Questions tagged [classifying-spaces]
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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The topos of maps that are étale at a given finite set of primes: What are its classifying spaces?
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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Classifying spaces in $\operatorname{Ét}_{\operatorname{Spec}(\mathbb{Z}[x_{1},...,x_{n}])}$?
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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Are two symplectic fibration(resp. Hamiltonian fibration) are smoothly fibration isomorphic if it holds continuously?
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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Which objects in a topos such as the étale topos are "classifying spaces" of their fundamental groups?
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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Holonomy of Simplicial $G$-bundle
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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A K(G,1) that is not homotopy equivalent to a CW complex
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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Stack quotient of a point vs. classifying spaces
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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Explicit homotopy equivalence $\operatorname{BDiff} \rightarrow \operatorname{BGL}$
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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Circle action on free loop space of a classifying space
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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Topology on the Milnor construction $EG$
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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Is this true of spinnable frame bundles $\operatorname{Fr}(M)$?
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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Why is this true of the classifying space functor on fibrations?
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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Explicit Data of Homotopy Fixed Points in Lurie's TFT
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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Reduction of structure group and classifying spaces
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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Is this true of the frame bundle $\operatorname{Fr}(M)$?
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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