Talk:Bundle structures and lifting problems (Ex)
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* 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 $O\to EO\to BO$ 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 $\pi_1(BSO)=0$. | + | * 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 $\pi_1(BSO)=0$. Of course $\pi_0(BSO)=0$. |
− | + | * The homotopy fiber of $p$: $BSO\to BO$ can be calculated from the long exact sequence asociated to the fibration | |
+ | $$ | ||
+ | \mathrm{hofib}(p)\to BSO\to BO | ||
+ | $$ | ||
+ | using the previous results. For all $n\geq1$ we obtain $\pi_n(\mathrm{hofib}(p))=0$ and $\pi_0(\mathrm{hofib}(p)))\mathbb{Z}/2\mathbb{Z}$. | ||
</wikitex> | </wikitex> |
Revision as of 19:37, 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 . Of course .
- The homotopy fiber of : can be calculated from the long exact sequence asociated to the fibration
using the previous results. For all we obtain and .