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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1answer
34 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 ...
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0answers
9 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
27 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.
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0answers
24 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
98 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
29 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
72 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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votes
1answer
75 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
84 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
94 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 ...
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0answers
67 views
Reduction of stucture group of tangent space [on hold]
We consider the exat sequence of groups
$$SO(n)\rightarrow O(n) \rightarrow Z_2$$
where the first map is the inclusion and the second is the determinant
and the induced sequence
$$BSO(n)\rightarrow ...
4
votes
0answers
87 views
The classifying space of a gauge group
Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
3
votes
1answer
88 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
126 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 ...
3
votes
1answer
218 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
186 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
75 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
91 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
148 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
135 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
94 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
176 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...
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votes
0answers
58 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
100 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
$$\begin{array}
& & K(BG)\\
&\nearrow &\downarrow\\
...
6
votes
1answer
178 views
How to correct a wrong proof about the Birman exact sequence?
I've given a proof of the exactness of the Birman exact sequence of groups: $$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
477 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
223 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
266 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
526 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 ...
