Based on this question: What is the homology groups of the torus with a sphere inside?
I'm trying to find the fundamental group of this space using the Seifert–van Kampen theorem. If $U$ is the torus and $V$ is the sphere, then $U\cap V$ is the circle, thus we have the following fundamental groups:
$\pi_1(U)=\mathbb Z\times\mathbb Z$
$\pi_1(V)=0$
$\pi_1(U\cap V)=\mathbb Z$
If we use the group presentation notation we have:
$\pi_1(U)=\langle\alpha,\beta\mid \alpha\beta=\beta\alpha\rangle$
$\pi_1(V)=\langle\emptyset\mid\emptyset\rangle$
$\pi_1(U\cap V)=\langle\gamma\mid\emptyset\rangle$
Thus using the Seifert–van Kampen theorem:
$\pi_1(X)=\langle\alpha,\beta\mid\alpha\beta=\beta\alpha,\beta\rangle$
Note that I added $\beta$ above because when we turn around the generator of $S^1$ which is $U\cap V$, we span one of the generators of the torus which is $U$.
Thus the fundamental group of this space is $\mathbb Z\times \{0\}$ which is $\mathbb Z$ itself.
My approach is correct?
Thanks a lot!
Best Answer
The approach is correct, but you can use a simpler method. The space you care about is homotopic to the wedge sum of two spheres and one circle. Thus the fundamental group is Z. To see why this space is homotopic to the space I said, you just check first that you can contract the upper half-sphere, thus the lower half-sphere becomes to a sphere, and the torus becomes a sphere with its south and north poles glued together. A sphere with its south and north poles glued together is homotopic to a sphere with a line connect its south and north poles. After these deformations you have a space which is what I said above.