Talk:Bundle structures and lifting problems (Ex)
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Exercise 2.1
Assume that there exists a lift : of . Then there exist maps : and : such that for all we have . Since we find that . Furthermore we have for all : and . Thus defines a homotopy from to a constant map.
Assume that there exists : such that for all we have and . Define : by . By the definitions of and of we find that is well defined and a lift of .
Exercise 4.1
- The map is given by . Since is contractible, every third term in the long exact sequence in homotopy is zero. Thus we obtain for all that . By the uniqueness of Eilenberg-MacLane spaces the first assertion follows.
- It is clear that composition of paths induces a group structure on . Thus the second assertion follows.
- Using the long exact sequence of the fibration and that is contractible we obtain for all that .
- Using the long exact sequence of the fibration we obtain for all that and . Of course .
- The homotopy fiber of : can be calculated from the long exact sequence asociated to the fibration using the previous results. For all we obtain and .