Talk:Bundle structures and lifting problems (Ex)
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By the definitions of $h$ and of $\mathrm{hofib}(p)$ we find that $\overline{f}$ is well defined and a lift of $f$. | By the definitions of $h$ and of $\mathrm{hofib}(p)$ we find that $\overline{f}$ is well defined and a lift of $f$. | ||
</wikitex> | </wikitex> | ||
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+ | '''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. |
Revision as of 19:20, 2 April 2012
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.