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

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.

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Thank you very much you helped not only with this question but with the question I linked about homology groups of this space. –  user42912 Jan 25 '13 at 11:16
one question, do you know where can I find a book with these constructions (except Hatcher's book)? these constructions aren't so obvious at the first glance. –  user42912 Jan 25 '13 at 11:18
@user42912 I was about to introduce you the Hatcher's book. You just have to learn the first two chapters you will get the idea. There are a lot of deformations of CW cplx. –  lee Jan 25 '13 at 11:40

Nitpicking, but I suppose you could be more explicit about the relation $\beta=\gamma$, which comes from the inclusion $\iota:U\cap V\to U$, $\iota_*(\gamma)=\beta$. So you would have

$$\pi_1(X)=\langle\alpha,\beta,\gamma|\alpha\beta=\beta\alpha,\beta=\gamma\rangle$$

As always, it depends how familiar your audience (and you!) are with the subject.

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