Homotopy quotient

In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group $G$ , is a specific bundle over a classifying space $BG$ , such that every bundle with the given structure group $G$ over $M$ is a pullback by means of a continuous map $M \to BG$ .

When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem.

We will first prove:

Proof. There exists an injection of $G$ into a unitary group $U (n)$ for $n$ big enough. If we find $EU (n)$ then we can take $EG$ to be $EU (n)$ . The construction of $EU (n)$ is given in classifying space for $U (n)$ .

The following Theorem is a corollary of the above Proposition.

Proof. On one hand, the pull-back of the bundle $π : EG \to BG$ by the natural projection $P \times G EG \to BG$ is the bundle $P \times EG$ . On the other hand, the pull-back of the principal $G$ -bundle $P \to M$ by the projection $p : P \times G EG \to M$ is also $P \times EG$

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