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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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\...
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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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Can $B\mathrm{Isom}(M,g)$ be viewed as the "moduli space of Riemannian metrics" on $M$?

Let $M$ be a smooth manifold. Pick some Riemannian metric $g_0$ on $M$, and let $G := \mathrm{Isom}(M,g_0) \subset \mathrm{Diff}(M)$. Now, by classifying space theory, for a manifold $X$ we have an ...
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When does the associated bundle construction induce an equivalence between $G$-bundles and $G$-structure bundles?

My question is about which kinds of continuous group representations / actions produce equivalences between $G$-bundles and groupoids of certain bundles with $G$-structure, through the associated ...
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Is a classifying space of $S^1$ the $S^1$ itself?

I almost surely believe that it is not, but can't find a mistake in my argument below: Fix a manifold $M$ and consider orientable plane bundles. These have $GL^+(2,\mathbb{R})$ structure group, which ...
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Details of the self-map on $BO_\infty$ which exchanges tangential and stable-normal structures?

According to Remark 2.14 in these notes, there is a self-map $s : BO_\infty \rightarrow BO_\infty$ which "exchanges the stable normal and tangential structures". As I understand it, this ...
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Is there a "universal connection" on the universal $G$-bundle?

Let $G$ a Lie group, $X$ a smooth manifold. Let $EG \rightarrow BG$ be the (topological) universal $G$-bundle. We know for every (topological) principal $G$-bundle $P \rightarrow X$ there is a (...
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Obtaining $\mathscr{B}G$ from the topological groupoid $BG$; which notion of "nerve" of a topological groupoid/category should be used?

For $G$ a group without topology (or a discrete topological group), let $BG$ denote the groupoid with one object and morphisms given by $G$. Then, as described at this nLab page, the geometric ...
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Is the universal $G$-bundle functorial in $G$?

We know that the classifying space construction $G \mapsto BG$ gives a functor from topological groups to spaces. I was wondering if the whole construction of the universal bundle is functorial in $G$?...
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Classifying space functor: why do different maps $U(n-1) \longrightarrow U(n)$ give maps between $BU$'s that classify different bundles?

I already started discussion there: One unclear detail in Baum 1968 paper "On the cohomology of homogeneous spaces". But I think that one is quite old, and this time the question is more ...
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Group/H-space structures for standard models of classifying spaces?

Let $G$ be a commutative topological group. May in his textbook, A concise course in algebraic topology, gives a model of the classifying space $BG$ so that it is a commutative topological group ...
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Is There a Smooth Approximation to Classifying Spaces $BG\,?$

If you look at https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism, right above contents the claim is made that we can approximate the classifying space by smooth manifolds. I am aware of at ...
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Homotopy equivalence of $BGL_n(\mathbb{R})$ and $BO_n(\mathbb{R})$

I have tried to prove the above thing. My idea was the following: $\iota:O_n(\mathbb{R})\to GL_n(\mathbb{R})$ be the inclusion map which is a group homomorphism. It induces a fibre bundle $B\iota:BO_n(...
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Example of classifying spaces [closed]

I'm new studying algebraic topology and I am studying classifying space for a group $G$, but I cannot find other examples different than $G = \mathbb{Z} / 2$ or $G = U(1)$. I'm interested in the ...
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Classifying space of Clifford Algebra

I am a physics student, and I am reading a paper in topological condensed matter employing the classifying space of Clifford Algebra.This paper In particular, I feel that I am a bit confused by the ...
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Why the homotopy orbit $(S^{2d+1})^n_{hNT}$ is homotopy equivalent to $(\mathbb{C}P^d)^n_{hS_n}$?

Firstly a few preliminares. Definition: Let $G$ denote a compact Lie group acting on a space $X$, its homotopy orbit space $X_{hG}$ is defined as the quotient of the product space $EG × X$ by the ...
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Colorblind test -- finding and classifying lines in 3-D space

I am creating a color blind test that uses RGB colors. RGB ranges [0-255] in all three colors to determine the final color. I would like to test the user to find which colors they struggle with. For ...
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How can the Chern-Simons form trivialize a non-trivial characteristic class?

I have a very basic confusion about Chern-Simons forms: On Wikipedia and other sources, it is stated that the Chern-Simons 3-form $$Tr(A\wedge dA+\frac23 A\wedge [A\wedge A])$$ trivializes $Tr(F^2)$, ...
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Can we use the algebraic definition of group cohomology for $H(BG, A)$ for compact Lie groups $G$?

