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 $G$.
282 questions
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Vector bundles with equivalent sphere bundles
Suppose that $E_0, E_1 \rightarrow M$ are two $k$-dimensional vector bundles over a manifold $M$ classified by maps $\phi_0, \phi_1: M \rightarrow BGL(k)$. If $\phi_0, \phi_1$ are homotopic, then $E_0,...
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What is the continuous group cohomology of Euclidean space? More generally, request for references [duplicate]
I am interested in understanding nilpotent Lie groups; the simplest nontrivial one has central extension
$$0 \to [N,N] \to N \to \mathbb{R}^m \to 0.$$
I have seen that such extensions (as Lie groups!) ...
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What did Quillen mean by $M_0=BM_1$?
The following is a paragraph from Quillen's Appendix Q: On the group completion of a simplicial monoid.
Let $M$ be a simplicial monoid and let $\text{Nerv}(M): (p, q) \mapsto (M_q)^p$ be the ...
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Fibers of morphisms of classifying stacks of group schemes
Given an exact sequence of topological / simplicial groups:
\begin{align*}
1\longrightarrow K\longrightarrow G\longrightarrow H\longrightarrow1
\end{align*}
we have a fibration of classifying spaces:
\...
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75
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Showing that $BG$ classifies $G$-torsor.
Let $G$ be a (derived) group scheme. View it as a stack in some fixed topology and define $BG$, the classifying stack, to be the geometric realization of the simplicial diagram
$$ \cdots \substack{\...
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2
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Why does $B\operatorname{det}\colon BU(n) \to BU(1)$ classify the determinant bundle?
Consider the "determinant bundle" construction that takes a $n$-dimensional vector bundle $p\colon X \rightarrow B$ and produces a line bundle so that each fiber $X_b$ becomes the exterior ...
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Bar Construction $BG$ for $G = U(1)$
Let $G$ be a discrete simplicial group. Then its classifying space $(BG)$ can
be realized from a simplicial set constructed via the bar
construction. In particular,
when $G$ is a discrete (ordinary) ...
- 1,982
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Characteristic classes of flat $G$-bundles, induced from $G$-invariant forms on $G/K$, where $G$ is a Lie group and $K$ its maximal compact subgroup
I'd like to solve or find a reference for Exercise 2 (a), Chapter 9 in the book Curvature and Characteristic Classes by Johan L. Dupont. https://mathscinet.ams.org/mathscinet/article?mr=500997
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3
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2
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Cohomology Isomorphism of Classifying Spaces and Equivalence of Compact Lie Groups
Let $R$ be a commutative ring with unity, and let $G$ and $H$ be compact Lie groups with $BG$ and $BH$ as their respective classifying spaces. If there exists isomorphism $H^j(BH; R) \to H^j(BG; R)$ ...
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3
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2
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On the functoriality of classifying spaces (for maps $B\mathbb{Z} \to B\mathbb{Z}_2$)
I am trying to understand what the classifying space functor $G \mapsto BG$ does to group homomorphisms $f : H \to G$ for $H, G \in \textbf{TopGrp}$. Is there any suggested reference on the ...
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Classifying spaces of spheres
I am currently reading Topology of fibre bundles by Steenrod in an effort to understand Milnor's exotic sphere paper. In section 19 we are given the classifying theorem for universal bundles that ...
- 147
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73
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Milnor construction on principal bundles
By way of example, suppose we have principal bundle of orthonormal frames on a manifold $M$
$$SO(n)\hookrightarrow PM\rightarrow M.$$
Does there exist a construction (I'm aiming at at least a ...
- 323
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Reference for "Classifying topological spaces up to homeomorphism is impossible"
On the Wikipedia page it is stated that there is no algorithm that classifies manifolds up to homeomorphism because of the "word problem". And therefore one can deduce that classifying ...
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Is (or can be) category theory used for inferring classification results? [closed]
There is category of groups and there is classification theorem of finite simple groups. There is category of topological spaces and the topological spaces can be classified by topological invariants.
...
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How to show that this space has same homotopy type as the classifying space of infinite unitary group
To show space $\bigcup_{n\geq 1} \frac{U_{2n}}{U_{n}\times U_{n}}$ has the same homotopy type as $BU = \bigcup_{k\geq 1}BU(k)$, where $BU(k)=\bigcup_{n\geq k} \frac{U_{n}}{U_{k}\times U_{n-k}}$ and $...
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