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 allwe obtain
and
.