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 contractibleis 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 framesof $n$ orthonormal vectors in ${ℝ}^{\mathrm{\infty }}$, and $BO(n)=G(\mathrm{\infty },n)$, theGrassmanian of $n$-planes in ${ℝ}^{\mathrm{\infty }}$. The projection map istaking the subspace spanned by a frame of vectors.