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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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Let $M$ be a smooth closed manifold. $N(M)$ be the set of normal invariants. $N(M) \cong [M,G/O]$ or $[M,G/Top]$ depending on context of $Diff$ or $TOP$ category. How does this isomorphism behaves ...
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Are there references that compute the integral motivic cohomology $\mathrm{H}^{p,q}(BG,\mathbb Z)$, for $G$ finite cyclic and where $BG$ is defined over the rationals. I am particularly interested in $...
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This is part of a series of follow-up questions to: Which objects in a topos such as the étale topos are "classifying spaces" of their fundamental groups?. See also Classifying spaces in $\...
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This is a follow-up to the question: Which objects in a topos such as the étale topos are "classifying spaces" of their fundamental groups? Let $S=\operatorname{Spec}(R)$ be an open ...
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Let $P, P'$ be symplectic(resp. Hamiltonian) fibrations over a base $B$. If two $P, P'$ are continuously symplectic fibration(resp. Hamiltonian fibration) isomorphic, then are they also smoothly ...
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Let $T$ be a topos over a site, such as the étale topos $\operatorname{Et}_{S}$ over a scheme $S$. This topos comes with a cohomology theory $H^{\bullet}$ and the notion of homotopy groups $\pi_{\...
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One can define a simplicial $G$-bundle $E\rightarrow M$ for simplicial manifolds $E=\{E_p\}$ and $M=\{M_p\}$ as a sequence of smooth $G$-bundles $\phi_p:E_p\rightarrow M_p$. We then define a ...
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Let $G$ be a discrete group. Since I'll be using the term in a more general way than is usual, let me spell out what I mean by a $K(G,1)$: it is a pointed space $(X,x_0)$ with the following three ...
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I want to work over the site of smooth manifolds with Grothendieck topology induced by open immersion. I am primarily concerned with the the following question: in what ways can we classify vector ...
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Let $\operatorname{BGL}(d)$ and $\operatorname{BDiff}(\mathbb{R}^d)$ be the simplicial Spaces defined as the nerves of the obvious topological groupoids. I am looking for an explicit weak homotopy ...
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It is a folklore result that if $G$ is a discrete group and $BG$ its classifying space, then the free loop space $L(BG)$ is homotopy equivalent to $EG\times_{Ad} G$ where $EG$ is the universal $G$-...
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Let $G$ be a compact Hausdorff group. Milnor in [1] uses the strongest topology. tom Dieck in [4] and A. Dold in [3] use the coarsest topology. I'm a little confused about this. Atiyah and Segal use ...
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On an orientable (Riemannian) $n$-manifold $M$, with orthonormal frame bundle $\operatorname{Fr}(M)$, its classifying map $\tau_M : M \to B{\operatorname O(n)}$ lifts (up to homotopy) to $\widetilde{\...
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Let $\mathbb{Z}_2, \operatorname{Spin}(n), \operatorname{SO}(n) \in \mathsf{TopGrp}$. Take the fibration $\mathbb{Z}_2 \hookrightarrow \operatorname{Spin}(n) \twoheadrightarrow \operatorname{SO}(n)$. ...
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Lurie's classification theorem [1, Theorem 2.4.26] classifies fully extended tfts for $G$-manifolds in terms of homotopy fixed points: Let $C$ be a symmetric monoidal $(\infty, n)$-category with ...
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Let $H, G$ be topological groups and $\phi : H \to G$ a group homomorphism. Let $M$ be a paracompact topological space. For any principal $G$-bundle $P \to M$, a reduction (or sometimes 'lift') of its ...
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On an orientable (Riemannian) $n$-manifold $M$, with orthonormal frame bundle $\operatorname{Fr}(M)$, we have that the tangent bundle classifying map $\tau_M : M \to B{\operatorname O(n)}$ lifts to $B{...
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For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort: There are two orientations on $M$. Is it ...
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The functor $k_{U(d)}$ of the set of isomorphism classes of numerable principal $U(d)$-bundles is represented by $BU(d)$ the Milnor construction of the classifying space. The restriction of $k_{U(d)}$ ...
