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Version 2 |
| Let $G$ be a topological group. A \emph{universal bundle} for $G$ is a principal bundle $p :EG \to BG$ such that for any principal bundle $\pi:E\to B$, with $B$ a CW-complex, there is a map $\vp:B\to BG$, unique up to homotopy, such that the pullback bundle $\vp^*(\pi)$ is equivalent to $p$, that is such that there is a bundle map $\vp'$. |
Let $G$ be a topological group. A \emph{universal bundle} for $G$ is a principal bundle $p :EG \to BG$ such that for any principal bundle $\pi:E\to B$, with $B$ a CW-complex, there is a map $\vp:B\to BG$, unique up to homotopy, such that the pullback bundle $\vp^*(\pi)$ is equivalent to $p$, that is such that there is a bundle map $\vp'$. |
| $$\xymatrix{E\ar[d]^{\pi}\ar[r]^{\vp'(E)}&EG\ar[d]^p\\ |
$$\xymatrix{E\ar[d]^{\pi}\ar[r]^{\vp'(E)}&EG\ar[d]^p\\ |
| B\ar[r]^{\vp'(B)=\vp}&BG}$$ |
B\ar[r]^{\vp'(B)=\vp}&BG}$$ |
| such that any bundle map of any bundle over $B$ extending $\vp$ factors uniquely through $\vp'$. |
such that any bundle map of any bundle over $B$ extending $\vp$ factors uniquely through $\vp'$. |
| The base space $BG$ is often called a {\em classifying space} of $G$, since homotopy classes of maps to it from a given space classify $G$-bundles over that space. |
The base space $BG$ is often called a {\em classifying space} of $G$, since homotopy classes of maps to it from a given space classify $G$-bundles over that space. |
| There is a useful criterion for universality: a bundle is universal if and only if all the homotopy groups of $EG$, its total space, are trivial. |
There is a useful criterion for universality: a bundle is universal if and only if all the homotopy groups of $EG$, its total space, are trivial. |
| In 1956, John Milnor gave a general construction of the universal bundle for any topological group $G$ (see \emph{Annals of Mathematics}, Second Series, Volume 62 Issue 2 and Volume 63, Issue 3 for details). His construction uses the infinite join of the group $G$ with itself to define the total space of the universal bundle. |
In 1956, John Milnor gave a general construction of the universal bundle for any topological group $G$ (see \emph{Annals of Mathematics}, Second Series, Volume 62 Issue 2 and Volume 63, Issue 3 for details). His construction uses the infinite join of the group $G$ with itself to define the total space of the universal bundle. |
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