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 $G$.

learn more… | top users | synonyms

1
vote
0answers
20 views
+50

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 ...
2
votes
1answer
43 views

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 ...
0
votes
0answers
12 views

Does combing features for Naive Bayes classification violates independence assumption?

I have ran some experiments using a naive bayes classifier and i have several features and I ran the experiments using only one feature at a time. The equation was: \begin{equation} P({ f }|{ t }_{ i ...
1
vote
1answer
39 views

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.
1
vote
0answers
29 views

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 ...
1
vote
1answer
111 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 ...
2
votes
1answer
30 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 ...
5
votes
0answers
76 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) ...
0
votes
1answer
77 views

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 ...
1
vote
0answers
50 views

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, ...
2
votes
1answer
89 views

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 ...
3
votes
1answer
95 views

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 ...
4
votes
0answers
89 views

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) = ...
3
votes
1answer
90 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 ...
4
votes
1answer
67 views

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 ...
5
votes
0answers
134 views

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 ...
5
votes
1answer
247 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 $\Sigma$ ...
3
votes
1answer
213 views

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 ...
0
votes
0answers
76 views

vapnik chervonenkis dimension for a circle and rhombus

my question is how to prove maximum VC dimension for points on a circle and a rhombus as a hyperplane Thanks. EDIT: Proving, that e.g. 7 points are inseparable (therefore VC < 7 but can't find ...
1
vote
0answers
92 views

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 ...
3
votes
2answers
153 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 ...
3
votes
1answer
140 views

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 ...
4
votes
1answer
98 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 ...
3
votes
1answer
180 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...
0
votes
0answers
61 views

classifying space of $p$-group

I want to know a model for the classifying space of a finite $p$-group $P$ (say of order $p^n$) and the mod $p$ cohomology algebra of $P$. In particular, what is the classifying space and mod $p$ ...
2
votes
0answers
105 views

Question involving the Chern character from the book “Fibre Bundles”

On page 311 of Dale Husemöller's book Fibre Bundles in Theorem 11.6 he has the following commutative diagram $$\begin{array} & & K(BG)\\ &\nearrow &\downarrow\\ ...
6
votes
1answer
179 views

How to correct a wrong proof about the Birman exact sequence?

I've given a proof of the exactness of the Birman exact sequence of groups: $$1\to\pi_1(S_{g,r}^s)\to MCG(S_{g,r}^{s+1})\overset{\lambda}{\to} MCG(S_{g,r}^s)\to 1$$ making use of classifying spaces ...
3
votes
3answers
496 views

Group structure on Eilenberg-MacLane spaces

How do we put a group structure on $K(G,n)$ that makes it a topological group? I know that $\Omega K(G,n+1)=K(G,n)$ and since we have a product of loops this makes $K(G,n)$ into a H-space. But what ...
2
votes
1answer
226 views

Maps between Eilenberg–MacLane spaces

I was re-reading an algebraic topology book the other day, and I came across the following problem: Suppose that $\pi$ and $\rho$ are abelian groups and $n\geq 1$. Determine ...
0
votes
0answers
121 views

Homotopy type of Eilenberg-MacLane spaces

From the path fibration we extract that $\pi_{i+1}(K(G,r))=\pi_{i}\Omega K(G,r)$, then for all $k$, $\pi_{k}(K(G,r-1))=\pi_{k}\Omega K(G,r)$. How can we conclude that $K(G,r-1)\simeq \Omega ...
11
votes
1answer
268 views

What functor does $K(G, 1)$ represent for nonabelian $G$?

For $G$ an abelian group, the Eilenberg-Maclane space $K(G, n)$ represents singular cohomology $H^n(-; G)$ with coefficients in $G$ on the homotopy category of CW-complexes. If $n > 1$, then $G$ ...
9
votes
4answers
537 views

Why is the cohomology of a $K(G,1)$ group cohomology?

Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular ...