Universal bundle

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 → BG. When the definition of the classifying space takes place within the homotopy of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem. We will first prove: Proposition. Let G be a compact Lie group. There exists a contractible space EG on which G acts freely. The projection EG → BG is a G-principal fibre bundle. 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. Theorem. If M is a paracompact manifold and P → M is a principal G-bundle, then there exists a map f : M → BG, unique up to homotopy, such that P is isomorphic to f ∗(EG), the pull-back of the G-bundle EG → BG by f. Proof. On one hand, the pull-back of the bundle π : EG → BG by the natural projection P ×G EG → BG is the bundle P × EG. On the other hand, the pull-back of the principal G-bundle P → M by the projection p : P ×G EG → M is also P × EG Since p is a fibration with contractible fibre EG, sections of p exist. To such a section s we associate the composition with the projection P ×G EG → BG. The map we get is the f we were looking for. For the uniqueness up to homotopy, notice that there exists a one-to-one correspondence between maps f : M → BG such that f ∗(EG) → M is isomorphic to P → M and sections of p. We have just seen how to associate a f to a section. Inversely, assume that f is given. Let Φ : f ∗(EG) → P be an isomorphism: Now, simply define a section by Because all sections of p are homotopic, the homotopy class of f is unique. The total space of a universal bundle is usually written EG.