Talk:Bundle structures and lifting problems (Ex)
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* 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. | * 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. | ||
* 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. | ||
+ | * Using the long exact sequence of the fibration $O\to EO\to BO$ and that $EO$ is contractible we obtain for all $n\geq1$ that $\pi_n(BO)\cong\pi_{n-1}(O)$. | ||
+ | * Using the long exact sequence of the fibration $BSO\to BO\to K(1,\mathbb{Z}/2\mathbb{Z})$ we obtain for all $n\geq2$ that $\pi_n(BSO)\cong\pi_n(BO)$ and $\pi_1(BSO)=0$. | ||
</wikitex> | </wikitex> |
Revision as of 19:33, 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.
- 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 .