I have the following question:
Let $X \subset \mathbb{R}^3$ denote the unit sphere with the disk in the $xy$-plane. What is $\pi_1(X)$?
I think that this is an application of Van-Kampen Theorem. Specifically, consider the open sets $U_1 = X-\{N\}$ and $U_2 = X – \{S\}$ where $N,S$ denote the north and south pole of the sphere, respectively. Notice that the intersection $U_1 \cap U_2 = X-\{N,S\}$ is path connected and $U_1 \cup U_2 = X$. We see that $U_1$ deformation retracts onto the half sphere with the disk which in turn deformation retracts onto the sphere (move the disk in the middle upward to form another half sphere). Thus, $\pi_1(U_1)$ is trivial and for the same reason $\pi_1(U_2)$ is trivial. By Van-Kampen Theorem $\pi_1(X) = \pi_1(U_1)*\pi_1(U_2)/N$ (where $N$ is the normal subgroup generated by identifying paths in the intersection as paths in $U_1$ and $U_2$). Since $\pi_1(U_1)$ and $\pi_1(U_2)$ are trivial so is $\pi_1(X)$.
Does this seem correct? Any comments or suggestions would be greatly appreciated.
Best Answer
Yes, this is a correct application of van Kampen.
A little bit nitpicking:
The second transformation is not a deformation retraction. You can only deformation retract onto a subset and clearly the sphere is not a subset of half sphere. Your half sphere is simply homeomorphic to the sphere via projection with a base point somewhere inside the half sphere.