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Homesome examples of universal bundles
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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.
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