# 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 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 $n\geq 0$. 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.

Title some examples of universal bundles SomeExamplesOfUniversalBundles 2013-03-22 13:12:05 2013-03-22 13:12:05 bwebste (988) bwebste (988) 11 bwebste (988) Example msc 55R15 msc 55R10 universal family of spaces CategoryOfQuantumAutomata Hilbert bundle