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If $E\xrightarrow{p}B$ is a principal $G$-bundle classified by $B\xrightarrow{f}BG$, then the image of any characteristic class of $B$ under $p^*$ is trivial by the naturality of $H^*$ and the fact ...
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Let $X$ be a variety over $k$ and $G$ be a finite abelian group. Then we know that $H_{fppf}^{2}(X,G)$ is in bijective correspondence with isomorphism classes of $G$-banded gerbes. Now we consider a ...
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I am using the Borel transgression theorem as given in Mimura and Toda's "Topology of Lie groups I and II", page 378, Theorem 2.7. I know that it applies when the fiber has the homotopy type ...
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Let $G$ be a topological group. Following Milnor one way of defining the total space of the universal bundle $EG$ of $G$ is to form the infinite join $$ EG = G^{\ast \infty} = G \ast G \ast \dots $$ ...
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Let $G$ be a compact group and $H$ be a closed subgroup of $G$. I know that if $G\rightarrow G/H$ is a principal $H$-bundle, then we can choose $E_{H}$ as $% E_{G}$, where $E_{G}$ is the Milnor space ...
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Let $G$ be a compact group (maybe non-Lie group). Let $B_{G}$ denote the classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$, then I think that $H^{\ast }( B_{G};\mathbb{Q} )$ is ...
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3d Chern-Simons gauge theories based on a Lie group $G$ are classified by an element $k_{CS}\in H^4(BG,\mathbb{Z})$, its level. Via the CS/WZW correspondence the theory is related with a 2d non-linear ...
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Let $M$ be a commutative monoid. Define the bar construction $BM$ as the thin geometric realization of $[p] \mapsto M^p$. I am looking for a reference for the fact that $BM$ is again a commutative ...
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Some days ago I came across the paper "Holomorphic fiber bundles over Riemann surfaces", by H. Rohrl. At the beginning of Section 1, the following theorem is quoted: Theorem. Every fiber ...
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Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups. For a discrete group $G$, we let $BG$ to be the classifying space of $G$. After reading this question, I was ...
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I've been reading some papers carefully, with their proofs (Notations are given at the end). The following comes from "Braids, mapping class groups and categorical delooping" by Song & ...
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The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces. I am looking for classes of examples of connected topological groups/...
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I understand that an internal abelian group in CW-complexes (while it is not the most extensive structure for which this can be done) produces a classifying space which itself has the structure of an ...
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I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where $${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
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Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. It's standard that the isom' classes of such principal $G$-bundles are ...
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I am a physicist knowing a bit of algebraic topology, and trying to answer the following question. This is perhaps not appropriate as a question on MO, in which case I apologize. I posted this ...
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In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this Corollary 5.17 ...
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$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let $$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$ be the homotopy ...
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Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that $$ H^i(G, ...
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Recall that Bott's obstruction for integrability [Bott70] asserts that: Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every ...
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I read somewhere that the classifying space $B_{G}$ for any topological group $G$ is paracompact and locally contractible. How can I prove this or can you give me a reference? Another question that I ...
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What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower? Namely, how do we know $$ K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)? $$ Naively -- in each step ...
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In what follows all the groups will be discrete, not necessarly finite. Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am ...
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As computed in various places (e.g. in Brown - The Cohomology of $B\mathrm{SO}_n$ and $B\mathrm O_n$ with Integer Coefficients), the integral cohomology ring of $B\mathrm{O}(n)$ (and similarly $B\...
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Let $\mathcal{G}$ be a strict topological $2$-group, i.e. a strict $2$-category with a single object, a space of invertible $1$-morphisms, a space of invertible $2$-morphisms and continuous structure ...
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First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...
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From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...
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Let $G$ be a finite abelian group, and its higher classifying space is $B^nG=K(G,n)$. For $n=1$ it is well known that $H^2(B G, \mathbb{R}/\mathbb{Z}) \cong H^2(G,\mathbb{R}/\mathbb{Z})$ is isomorphic ...
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Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
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For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
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The dihedral group $D_8$ can be presented as $(\mathbb{Z}_2\times \mathbb{Z}_2)\rtimes _{\rho}\mathbb{Z}_2$, where the last factor acts on $\mathbb{Z}_2\times \mathbb{Z}_2$ as $$ \rho_1(a,b)=(b,a) \ . ...
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