Talk:Bundle structures and lifting problems (Ex)
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'''Exercise 4.1''' | '''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 $\gamma\mapsto\gamma(1)$. | + | * 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 first assertion follows. |
− | 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 first assertion follows. | + | |
* It is clear that composition of paths induces a group structure on $[Y,\Omega X]$. Thus the second assertion follows. | * It is clear that composition of paths induces a group structure on $[Y,\Omega X]$. Thus the second assertion follows. | ||
</wikitex> | </wikitex> |
Revision as of 19:27, 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 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.