# universal bundle

Let $G$ be a topological group^{}. A *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 $\phi :B\to BG$, unique up to homotopy^{}, such that the pullback bundle ${\phi}^{*}(p)$ is equivalent^{} to $\pi $, that is such that there is a bundle map^{} ${\phi}^{\prime}$.

$$\text{xymatrix}E\text{ar}{[d]}^{\pi}\text{ar}{[r]}^{{\phi}^{\prime}(E)}\mathrm{\&}EG\text{ar}{[d]}^{p}B\text{ar}{[r]}^{{\phi}^{\prime}(B)}\mathrm{\&}BG$$ |

with ${\phi}^{\prime}(B)=\phi $, such that any bundle map of any bundle over $B$ extending $\phi $ factors uniquely through ${\phi}^{\prime}$.

As is obvious from the universal property^{}, the universal bundle for a group $G$ is unique up to unique homotopy equivalence^{}.

The base space $BG$ is often called a 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. This allows us to construct the universal bundle any subgroup from that of a larger group. Assume $H\le G$ and that $p:EG\to BG$ is a universal bundle for $G$. Then $H$ also acts freely on $EG$ which is contractable so ${p}_{H}:EH=EB\to BH=EB/H$ must be a universal bundle for $H$.

In 1956, John Milnor gave a general construction of the universal bundle for any topological group $G$ (see *Annals of Mathematics*, Second Series, Volume 63 Issue 2 and 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.

Title | universal bundle |
---|---|

Canonical name | UniversalBundle |

Date of creation | 2013-03-22 13:07:21 |

Last modified on | 2013-03-22 13:07:21 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 14 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 55R10 |

Classification | msc 55R15 |

Defines | classifying space |