The Classifying Space of a Group

For any topological group F one can construct a contractible space ET on which the group F acts freely. This space ET is called the total space of the universal bundle of F. The terminology comes from the fact that one has a principal T-bundle with the base space FiF/F and the total space ET, which one calls the universal bundle of F. 

There are various ways to find the space ET with desired properties. A prominent one is the so-called Milnor construction  

Proposition 8.14. The space EF obtained by the Milnor construction is contractible. 

It is easy to see that a join of a connected space with a nonempty one is always k + l)-connected, and that a join of any two nonempty spaces has to be connected. It follows that an n-fold join of F with itself is (n — 2)-connected, and therefore all the homotopy groups of EF are trivial. Since EF has a natural structure of a CW complex, it follows by Corollary 6.32 of Whitehead s theorem that EF is contractible. 

It is also easy to present a concrete contraction of the space EF. One can construct a homotopy from the identity map to the map that maps the entire space EF to some point in EF in two steps  

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