# Tagged Questions

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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### What is the classifying space G/Top?

I simply can't find the definition(except in one book on surgery where a definition was not actually given but instead they alluded to what the definition is) and I have spent an hour and half looking....
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### Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
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### Relation between two notions of $BG$

The following is something that's always niggled me a little bit. I usually think about stacks over schemes, so I'm a bit out of my element—I apologize if I say anything silly below. Let $G$ be a ...
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### A category whose classifying space has nontrivial higher homotopy groups

The classifying space of a category $\scr{C}$ is obtained by taking its nerve $N\scr{C}$, which is the simplicial set defined by $$N\mathscr{C}_n:= \mathrm{Fun}([n],\mathscr{C})$$ and the ...
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### Are there multiple non-isomorphic principal $G$-bundles on Euclidean space? [duplicate]

I'm pretty sure the answer is out there, see this MathOverflow question, but that is unfortunately way over my head :). I'm interested in the case that $G$ is a Lie group (e.g. $U(1)$), but I don't ...
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### Cohomology of a classifying space

I would like some advice on the following problem: I have a topological group $G=\langle H,g\rangle$, where $H$ is a subgroup and $h$ is an element. Using a result of classifying spaces $Bi:BH \to BG$...
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### Construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$? [closed]

Let $\pi$ be any group. How do I construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$?
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### Infinite torsion group with K(G,1) of finite type

I am wondering whether any group $G$ that is torsion and has a $K(G,1)$ of finite type (i.e. there are finitely many cells in each dimension) is already finite. The condition of having a $K(G,1)$ of ...
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### Is the classifying space a fully faithful functor?

Given a topological group $G$, we can form its classifying space $BG$; suppose we have chosen some specific construction, say the bar construction. $B$ is a functor - given any homomorphism $G \to H$, ...
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### configuration-spaces and iterated loop-spaces

In the paper Configuration-Spaces and Iterated Loop-Spaces. Graeme Segal, page 213-214, it is obtained that the labelled configuration space $C_n$ is homotopy equivalent to a topological monoid $C'_n$....
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In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on $M_\... 1answer 54 views ### homomorphism of$H$-spaces between a monoid and loop space of its classifying space Let$M$be a topological monoid.$M$can be considered as a category internal to topological spaces and has a simplicial space$N_\bullet(M)$as its nerve. The geometric realization$BM=|N_\bullet(M)|$... 1answer 67 views ### Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure? So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ... 0answers 70 views ### the geometric realization of a simplicial set is contractible Let$M$be a monoid up to homotopy. The simplicial set$WM$is defined by setting $$WM_n=M^{n+1}=\{(g_0, g_1,\cdots,g_n)\mid g_i\in M\}$$ with faces and degeneracies given by \begin{eqnarray*} d_i(... 0answers 35 views ### universal bundle of topological monoids Let$M$be a topological monoid. There is a classifying space$BM$(cf. canonical map of a monoid to its classifying space). When$M$is a group$G$, there is a principal$G$-bundle$EG\to BG$such ... 1answer 80 views ### question about “Homology fibrations and the group completion theorem” In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18: we have a fibre bundle$M_\infty\to (M_\infty)_M\to BM$with$(M_\infty)_M$constractible. In ... 0answers 48 views ### configuration space model for classifying space of monoid Let$M$be a monoid and$BM$be its classifying space. There is a model for$BM$based on labelled configuration spaces of the line$[0,1]$. Points of the configurations are labelled by elements of$m$... 0answers 54 views ### Different groups with the same classifying space. Let$G$be a topological group and$BG$its classifying space. From the LES of the universal bundle, we get$\pi_i(BG)\cong\pi_{i-1}(G)$, so given the classifying space, we know all homotopy groups of ... 1answer 38 views ### explicit equivalent relation in the expression of the classifying space of a monoid Let$M$be a topological monoid.$M$can be considered as a category internal to topological spaces and has a simplicial space$N_\bullet(M)$as its nerve. (It's also called the internal nerve.) The ... 1answer 186 views ### canonical map of a monoid to its classifying space Every monoid$M$is a category with one object$M$and morphisms the elements of$M$. [Martin Brandenburg.] Every small category$C$has a classifying space$BC$, defined as the geometric realization ... 1answer 195 views ### what is the classifying space of a monoid In the paper Homology Fibrations and the "Group-Completion". Theorem. McDuff, D.; Segal, G., 1976, the first line: A topological monoid$M$has a classifying space$BM$. I do not understand this ... 1answer 57 views ### Is a finite cyclic group a Poincare duality group? I am trying to understand whether the finite cyclic group of order$n$,$C_n$is a Poincare duality group, i.e. whether it's classifying space$K(C_n,\,1)$is a Poincare complex. I know that the ... 0answers 76 views ### Conjugate group homomorphisms induce homotopic maps on classifying spaces Let$G$be a group and let$\phi: G \to G$be the inner automorphism given by conjugation by an element$g' \in G$, i.e.,$\phi(g) = g'^{-1} g g'$. I want to show that the induced map on classifying ... 0answers 82 views ### Proof of the isomorphism$[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2).$I want to prove the following:$[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2)$via the map$[f] \mapsto f^* w_1(\gamma^1)$where$\gamma^1$denotes the tautological line bundle over$\mathbb{...
Consider a group $G$. We want to determine the universal space $EG$. Is it true that all universal spaces are homotopy equivalent. That is, to find $EG$ we only need to find a weakly contractible ...