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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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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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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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?
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Free loop space of classifying space as a disjoint union of classifying spaces of centralizer proof reference request.

I am looking for a reference for the proof or explanation of why for a discrete group $G$ we have that the free loop space of its classifying space is the disjoint union of centralizeers of $g$ where ...
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Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities). the isomorphism classes of complex line bundles over $X$ the homotopy classes of ...
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Stably equivalent bundles and stable homotopy classes of maps

We know that there is a one-to-one correspondence between isomorphism classes of principal $G$-bundles over a base space $M$ and homotopy classes of maps $M \to BG$, where $BG$ is the classifying ...
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classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...
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Analogue in algebra for characteristic classes?

By Swan's Theorem, we know that projective modules over a ring are an algebraic analogue of vector bundles over a base space. Is there some sort of cohomology theory of rings (or modules? or schemes, ...
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86 views

Homotopical classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$

Which are homotopy classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$ for $n < m$? In real case, even for any cellular complex $X$ with $\dim X<m$ homotopy classes of mappings $X \to ...
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22 views

How can I find the vector between two sets of data?

I need to identify the vector between two sets of data. The goal is to correctly "guess" whether a new piece of data is in group A or ...
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75 views

Eilenberg-Maclane space $K(G\rtimes H, 1)$ for a semi-direct product.

We know that $K(G\times H, 1)=K(G,1)\times K(H,1)$. Do we know something like this for a semi-direct product, where $K(G,1)$ denotes the Eilenberg-Maclane space.
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Constructing the classifying space for a groupoid G

I am working with the classifying space BG of the groupoid G. One definition is as follows: $$ BG = \bigsqcup _n (G_n \times \Delta ^n) /(d_i(g),x) \sim (g, \delta _i (x)). $$ Where the $d_i$ are ...
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Unit or non-zero octonions form an $A_\infty$-space?

If I have a Moufang loop, can it have a classifying space? I'm thinking of the unit octonions, if that's too general, so perhaps a better question is: are the unit (or non-zero) octonions an ...
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Eilenberg-Mac Lane and classifying spaces

What can we say about An Eilenberg-Mac Lane space $K(G,n)$ is a classifying space $BG$. When it could be true? For what kind of $G$? For what values of $n$? References are welcomed.
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A group-like topological monoid is a loop space

I am looking for an elementary reference for the following fact. Let $M$ be a topological monoid and suppose moreover that it is group-like, ie. $\pi_{0}(M)$ is a group. Then the canonical map $M ...
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242 views

Universal quotient bundles of $G(2,4)$ and $\mathbb{G}(1,\mathbb{P}^3)$

Let $V$ be an $n$-dimentional complex vector space, $G=G(k,V)$ the Grassmannian of $k$-planes in $V$, and let $\mathcal{V}:=V \otimes \mathcal{O}_G$ the rank-$n$ trivial vector bundle on $G$. We ...
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58 views

Creating a lift chart for a classification tree

This is likely a simple question but I'm new to data mining techniques and am trying to compare two different predictive models. I've created a logistic regression and a classification tree and would ...
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124 views

Why is this space aspherical?

Let $X = Y \cup Z$ be a connected, path-connected Hausdorff space. Suppose that $Y$, $Z$, and $Y\cap Z$ are all connected, path-connected, and aspherical, and that the homomorphism $\pi_1(Y\cap Z) ...
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Group Extension and Classifying Space

If $$ 0 \to H \to G \to G/H \to 0\ $$ is a group extension, under what conditions do we have a fibration of the form $$ BH \to BG \to B(G/H), $$ where $BG$ is a classifying space of $G$? Suppose ...
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What are the non-degenerate faces of $N\mathbb{Z}_2$

I don't understand the nerve construction. For $\mathbb{Z}_2$, Wikipedia says $\bullet \overset{1}\longrightarrow \bullet \overset{1}\longrightarrow \bullet$ should produce a nondegenerate 2-simplex, ...
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Classifying space for finite-dimensional torus

Note that $K({\bf Z},1)=S^1$ but $BS^1 = {\bf CP}^\infty$. For finite groups $H$, $G$, $$K(G\times H,1) = K(G,1)\times K(H,1)$$ Does it works for classifying spaces of continuous groups ? As far ...
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Can one apply the classifying space functor $B$ more than once?

For a topological monoid $M$, the classifying space $BM$ is at least a pointed topological space as far as I know. From where to where is the construction $B$ a functor actually? Can I plug in an ...
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The classifying space of a gauge group

Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by $$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
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110 views

Classifying space of the reals

What's the classifying space $B\mathbb{R}$ of the additive group of real numbers provided with the Euclidean topology ? By the extension $\mathbb{Z} \hookrightarrow \mathbb{R} \twoheadrightarrow ...
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Cohomology of $\Bbb CP^{\infty}=BU_1, BU_2,\dots$ : A reference request

Where can I find the calculation of the cohomology rings of the classifying spaces $BU_n,~BO_n$ and $BO,~BU$? I took a class where extensive use was made of these cohomology rings, but I missed the ...
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Which is the correct universal line bundle: the tautological bundle or its dual?

With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$ In ...
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436 views

Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?

I'm looking for a reference for the following result: If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ ...
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Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
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VC dimension of an oriented hyperplane

What is VC dimension (Vapnik-Chervonenkis dimension) of an oriented hyperplane? I know that VC dimension of set of oriented hyperplanes is $n+1$. Is it the same? I came across this question ...
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204 views

Is there a classifying space for covering maps?

It is often said that a sheaf on a topological space $X$ is a "continuously-varying set" over $X$, but the usual definition does not reflect this because a sheaf is not a continuous map from $X$ to ...
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1answer
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Why is the group of covering transformations relative to the quotient map isomorphic to a subgroup of the Fundamental Group?

I'm trying to prove the classification theorem for covering spaces. I've got to the stage where I need to show the following: If $H$ a subgroup of $\Pi_1(X,x_0)$ then $\exists Y$ covering space of ...
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120 views

Homotopy-Fibre Sequence of Classifying Spaces

Let $G$ be a topological group and $H$ be a normal subgroup of $G$ (I think $H$ is required to be admissible in the sense that the quotient map $G\to G/H$ is a principal $H$-bundle, am I right?). Then ...
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261 views

Finite dimensional Eilenberg-Maclane spaces

Given a positive integer $n\geq 2$ and an abelian group $G$, is it possible to find a finite dimensional $K(G,n)$? In case it does, which are some examples? Thanks...