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.

• $G={\mathbb{Z}}_2$: $E{\mathbb{Z}}_2=S^{\mathrm{\infty }}$ and $B{\mathbb{Z}}_2=ℝP^{\mathrm{\infty }}$.

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

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

• More generally, if $G$ is any discrete group then one can take $BG$ to be any Eilenberg-Mac 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 $n\ge 0$. This example includes the previous three and many more.

• $G=S^1$: $E{S}^1=S^{\mathrm{\infty }}$ and $B{S}^1=ℂP^{\mathrm{\infty }}$.

• $G=SU(2)$: $ESU(2)=S^{\mathrm{\infty }}$ and $BSU(2)=ℍP^{\mathrm{\infty }}$.

• $G=O(n)$, the $n$-th orthogonal group: $EO(n)=V(\mathrm{\infty },n)$, the manifold of frames of $n$ orthonormal vectors in ${ℝ}^{\mathrm{\infty }}$, and $BO(n)=G(\mathrm{\infty },n)$, the Grassmanian of $n$-planes in ${ℝ}^{\mathrm{\infty }}$. The projection map is taking the subspace spanned by a frame of vectors.

Title	some examples of universal bundles
Canonical name	SomeExamplesOfUniversalBundles
Date of creation	2013-03-22 13:12:05
Last modified on	2013-03-22 13:12:05
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	11
Author	bwebste (988)
Entry type	Example
Classification	msc 55R15
Classification	msc 55R10
Synonym	universal family of spaces
Related topic	CategoryOfQuantumAutomata
Defines	Hilbert bundle