there something similar for ...
5
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1answer
257 views

Stable homotopy groups of $QX$

If $X$ is a space, we can form $QX=\varinjlim \Omega^n\Sigma^nX$ which is an infinite loop space with homotopy groups $\pi_i(QX)=\pi^{s}_i(X)$ the stable homotopy groups of $X.$ But these are the ...
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143 views

classifying space of algebraic groups

Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a Borel pair $(B,T)$. Let $BG$ be the classifying space of $G$. Can we say that $BG$ is the homotopy colimit of all $BP$ for $P$ a ...
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146 views

fibre sequence of classifying space

I read Steve Mitchell's Notes on principal bundles and classifying spaces. There is a theorem: Let $G$ be any topological group, $H$ an admissible normal subgroup. Then there is a homotopy-fibre ...
2
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1answer
489 views

Infinite Grassmannian does not have the homotopy type of a finite-dimensional complex

Is there a proof that $BO(k)$ is not of the homotopy type of a finite dimensional complex? The Grassmannian $BO(k) := \{ k\text{-dim subspaces of } \mathbb{R}^\infty \}$ classifies the $k$-...
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1answer
240 views

Elementary question: Intuition for equivariant cohomology

A group $G$ acts freely on a manifold $M$, then $H^*_G(M)=H^*(M/G)$. Why is $H^*_G(M)$ a torsion $H^*_G$-module, where $H^*_G=H^*_G(pt)=H^*(BG)$? If $G=T=(S^1)^{n+1}$ is a torus then $H^*_G=H^*...
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Are the unwound thin realization and fat realization homotopy equivalent?

This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces Recall some definitions first: Given a category $\mathcal{C}$ internal in $\...
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1answer
202 views

Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map $f:\mathbb{C}P^2\...
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206 views

The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group

Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...
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1answer
749 views

Homotopy fiber of a map between classifying spaces

I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong): Let $G$ and $H$ be topological groups and $f : G \to H$ be ...
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1answer
243 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 ...
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1answer
211 views

Chern classes of PU(n)-bundles

Let $PU(n) = U(n)/U(1)$ be the projective unitary group and denote by $BPU(n)$ its classifying space. Consider the algebra $M_n(\mathbb{C})$ as an $n^2$-dimensional Hilbert space equipped with the ...
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1answer
226 views

Are there familiar expressions for (finite) joins of finite groups?

Milnor construction of the classifying space of a topological group $G$ is given in terms of infinite joins of $G$. Schwarz then proved that the $k+1$ iterated self join of a group $G$ classifies $G$-...
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160 views

Is every simply connected finite complex the classifying space of a finite monoid

On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...
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2answers
265 views

comparing homology of a space and homology of the classifying space of its fundamental group

Let $X$ be a (connected) closed $n$-manifold and $G=\pi_1(X)$ be the fundamental group of $X$. There is a classifying map $f: X \rightarrow K(G, 1)$ which induces an isomorphism on $\pi_1$. I would ...
6
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2answers
645 views

quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map $$ K(\pi,1)\longrightarrow K(\pi,1)/G. $$ ...