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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Find $\mathscr X$ such that $\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $\mathscr X(\Omega G)$ for any $G$

For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category of $G$-spaces and the homotopy category of ...
მამუკა ჯიბლაძე's user avatar
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Presentations of smooth manifolds

A presentation of a affine complex variety consists of finitely many polynomials $f_1,...,f_m$ in $\mathbb{C}[x_1,...,x_n]$. A presentation of a projective complex variety consists of finitely many ...
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A numerable G-principal bundle $E \rightarrow B$ is universal iff E is contractible

I am having some trouble understanding aspects of the proof that shows that a numerable $G$-principal bundle $E\rightarrow B$ is a universal $G$-bundle if $E$ is contractible. The proof starts off by ...
Topological Sigma Grindset's user avatar
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Chern-Weil theory in terms of pullback along classifying map

Let $G$ be a semisimple real Lie group and $M$ be a smooth manifold. The map on cohomology $H^{\ast}(BG,\mathbb{C}) \rightarrow H^{\ast}(BT,\mathbb{C})\simeq \mathbb{C}[\mathfrak{h}]$ induced by the ...
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Fibration coming from a group extension

I am trying to solve the following exercise about classifying spaces (5.1.28) in the book "Algebraic K-Theory and its applications" by Rosenberg: Let $$1 \longrightarrow N \longrightarrow G \...
The_Rookie's user avatar
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Plus construction and classifying space

Suppose $G$ is a perfect group, and let us consider $BG$ and its plus construction (We consider $BG$, the classifying space, by the nerve construction). By following Hatcher’s book (Proposition 4.40, ...
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Homotopy long exact sequence and image of connecting map in the center of some group [duplicate]

For context I am working on Weibel's K-book, chapter IV, my question comes from the proof of proposition 1.7. In this he claims that for the long exact sequence of a fibration with acyclic fiber $F\...
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Cellular prism operator

I have a question. Given a nulhomotopic map $f : X \rightarrow Y$, we can define a prism operator $P : C_n(X) \rightarrow C_{n+1}(Y)$ between singular chains. Then do we have a cellular prism operator ...
May's user avatar
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Cohomology of $BU(n)$

The cohomology ring for $BB\mathbb{Z}$ is $\mathbb{Z}[[x]]$, where $x$ lies in degree $2$. My question is about $\mathbb{Z}[[x_1, \dots, x_n]]$. I was trying to find a topological group (ideally an ...
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Why $EG \to EG/G$ is a fibration?

I am trying to prove that for any group $G$, if $EG$ is a contractible CW-complex on which $G$ acts freely, and for which the action of $G$ is cellular and free on the cells of $EG$, then the ...
Antoine Rodrigues's user avatar
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How to understand all characteristic classes as generators of classifying space cohomologies

Context: I'm somewhere in the middle of my study of differential geometry and starting to learn about characteristic classes. I like to have a general intuitive understanding of a concept before ...
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B of a pseudo-group

I am thinking about the classifying space construction. The loop space functor Ω : Grpd₀ ⭢ Grpd sends a space to an internal pseudo group. Elsewhere I have seen people write BΩX for a based connected ...
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multiple for first fractional Pontryagin class

Consider the Whitehead tower of the classifying space of the special orthogonal group $BSO(n)$. $$\cdots\to BString(n)\to BSpin(n)\to BSO(n)$$ There is an obstruction to lifting classifying maps into $...
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Why does the bar construction model the classifying space in both topology and AG?

For a topological group $G$, we can construct the classifying space $BG$ as the geometric realization of the nerve of $G$. I have seen a very similar assertion in the context of algebraic geometry: ...
Hyunbok Wi's user avatar
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what is meant by the space $BSU(2)$,

I am aware of the group, $SU(2)$, but I came across a space, $BSU(2)$, but I have no idea of what this stands for? I noticed that $BSU(2) \cong \mathbb{HP}^\infty$, can any one please explain how is ...
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Diagonal action, universal bundle and Lie subgroup

Let us a consider a Lie group $G$, $S$ be a Lie subgroup and $EG$ the total space of the universal bundle $EG\twoheadrightarrow BG$. Let us consider the diagonal action $d$ of $G$ on the product $EG\...
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Confusion in identification of quasicoherent sheaves on $BG$ and $G$-representations

Let $k$ be some field, say of characteristic 0, and let $G$ be a finite group considered as a discrete algebraic group. Then we get the classifying Deligne-Mumford stack $BG$ and its etale ...
Sergey Guminov's user avatar
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Principal bundles: Reductions and lifts of classifying maps

I am currently trying to understand how reductions of structure groups of principal bundles correspond to lifts of classifying maps. Some definitions: Let $\rho: H \rightarrow G$ be a homomorphism of ...
Daniel Ruhland's user avatar
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How do I view the tangent bundle of 2-sphere as pullback from the universal bundle?

