You are here
Homesome examples of universal bundles
Primary tabs
some examples of universal bundles
The universal bundle for a topological group $G$ is usually written as $\pi:EG\to BG$. Any principal $G$bundle for which the total space is contractible is universal; this will help us to find universal bundles without worrying about Milnor’s construction of $EG$ involving infinite joins.

$G=\mathbb{Z}_{2}$: $E\mathbb{Z}_{2}=S^{\infty}$ and $B\mathbb{Z}_{2}=\mathbb{R}P^{\infty}$.

$G=\mathbb{Z}_{n}$: $E\mathbb{Z}_{n}=S^{\infty}$ and $B\mathbb{Z}_{n}=S^{\infty}/\mathbb{Z}_{n}$. Here $\mathbb{Z}_{n}$ acts on $S^{\infty}$ (considered as a subset of a separable complex Hilbert space) via multiplication with an $n$th root of unity.

$G=\mathbb{Z}^{n}$: $E\mathbb{Z}^{n}=\mathbb{R}^{n}$ and $B\mathbb{Z}^{n}=T^{n}$.

More generally, if $G$ is any discrete group then one can take $BG$ to be any EilenbergMac Lane space $K(G,1)$ and $EG$ to be its universal cover. Indeed $EG$ is simply connected, and it follows from the lifting theorem that $\pi_{n}(EG)=0$ for $n\geq 0$. This example includes the previous three and many more.

$G=S^{1}$: $ES^{1}=S^{\infty}$ and $BS^{1}=\mathbb{C}P^{\infty}$.

$G=SU(2)$: $ESU(2)=S^{\infty}$ and $BSU(2)=\mathbb{H}P^{\infty}$.

$G=O(n)$, the $n$th orthogonal group: $EO(n)=V(\infty,n)$, the manifold of frames of $n$ orthonormal vectors in $\mathbb{R}^{\infty}$, and $BO(n)=G(\infty,n)$, the Grassmanian of $n$planes in $\mathbb{R}^{\infty}$. The projection map is taking the subspace spanned by a frame of vectors.
Mathematics Subject Classification
55R15 no label found55R10 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Lorenz system by David Bankom
Oct 19
new correction: examples and OEIS sequences by fizzie
Oct 13
new correction: Define Galois correspondence by porton
Oct 7
new correction: Closure properties on languages: DCFL not closed under reversal by babou
new correction: DCFLs are not closed under reversal by petey
Oct 2
new correction: Many corrections by Smarandache
Sep 28
new question: how to contest an entry? by zorba
new question: simple question by parag
Sep 26
new question: Latent variable by adam_reith