I have the following question:
Let X $\subseteq$ $\mathbf{R^3}$ denote the unit sphere with the circle in the xy-plane. What is $π_1$(X)?
I think that this is an application of Van-Kampen Theorem. Specifically, consider the open sets $U_1$= X−{N} and $U_2$ =X−{P} where N,P denote the north pole of the sphere and P is a point on the circle , respectively. Notice that the intersection $U_1$ $\cap$ $U_2$= X−{N,P} is path connected and $U_1$$\cup$ $ U_2$=X . We see that $U_1$ deformation retracts onto the circle Thus, $π_1$($U_1$) is $Z$ and $U_2$ deformation retracts onto the sphere Thus $π_1$ ($U_2$) is trivial By Van-Kampen Theorem $π_1$(X)= $π_1$($U_1$)∗$π_1$($U_2$)/N (where N is the normal subgroup generated by identifying paths in the intersection as paths in $U_1$ and $U_2$) so $π_1$(X) is $Z$
Does this seem correct? Any comments or suggestions would be greatly appreciated.
This question was asked in this link but without using Van-Kampen Theorem