I am trying to visualize that any vector bundle is a pullback from the Tautological bundle of Grassmannian. If the tangent bundle of $S^2$ is not a simple example, is there a/the simplest example? The ...
Alex's user avatar
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What are the loops in BG?

I'm having trouble with the following elementary(?) thing, but its confusing me alot. For a category $\mathcal{C}$, we can define the classifying space in terms of nerve. Similarly, for a topological ...
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Comparing the Segal and Milnor Models for BG

In Segal's paper "Classifying Spaces and Spectral Sequences" he claims that Milnor's join construction for the classifying space of a topological group is homeomorphic to taking the ...
Andrew Davis's user avatar
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Is the universal cover of a classifying space always $\mathbb{R}^n$ if it is finite dimensional?

If $G$ is a discrete group, and if $BG$ can be represented by a (finite dimensional) manifold, is it always true that $EG$ can be chosen to be $\mathbb{R}^n$, for some $n?$ I'm guessing the answer is ...
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Operad structure on the finite covers of a space

Let $(\Sigma_\bullet)$ be the collection of the symmetric groups. These have a structure of an operad in $\mathsf{Set}$ (it is in fact the operad $Ass$ encoding monoids). The collection of the ...
Nicolas Guès's user avatar
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Long fibre sequence of classifying spaces of finite abelian groups and equivalence of the algebraic and the geometric definition of the Bockstein map

$X$ a manifold. A short exact sequence of finite abelian groups $$ 1\rightarrow A_1\rightarrow A_2\rightarrow A_3\rightarrow 1 $$ induces a long exact sequence of cohomology groups of $X$ with ...
Andrea Antinucci's user avatar
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2 answers
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Group structure on the classifying space of abelian groups

Let $G$ be a topological abelian group. Its classifying space $BG$ is (at least sometimes) also a topological group. For example if $G$ is a finite abelian group, higher Eilenberg-MacLane spaces are $...
Andrea Antinucci's user avatar
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Simple explicit example of higher Eilenberg-MacLane space

Given an Abelian group $G$ and a positive integer $n$, the Eilenberg-MacLane space $K(G,n)$ is a topological space such that $\pi_n(K(G,n))=G$, while $\pi_m(K(G,n))=0$ if $m\neq n$. For $n=1$ this ...
Andrea Antinucci's user avatar
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1 answer
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Classifying Spaces of Matrix Lie Groups

I’m currently studying the classifying spaces of some of the matrix Lie groups. I’ve come across a post here that describes the classifying spaces for $SO(n)$, $SU(n)$, $GL(n)$, and $Sp(n)$. I’ve also ...
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Maps between classifying spaces induced from group homomorphism

I was trying to undetstand the statement that vanishing of second Stiefel whitney class implies the existence of Spin structure. I came across the answer https://math.stackexchange.com/a/808396/474870....
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How to construct an explicit homotopy equivalence between these classifying spaces?

Let $E \to G$ be the tautological bundle of $n$-planes, and let $p : Fl(E) \to G$ be the bundle of complete flags in $E$. Tautologically, the pullback $p^\star E \to Fl(E)$ is the universal vector ...
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Pullback of a covering map along the covering map.

Consider $H$ as a subgroup of $G$, we can push them to the level of classifying spaces $p: BH \rightarrow BG$. This is a covering space with fiber $[G:H]$. What is the pullback of this covering map ...
Rookiecookie's user avatar
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2 answers
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Classification of surface bundles and an isomorphism between $[S^n, BDiff(X)]$ and $\pi_{n-1}(Diff(X))/\pi_0(Diff(X))$

I'm trying to learn about the classification of surface bundles (in the smooth case, over a circle), and I might be missing some prerequisites. I am somewhat familiar with classifying spaces and ...
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Homotopy Types of Fibrations in Postnikov Tower encoded by Classifing map

A Postnikov system of a path-connected space $X$ is an inverse system of spaces $$ \cdots \to X_{n}\xrightarrow {p_{n}} X_{n-1}\xrightarrow {p_{n-1}} \cdots \xrightarrow {p_{3}} X_{2}\xrightarrow {...
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Construct a trivializing cover of universal principal $G$-bundle $EG \to BG$

In Hatcher's book on algebraic topology (p 89) the universal bundle $EG$ ($G$ discrete group) carries structure of a $\triangle$-complex whose $n$-simplices are the ordered $(n + 1)$ tuples $[g_0, ......
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construct map $ X \to BG $ associated to principal $ G $ bundle $ F \to X $ explicitly combinatorically

Let $ G $ be a discrete group and $ X $ paracompact topological space. The classifying space $ BG $ classifies isomorphy classes principal bundles over $ X $ via $$ [ X, BG] \cong ...
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How to compute $i^*:H^*(U^{m+n})\rightarrow H^*(U^m)\bigotimes H^*(U^n)$?

From $i:U^n\times U^m\rightarrow U^{m+n}$, one has $Bi:BU^n\times BU^m\rightarrow BU^{m+n}.$ The target is to compute $Bi^*.$ By the naturality of spectral sequence of $G\rightarrow EG\rightarrow BG,$ ...
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What is known about the classification of real functions?

