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 assertion follows.