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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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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Classifying space of finite group as a complex manifold
Suppose $G$ be a finite group. Then, is it always possible to construct a classifying space which is a finite dimensional(if not, possibly infinite dimensional) complex manifold?
More precisely, are ...
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Algebraic topology of Lie group, classifying space, and homogeneous space
I'm interested in the homotopy theory (homotopy group, homology group, homotopy type, etc.) about compact Lie groups, their classifying spaces, and homogeneous spaces, as well as the interplay among ...
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The universal bundle $\gamma_k\rightarrow BO_k$ is a real vector bundle
For all $n$, let $\gamma_k^n$ bet the tautological bundle over $Gr_k(\mathbb R^n)$, i.e.
$$\gamma_k^n=\{(V,v):V\in Gr_k(\mathbb R^n), v\in V\}$$
This is also naturally identified with the associated ...
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On the topology of $BO_k$
Let $BO_k$ be the classifying space given by:
$$BO_k=\varinjlim_{\mathbb N\ni n}Gr_k(\mathbb R^n)$$
I am trying to determine aspects about the topology of this space, but cannot find any sources that ...
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Find $\mathscr X$ such that $\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $\mathscr X(\Omega G)$ for any $G$
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category of $G$-spaces and the homotopy category of ...
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Presentations of smooth manifolds
A presentation of a affine complex variety consists of finitely many polynomials $f_1,...,f_m$ in $\mathbb{C}[x_1,...,x_n]$. A presentation of a projective complex variety consists of finitely many ...
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A numerable G-principal bundle $E \rightarrow B$ is universal iff E is contractible
I am having some trouble understanding aspects of the proof that shows that a numerable $G$-principal bundle $E\rightarrow B$ is a universal $G$-bundle if $E$ is contractible. The proof starts off by ...
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Chern-Weil theory in terms of pullback along classifying map
Let $G$ be a semisimple real Lie group and $M$ be a smooth manifold. The map on cohomology
$H^{\ast}(BG,\mathbb{C}) \rightarrow H^{\ast}(BT,\mathbb{C})\simeq \mathbb{C}[\mathfrak{h}]$
induced by the ...
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Fibration coming from a group extension
I am trying to solve the following exercise about classifying spaces (5.1.28) in the book "Algebraic K-Theory and its applications" by Rosenberg:
Let
$$1 \longrightarrow N \longrightarrow G \...
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Plus construction and classifying space
Suppose $G$ is a perfect group, and let us consider $BG$ and its plus construction (We consider $BG$, the classifying space, by the nerve construction).
By following Hatcher’s book (Proposition 4.40, ...
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Homotopy long exact sequence and image of connecting map in the center of some group [duplicate]
For context I am working on Weibel's K-book, chapter IV, my question comes from the proof of proposition 1.7.
In this he claims that for the long exact sequence of a fibration with acyclic fiber $F\...
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Cellular prism operator
I have a question.
Given a nulhomotopic map $f : X \rightarrow Y$, we can define a prism operator
$P : C_n(X) \rightarrow C_{n+1}(Y)$ between singular chains.
Then do we have a cellular prism operator ...
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Cohomology of $BU(n)$
The cohomology ring for $BB\mathbb{Z}$ is $\mathbb{Z}[[x]]$, where $x$ lies in degree $2$. My question is about $\mathbb{Z}[[x_1, \dots, x_n]]$. I was trying to find a topological group (ideally an ...
user900250