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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Classifying spaces and smooth bundles

I feel like this should already have been asked on SE, but I cannot seem to find anything on the matter. It is well-known that for well-behaved spaces $X$, e.g. paracompact spaces, one can construct ...
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Classifying spaces of $E_1$-spaces

I'm a newbie trying to understand May's recognition principle on $E_1$-spaces. In May's paper The Geometry of Iterated Loop Spaces, the classifying space of an $E_1$-space $X$ is defined to be $B(\...
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Classifying space level description of the tensor product of vector bundles?

Let's say $\xi: X \to BO(n), \eta : X \to BO(m)$ are two vector bundles over $X$. If I want to take the sum of these two vector bundles, then at the level of classifying spaces, I have the map $\...
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$\mathbb{C}P^n$-bundle over $BU(n)$

The inclusion $U(1)\times U(n-1)\to U(n)$ induces a map $BU(1)\times BU(n-1)\to BU(n)$. This source claims that the fiber of this map is $\mathbb{C}P^n$. Could someone please explain why this is true?
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Isomorphism between a principal bundle and a pullback bundle.

I have seen in many texts on the classification of main bundles that, given two homotopically equivalent X and Y spaces, this equivalence being the function $f: Y \rightarrow X$, given a group G, if $...
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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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Which integral Steifel-Whitney classes are universally $0$?

Let $BO(n)$ denote the classifying space of the orthogonal group $O(n)$. Then there is the well-known ring isomorphism $$H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\dots,w_n] $$ where $w_i \in H^...
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Relation/Difference between moduli spaces and classifying spaces.

From what I have read so far, a classifying space is a representing object of some (co)representable functor. For example, the $n^\text{th}$ Eilenberg–MacLane space is the classifying space for the $...
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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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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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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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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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Cohomology of the Eilenberg-Maclane space $K(\mathbb{R},1)$

Let $\mathbb{R}$ be the reals as an abelian group. A connected topological space $X$ is called an Eilenberg–MacLane space of homotopy type $K(\mathbb{R},1)$, if it has fundamental group isomorphic to $...
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Is the classifying space of a group infinite-dimensional?

According to the textbook by Allen Hatcher, the classifying space B$\mathscr{C}$ of a category ${\mathscr C}$ is constituted with a set of simplices of which $n$-simplices are the strings of morphisms....
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Good Resource on Universal Bundles?

I am looking to learn about universal bundles and classifying spaces as I came across a question which a professor I'm working with suggested is related to these objects, but they don't appear in any ...
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Classifying space of a Group: Reference Request

I am planing to self-study the notion of classifying spaces. Do any of you know a self contained reference (Books, lecture notes, or videos) on this subject suitable for a beginner? Thank you in ...
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Can morphisms of vector bundles be expressed in terms of classifying spaces?

A vector bundle on a space $X$ can be encoded as a map $X \to BU(n)$. Does a similar thing occur for morphisms? One very optimistic interpretation would be as follows. If I have two vector bundles on $...
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homotopy equivalent structure groups have h.e. classifying spaces (if $G\simeq H$ then $BG\simeq BH$)

I believe there is a theorem like if $G$ and $H$ are homotopy equivalent topological groups, then $G$-bundles can have their structure group reduced to $H$ or equivalently (?) the classifying space $...
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Total Space of Classifying Space Contractible

I have a question about the topological intuition that the total space $EG$ of the classifying space $BG$ is contractible. Danny Calegari's "NOTES ON FIBER BUNDLES" on page 4 there is suggested ...
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How much the discrete subgroups of $\mathrm{O}(n)$ can be complicated?

I have no intuition about the difficulty of classification of discrete subgroups of $\mathrm{O}(n)$ and I wanna to know about it. How much the discrete subgroups of $\mathrm{O}(n)$ can be ...
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Classification and Parametrization in Algebraic Geometry and Algebraic Topology - Book recommendation

Logicians, in particular computability theorists, are interested in the distinction between a family of problems being solvable vs. solvable uniformly; in computable analysis and topology we often ...
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1answer
158 views

K-theory of classifying spaces

Can someone help me calculate the following groups in $ K $-theory 1) $ KU^0 (B\mathbb{S}^1) $ 2) $ KU^0 (\mathbb{RP}^\infty) $ where $ B \mathbb{S}^1$ is the classifying space of $ \mathbb{S}^1 $ ...
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119 views

Cohomology of classifying space

I would like to know if anyone knows how to calculate the cohomology of the following spaces, especially in the case of classifying spaces: 1) $ H^\ast (BSU(2), \mathbb{Z}) $ 2) $ H^\ast (BO(3), \...
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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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157 views

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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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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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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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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Classification of circle bundles over $\mathbb{RP}^2$

I am trying to understand isomorphism classes of bundles $$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$ These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$. ATTEMPT 1. $...
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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 of a Category Contractible [duplicate]

My question refers to a statement in Laures' and Szymik's "Grundkurs Topologie" (page 233). Sorry, there exist only a German version. Here the relevant excerpt: My question is why a category having a ...
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Different universal bundle definitions

When reading about classifying spaces and universal (principal) bundles, multiple definitions seem to occur. One has the following possibilities (as far as I can see): The universal $G$-bundle $EG\...
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Milnor construction and deloopings

To construct a classifying space (and universal bundle) of a topological group $G$ one can use the well-known Milnor construction based on the infinite join of $G$. On the other hand one can (at ...
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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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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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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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$\mathrm{Οut}(G) = [BG, BG]$?

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 ...
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Classifying map for the hyperplane bundle

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 ...
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Classifying space of a monoid with 5 elements

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 ...
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Serre spectral sequence of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2=\mathbb{R}P^\infty$

As the title shows, we have a fibration of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2\sim\mathbb{R}P^\infty$. I am trying to check my understanding of Serre spectral sequence with ...
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A 2-group $\mathbb{G}$, so that always exists $0 \to BG_b \to \mathbb{G} \to G_a \to 0?$

If $\mathbb{G}$ is a 2-group, does there always exists a short exact sequence for this $\mathbb{G}$, such that $$ 0 \to BG_b \to \mathbb{G} \to G_a \to 0? $$ where both $G_a$ and $G_b$ are nontrivial ...
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map between classifying spaces

Assume that $H$ is a closed subgroup of a Lie Group $G$ and let $EG \rightarrow BG$ the universal bundle. It is well known the fact that there is an induced map $BH \rightarrow BG$ with homotopy ...
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$\mathbf{B}A$ as a 2-group in a long fiber sequence

I am trying to digest the following statement about 2-group: From nlab Observation 4.2: "Let $A \to \hat G$ be the inclusion of a subgroup, exhibiting a central extension $A \to \hat G \to G$ ...
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What is the total space of the universal bundle over $B\mathbb{Q}$?

What is the total space of the universal bundle over $B\mathbb{Q}$, i.e. what is $E\mathbb{Q}$ for $B\mathbb{Q}=E\mathbb{Q}/\mathbb{Q}$ where $B\mathbb{Q}$ is the classifying space? Thoughts/Attempt: ...
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A direct proof of the Chern-Weil isomorphism

Given a principal $G$-bundle $P \to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism $$S^*(g)^G \to H_{DR}^*(M)$$ associated to any invariant polynomial on $...