In today's article, we will delve into the fascinating world of Universal bundle. From its beginnings to the present, Universal bundle has been a topic of interest that has captured the attention of many people around the world. Throughout this article, we will explore the various aspects of Universal bundle, including its history, evolution, impact on society, and its relevance today. Through a detailed analysis, we will discover the reasons why Universal bundle has generated so much interest and how it has influenced different areas of daily life. Get ready to immerse yourself in the exciting world of Universal bundle and discover everything this theme has to offer!
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 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 → 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. These spaces are of interest in their own right, despite typically being contractible. For example, in defining the homotopy quotient or homotopy orbit space of a group action of G, in cases where the orbit space is pathological (in the sense of being a non-Hausdorff space, for example). The idea, if G acts on the space X, is to consider instead the action on Y = X × EG, and corresponding quotient. See equivariant cohomology for more detailed discussion.
If EG is contractible then X and Y are homotopy equivalent spaces. But the diagonal action on Y, i.e. where G acts on both X and EG coordinates, may be well-behaved when the action on X is not.