Talk:Bundle structures and lifting problems (Ex)
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 $P(K(n+1,\mathbb{Z}/2\mathbb{Z}))\to K(n+1,\mathbb{Z}/2\mathbb{Z})$ is given by $\gamma\mapsto\gamma(1)$. Since $P(K(n+1,\mathbb{Z}/2\mathbb{Z}))$ is contractible, every third term in the long exact sequence in homotopy is zero. Thus we obtain for all $m\geq1$ that $\pi_m(K(n+1,\mathbb{Z}/2\mathbb{Z}))\cong\pi_{m-1}(\Omega K(n+1,\mathbb{Z}/2\mathbb{Z}))$. By the uniqueness of Eilenberg-MacLane spaces the assertion follows.