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
.