Let be a connected and locally connected based space and a covering map. We will denote , the fiber over the basepoint, by , and the fundamental group by . Given a loop with and a point there exists a unique with such that , that is, a lifting of starting at . Clearly, the endpoint is also a point of the fiber, which we will denote by .
Let , two loops homotopic relative and their liftings starting at . Then there is a homotopy with the following properties:
According to the lifting theorem lifts to a homotopy with . Notice that (respectively ) since they both are liftings of (respectively ) starting at . Also notice that that is a path that lies entirely in the fiber (since it lifts the constant path ). Since the fiber is discrete this means that is a constant path. In particular or equivalently .
By (1) the map is well defined. To prove that it is an action notice that firstly the constant path lifts to constant paths and therefore
Secondly the concatenation of two paths lifts to the concatenation of their liftings (as is easily verified by projecting). In other words, the lifting of that starts at is the concatenation of , the lifting of that starts at , and the lifting of that starts in . Therefore
This is a tautology: fixes if and only if its lifting starting at is a loop.
|Date of creation||2013-03-22 13:26:20|
|Last modified on||2013-03-22 13:26:20|
|Last modified by||mathcam (2727)|