associated bundle construction
Let be a topological group, a (right) principal -bundle, a topological space and a representation of as homeomorphisms of . Then the fiber bundle associated to by , is a fiber bundle with fiber and group that is defined as follows:
The total space is defined as
where the (left) action of on is defined by
The projection is defined by
where denotes the –orbit of .
The above is well defined and defines a –bundle over with fiber . Furthermore has the same transition functions as .
Sketch of proof.
To see that is well defined just notice that for and , . To see that the fiber is notice that since the principal action is simply transitive, given any orbit of the –action on contains a unique representative of the form for some . It is clear that an open cover that trivializes trivializes as well. To see that has the same transition functions as notice that transition functions of act on the left and thus commute with the principal –action on . ∎
Notice that if is a Lie group, a smooth principal bundle and is a smooth manifold and maps inside the diffeomorphism group of , the above construction produces a smooth bundle. Also quite often has extra structure and maps into the homeomorphisms of that preserve that structure. In that case the above construction produces a “bundle of such structures.” For example when is a vector space and , i.e. is a linear representation of we get a vector bundle; if we get an oriented vector bundle, etc.
|Title||associated bundle construction|
|Date of creation||2013-03-22 13:26:46|
|Last modified on||2013-03-22 13:26:46|
|Last modified by||rspuzio (6075)|