Universal Bundle
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High Quality Content by WIKIPEDIA articles! 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. The total space of a universal bundle is usually written EG. These spaces are of intere...
High Quality Content by WIKIPEDIA articles! 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. 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.
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