For every possible mapping on a set of real numbers to a set of real numbers, is there any overarching theory of their classification? For example, some mappings aren’t defined by a formula but are ...
Julius Hamilton's user avatar
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homology of finitely generated group

The homology groups of every finite group are finite and those of $\mathbb{Z}^n$ with coefficients in $\mathbb{Z}$ are finitely generated. I was wondering whether for some finitely generated group ...
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Classifying space of real line bundles

I have read that $\mathbb{R}P^{\infty}$ is the classifying space of real line bundles. But I don't understand what that means. From all I know, a classifying space refers to a group. Is this meant to ...
Candyblock's user avatar
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Is Milnor's join the realization of a simplicial set?

I am reading the famous papaer Classifying spaces and spectral sequences by Segal and I am a little confused by something. I am familiar with Milnor's join construction of classifying spaces. Let us ...
Federico R.'s user avatar
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Principal $PU(\mathcal H)$-bundle from a principal $\mathbb Z$-bundle and a principal $S^1$-bundle.

Since there is a cup product map $H^1(X;\mathbb Z) \times H^2(X;\mathbb Z) \to H^3(X;\mathbb Z)$, one should be able to produce from a principal $\mathbb Z$-bundle and a principal $S^1$-bundle a ...
Motmot's user avatar
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Calculating $K(\mathbb{Z}, 1)$ in a way which canonically produces $\mathbb{C}^\times$

I am thinking about the Eilenberg-MacLane spaces $K(\mathbb{Z}, 1)$ and $K(\mathbb{Z}, 2)$. I understand the usual construction of these spaces involves $EG$ and $BG$, $G = \mathbb{Z}$. $EG$ is a ...
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Extensions of classifying spaces and higher-groups

$\newcommand{\B}{\mathrm{B}}\newcommand{\U}{\mathrm{U}}$The extension of a group $G$ by $\B\U(1)$, where $\B$ denotes the classifying space specifies a 2-group, $\tilde{G}_2$ $$ 1\to\B \U(1)\to \tilde{...
ɪdɪət strəʊlə's user avatar
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Reference describing long fiber sequence of classifying spaces from a short exact sequence of topological Abelian groups

This nLab page uses the fact that from a short exact sequence $$ 0\rightarrow K \rightarrow L \rightarrow M \rightarrow 0 $$ of (topological, possibly discrete) Abelian groups, we get a long homotopy ...
Indraneel Tambe 2's user avatar
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Showing vanishing 2nd Stiefel-Whitney class implies existence of spin structure

Let $V \rightarrow X$ be an oriented rank-$n$ real vector bundle with equipped with bundle metric. Therefore it gives a classifying map $X\xrightarrow{u} BSO(n)$. I would like to show that if the 2nd ...
Indraneel Tambe 2's user avatar
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1 answer
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Is there a relation obeyed by the 2nd universal Chern class, analogous to the 2nd universal Stiefel-Whitney class?

The Stiefel-Whitney classes. 1.1: There exist $w_j \in H^j(BO(n),\mathbb{Z}_2)$ for $j\in\{1,2,\ldots,n\}$ such that $H^*(BO(n),\mathbb{Z}_2) = \mathbb{Z}_2[w_1,w_2,\ldots,w_n]$. 1.2: The map $BO(n) ...
Indraneel Tambe 2's user avatar
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Confusion regarding definition of integrability of $G$-structure

Given a continuous group homomorphism $\rho : G\rightarrow \mathrm{GL}(k,\mathbb{R})$, by a $G$-structure (or $\rho$-structure if we want to be specific) on a given rank-$k$ real vector bundle $V\...
Indraneel Tambe 2's user avatar
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The "connecting homomorphism" for the LES of iterated classifying spaces for an SES of topological Abelian groups

Assume we are working with a "nice" category $\mathrm{Top}$ of topological spaces closed under the categorical constructions we'll use. Let $B : \mathrm{TopGrp} \rightarrow \mathrm{Top}_*$ ...
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Explicit description of classifying space as a "delooping"?

Consider the "classifying-space functor" $\mathscr{B} : \mathrm{TopGrp} \rightarrow \mathrm{Top}$, constructed in the standard way as a geometric realization of a nerve (in TopCat if ...
I.A.S. Tambe's user avatar
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Cohomology over "discrete reals" vs "continuous reals"?

The canonical morphism $\mathbb{R}_{\text{discrete}} \rightarrow \mathbb{R}_{\text{continuous}}$ of Abelian topological groups should induce morphisms $B^n\mathbb{R}_\text{discrete} \rightarrow B^n\...
I.A.S. Tambe's user avatar
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2 votes
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What is the correct moduli space of inner products on $\mathbb{R}^n$?

Two possible candidates for the moduli space of inner products on $\mathbb{R}^n$ come to mind: the space $S_n$ of positive-definite symmetric real $n\times n$ matrices; or, the classifying space $BO(...
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