# 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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### Homotopy fiber of the second Stiefel-Whitney class

The second Stiefel-Whitney class $w_2$ can be identified with a (homotopy class of) map $BSO\to B^2\mathbb Z_2$ (I write $\mathbb Z_2$ the cyclic group of order 2, and omit the dimension assumed to be ...
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### Classifying space BG and contractable space EG

Choose a arbitrary discrete group $G$. The classifying space $BG$ of $G$ is constructed by forming a certain contractable $\Delta$-complex $EG$ (on concrete construction of $EG$: see below) endowed ...
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### How can an inclusion of finite groups induce a fibration of classifying spaces?

Let $G$ be a compact Lie group and $H$ be a closed subgroup. The inclusion $H \rightarrow G$ induces a homotopy fibration $G/H \rightarrow BH \rightarrow BG$. In particular, this must hold if $G$ and ...
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### Stable homotopy groups of unitary groups vs homotopy group of union of unitary groups

There is a canonical inclusion $i$ of $U(n-1)$ into $U(n)$. Using the fibration $U(n-1)\rightarrow U(n)\rightarrow S^{2n-1}$ and the associated long exact sequence of homotopy one gets that $i_*$ ...
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### How can I evaluate the universal Stiefel-Whitney class on a given simplex?

Let $\gamma:EO(n)\to BO(n)$ denote the universal n-plane bundle and $w_i(\gamma)$ the universal Stiefel-Whitney class. Since $w_i(\gamma)$ is an element of $H^i(BO(n),\mathbb{Z}/2\mathbb{Z})$, it ...
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### Classifying PDE and transforming to real inner space from bilinear def

$Φ(x, y) = 5x1x2−x1y2−x2y1+5y1y$ $∀ x = ( x1, y1 )^T ∈ R^ 2$ $∀ y = ( x2, y2 )^T ∈ R^ 2$ We need to show that it's (1) symmetric (2)quadratic form (3) classify it (4) there is this one ...
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### Simplicial space of a total space of a classifying bundle for $G$

I am reading lecture notes on topology and the total space $E(U(N))$ is given as a geometric realization of a simplicial space $$E(U(N))=|[n]\rightarrow U(N)^{n+1}|$$ Here I am confused because 1) ...
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### The loop space of the classifying space is the group: $\Omega(BG) \cong G$

Why does delooping the classifying space of a topological group $G$ return a space homotopy equivalent to $G$. In symbols, why $\Omega(BG) \cong G$, where $G$ is a topological group and $BG$ its ...
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### Classification of the compact surfaces: Example

What compact surface is represented by the regular $10$-gon with edges identified in pairs, as indicated by the symbol: $$abcdec^{-1}da^{-1}b^{-1}e^{-1}?$$ It's an exercise (Exercise 8.7 of chapter 1)...
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### Weyl group action on maximal torus

This is a classical result in the theory of Lie groups, but I am stuck on trying to understand the action of the Weyl group on the classifying space of the torus. Namely, Let $G$ be a compact ...
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### A stable system of vector bundles can be obtained by pulling back the universal stable system

I am trying to solve the problem 180 on Davis-Kirk (p.269). However, the authors give some special definitions, so-called "stable system of vector bundles", which I have never seen on other ...
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### What is relation between the path fibration of classifying map and the Borel construction

I am trying to solve exercise 179 on Davis-Kirk: Show that given a principal $G$-bundle $E\to B$, there is a fibration $$E\hookrightarrow EG\times_G E\to BG$$ where $EG\times_G E$ denotes ...
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### The cohomology $H^*(BU(n); R)$

I am reading this post, Prop. 2.1. It seem that none of the argument is dependent on the we are working with coefficient in $\Bbb Z$. Hence, let $R$ be a unital commutative ring. (I) Do the results ...
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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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### 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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### 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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### 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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### 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 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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### Is the classifying space a fully faithful functor?

Given a topological group $G$, we can form its classifying space $BG$; suppose we have chosen some specific construction, say the bar construction. $B$ is a functor - given any homomorphism $G \to H$, ...
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### Example of building a classifying space

I'm reading some things about algebraic topology, and they mention the classifying space of a group $G$ as $BG$, but they doesn't build one, so I want to ask if someone knows where can I find the way ...
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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 algebra ...
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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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### 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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### Chern-weil theory on the classifying bundle

Given a $G$-principal bundle $P \to M$, and an invariant polynomial $\mathcal{P}$ on the Lie algebra, we can define an element of the cohomology of the base space by plugging the curvature (of an ...
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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 ...
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 ...
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 ...