For any discrete group $G$, the classifying space $BG$ is a $K(G, 1)$ and can be constructed as a simplicial set with $|G|^i$ $i$-simplices. Accordingly, elements of $H^i(BG, A)$ can be represented by ...
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Two models of $BT^n$

Let $T^n$ be the $n$-torus. We have (at least) two nice models of the classifying space $BT^n$: $(\mathbb{CP}^\infty)^n$; The flag manifold $F_n(\mathbb{C}^\infty)=V_n(\mathbb{C}^\infty)/T^n$, where $...
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How is the classifying space BG related to Spec(k[G]) for some ring k, for an Abelian group G?

Given an Abelian group G, we can construct two kinds of “spaces”: BG: a homotopy type, an Elilenberg-Maclane space, the base of a universal G-bundle, e.t.c. Spec(Z[G]): the spectrum of the group ...
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Classifying space for the subgroup of $GL(n,\mathbb{C})$ preserving a hyperplane

so my knowledge on computing classifying spaces is very limited. I am interested in the classifying space of a particular subgroup of $GL(n,\mathbb{C})$. Fix a hyperplane $H \subset \mathbb{C}^n$ and ...
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Why is the fundamental group of the Volodin space $X(R)$ the Steinberg group $St(R)$?

The Volodin space $X(R)$ is defined in A.A. Suslin's "On the Equivalence of K-Theories" (https://www.tandfonline.com/doi/abs/10.1080/00927878108822666) as the union of classifying spaces $\...
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Classifying space of non-connected Lie groups

Let $G$ be a Lie group and $G_1$ the connected component of the identity. Is there anything we can say about the classifying groups $BG$ and $B(G_1)$? I suspect no since we could take a discrete group ...
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Deducing vanishing of a cohomology class from pairings (follow-up)

This is a follow-up to this question. Basically, I would like to know whether the desired vanishing in question can be deduced if we are allowed to vary $G$. Suppose $M$ is an oriented closed manifold....
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Deducing vanishing of a cohomology class from pairings

Let $M$ be an oriented closed manifold and $G$ a group. Let $x$ be a class in $H^*(M;\mathbb{Q})$. Suppose that for every class $y\in H^*(BG,\mathbb{Q})$ and every continuous map $$f\colon M\to BG,$$ ...
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Does a finite-dimensional Grassmannian classify the subbundles of a trivial vector bundle?

It is known that the universal vector bundle over the infinite-dimensional Grassmannian, $$ E \longrightarrow Gr_n(\mathbb{R}^{\infty}), $$ classifies the rank $n$ vector bundles in the sense that any ...
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Configuration spaces to moduli spaces

In Segal's paper on Mapping Configuration spaces to moduli spaces, I'm not understanding what the map $\Phi$ is, explicitly. Also in section 2, he goes on to say $M_{g,2} \simeq BHomeo^{+}(F_{g,2}; \...
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Examples of classifying spaces of monoids

Definition. The classifying space functor for monoids is the functor $|{-}|\colon\mathsf{Mon}\to\mathsf{Top}$ defined as the composition $$ \mathsf{Mon} \xrightarrow{\mathbf{B}} \mathsf{Cats} \...
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Algebraic torus over $\mathbb C$ and line bundle on its classifying space

Let $G=(\mathbb C^\times)^n$ the algebraic torus, I am trying to understand the isomorphism between its character group and the second cohomology of its classifying space $M(G)\cong H^2(BG,\mathbb Z)$....
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Computing the automorphism group of a $G$-bundle on a connected manifold for a finite group $G.$

Fix a finite group $G$ and a connected manifold $M$. Let $Bun_G(M)$ denote the groupoid of principal $G$ bundles on $M$. We know the classification theorem for principal G-bundles which states that $\...
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Relationship among $BG, K(G,1)$ and Čech cohomology

(1) If $G$ is an abelian group. We know Čech cohomology equals singular cohomology, $\check{H^1}(X,G)=H^1_{sing}(X,G)$. In addition, $\check{H^1}(X,G)$ classifies the isomorphism classes of principal $...
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Quotient by the action of a group commuting with sequential colimits

