Fundamental group of this space

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!

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It looks like that one generator of fundamental group of torus vanishes since you can collapse a close path around the equator. –  Sigur Jan 25 '13 at 2:02
@Sigur yes, you're right. –  user42912 Jan 25 '13 at 2:10
@Sigur but what can you say about my proof? –  user42912 Jan 25 '13 at 2:11