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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How many closed surfaces (up to homeomorphism) are there with Euler characteristic -2? [on hold]

I was thinking of splitting up the cases of orientable and non-orientable surfaces.
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Contradiction in spectral sequence calculation of $H_*(BO(2))$

$\newcommand{\Z}{\mathbb{Z}}$ For this post I am going to assume the answer namely $H_*(BO(2))=\Z_2[w_1,w_2]$. Consider the fibration $S^1 \hookrightarrow BO(1) \to BO(2)$. The $E^2$ page has ...
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Construct a homemorphism $\phi : T^2/A \rightarrow X/B $

Construct a homemorphism $\phi : T^2/A \rightarrow X/B $ $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by $A=S^1 \times\{1\}$. $X=S^1 \times [-1, 1]$ and $B = S^1 \times\{-1, 1\}$. ...
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Classifying spaces of different finite groups that are stably homotopy equivalent

Can one find non-isomorphic finite groups $G$ and $G'$ such that there is a homotopy equivalence $f:\Sigma^k BG \rightarrow \Sigma^k BG'$ for some $k$? This would be impossible if we assumed $f$ was ...
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Which of these graphs are homotopic but not homeomorphic?

I am struggling with the 'homotopic' part of the question Which of these are homotopic but not homeomorphic? The number of vertices of degree $\neq 2$ is a topological invariant, thus is true ...
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identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
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Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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The classifying space of open covers of a manifold

Let $M$ be a manifold of dimension $d$ and let $\mathsf{Disk}_{/M}$ be the category of open subsets of $M$ that are diffeomorphic to $\mathbb{R}^d$ with morphisms given by inclusions. Let $\mathrm{B} ...
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Why are bordism groups of a point nontrivial

My definitions of Bordism are from Tom Dieck's Algebraic topology book. Briefly, a singular manifold, $M \xrightarrow{f} pt $, for a closed smooth oriented manifold $(M, \omega)$ without boundary is ...
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What is the map from $H_j( \Sigma MSO(k)) \to H_{j-k}(BSO(k))$ on Tom Dieck page 537

I am reading Tom Dieck's page 537 and I am not sure what the vertical map that I put in the title is in the diagram in the bottom of the page. This map is labeled Thom Isomorphism. Here $MSO(k)$ is ...
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Why is $BSO(n-1)$ the sphere bundle of the tautolocigal bundle on $BSO(n)$?

To try to show this I wrote down explicitly what the classifying spaces can be realized as. I am realizing the classifying spaces $BSO(n-1)$ as $V^\infty_n \times_{SO(n-1)} pt$, where ...
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Fiber bundle and fibration of classifying space

Let $BG$ is classifying space of $G$ topological group. If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the inclusion map $i:H\rightarrow G$ induces \begin{equation*} ...
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Why is the fibration $BSO(n-1) \to BSO(n)$, $n-1$ connected?

In other words I want to show that the induced map $\pi_k BSO(n-1) \to \pi_k BSO(n)$ is 0 for $k\leq n-1$. The fiber of this fibration is $S^{n-1}$. I am having trouble with the case $k=n-1$. I am ...
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When is $BG$ a topological group?

Let $G$ be a topological group, then it has a classifying space $BG$. When is $BG$ a topological group? My motivation for asking this question is that I was thinking about the $B$-analogue of ...
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How to compare the K-theory and singular cohomology of a classifying space

In their foundational paper "Vector bundles and homogeneous spaces," Atiyah and Hirzebruch show, among many other things, that for $G$ a compact, connected Lie group, the K-theory of the classifying ...
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Source request for $H^*(B\mathrm{TOP},\mathbb{Q})\cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...
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classifying space of a category

In his K-book, chapter IV, Weibel states the following as a “a straight forward application of Van Kampen's Theorem”: Lemma 3.4 Suppose that $T$ is a maximal tree in a small connected category ...
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Example of building a classifying space

I'm reading some things about algebraic topology, and they mention the classifying space of a group $G$ as $BG$, but they doesn't build one, so I want to ask if someone knows where can I find the way ...
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Classification of line bundles by group homomorphisms from the fundamental group to $\mathbb{Z}_2$

Let $X$ be a path-connected manifold. Then by the classification of vector bundles, the collection of all (real) line bundles on $X$ is $$ \text{Vect}^1(X)\cong ...
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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 ...
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The oriented Grassmannian $\widetilde{\text{Gr}}(k,\mathbb{R}^n)$ is simply connected for $n>2$

I saw this result mentioned a lot in many references, but it is always stated as a fact or an exercise. My approach would be to see the oriented grassmannian as the quotient ...
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Classifying vector bundles with a reduction of its structure group

Let $Bun(X)$ denote the set of equivalence classes of complex rank 2 vector bundles with a reduction of its structure group to $\mathbb{H}^*$. How can I proof that there is a bijection between ...
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Does a group homomorphism up to homotopy induce a map between classifying spaces?

Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces. If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces ...
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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 ...
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action of a monoid on a mapping telescope

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 ...
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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 ...
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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 ...
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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*} ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
167 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Universal spaces are homotopy equivalent

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 ...
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quotient of fiber bundles

Let $G$ a topological group and consider $P$ a $G$-fiber bundle over $B$. Let $H$ any subgroup of $G$ and let $Q$ an $H$-fiber bundle over $B$. Let $P/Q$ be quotient over $B$, such that the fiber has ...
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55 views

What are classifying spaces actually classifying?

Let $G$ be a group. When we say the classifying space of $G$ we are actually meaning the classifying space of the principal $G-$bundles because the notion of classifying spaces is about classifying ...
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cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
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Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...
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Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?