Let $G$ be a group and suppose $X_1 \to X_2 \to X_3 \to \cdots$ is a sequence of topological $G$-spaces and continuous $G$-maps. The colimit of this sequence inherits then a $G$-action. Under what ...
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The Levi's decomposition in Lie theory and the problem of classification Lie algebras

It is well-know that: "any finite-dimensional Lie algebra over a field of characteristic zero can be expressed as a semidirect sum of a semi-simple subalgebra and its radical (its maximal ...
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Milnor's construction

In the Milnor's construction, $$ \mathcal{J}(G):=\underrightarrow{\lim}G^{*(k+1)} $$ where $G$ is a topological group. I know that there is a natural freely (right) $G$-action on $\mathcal{J}(G)$ and ...
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maps between classifying spaces of homotopy associative H-spaces

Let $f\colon X\to Y$ be an H-map between homotopy associative H-spaces in the homotopy category of based CW-complexes. It's well-known that there is an induced H-map from the homotopy fibre $F$ of $f$ ...
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Universal property of Classifying spaces encoded in $1$-cocycle of $EG \to BG$

Let $G$ be a topological group (for sake of simplicity I think it suffice to assume that $G$ is finite group. It is well known that for any homotopy class of a map $\phi: X \to BG$ (with $X$ nice '...
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Construction of a Covering Space by 'Twisting'

I have a question about the explanantion of the idea behind the classiyfing spaces $BG$ with respect (topological) group $G$. In wikipedia is stated that The classifying property required of $BG$ in ...
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classifying space of the normalizer of maximal torus and borel construction

I want to show that $B(NT) \simeq (BT)_{hW}$, where $NT$ is the normalizer of the maximal torus, $W$ is the Weyl group (that is $W = NT/T$) and $(BT)_{hW}$ is the homotopy orbit/Borel construction, $(...
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Connecting fiber sequences for classifying spaces

My original question was: Is there a long exact sequence for classifying spaces of topological groups? It is solved by a comment from @JHF , since $BA$ is not usually a group. The sequence cannot go ...
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Orientability of vector bundles and the map $BSO(n) \to BO(n)$

Edit: (1) below is now clear, but I'm still looking for an answer for (2). There are two issues I have when trying to understand the classifying spaces of (oriented) vector bundles: (1) Let $E \to X$ ...
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4 votes
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Trying to prove that classifying space(BG) is unique up to homotopy

I'm trying to prove that the classifying space(BG) of a group $G$ is unique in the homotopy category of CW-complexes. Here I define "classifying space" as a topological space $X/G$ s.t. $X$ ...
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classifying space induces a equivalence of categories between 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 https://arxiv.org/abs/1705.05171 (Remark 2.3 d) that there is an equivalence of categories $$ \Pi (M,BG) ...
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Principal $G$-bundle maps and sections of Associated Bundles

These days, I am making my way towards the classification result about homotopy classes of maps from a CW-complex $Y$ to $BG$ and isomorphism classes of principal $G$-bundles over $Y$, where $G$ is a ...
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Reference request for functorial approach to classifying spaces

In the introduction to Bott and Tu's Differential Forms in Algebraic Topology, there was mention that ...there was no time left within the scope of our book to explain the functorial approach to ...
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Can the classifying space of a Lie group be also a Lie group?

How to show a classifying space of a Lie group is also a Lie group or not a Lie group? For example, $U(1)$ is a Lie group, let us consider the following classifying spaces: so $BU(1)=\mathbf{CP}^\...
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2 votes
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understanding construction and definition of classifying space BG

Let $G$ be a discrete group. $EG$ is defined as the $\triangle$-complex (Hatcher p.102) whose $n$-simplices are given by $[g_0,g_1,...,g_n]$ glued together in the obvious way. Then define $BG=EG/G$. ...
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Fibre sequences involving $BO(2)$, $BSO(3)$ and $BSU(2)$

In an answer to another question, it is remarked that there are fibre sequences $$\mathbf{RP}^2 \to BO(2) \to BSO(3),$$ $$\mathbf{RP}^\infty \to BSU(2) \to BSO(3).$$ The second one is explained by ...
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Explicit realization of Eilenberg-MacLane spaces and TOP/PL [closed]

What is the explicit realization of following Eilenberg-MacLane spaces? There are some examples I know but not for the higher $${\displaystyle K(\mathbb {Z} ,1)}=S^1$$ $${\displaystyle K(\mathbb {Z} ...
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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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