Counting the genus of a model

I was reading a webpage on Euler-Poincaré Formula and it has a question that I couldn't figure out. It should be simple, but I am not getting the right answer.

The question on the webpage goes like this:

Consider the following model which is obtained by taking out a torus and tube from the interior of a sphere. What is the genus (penetrating hole) of this model?

So, the shape should look something like a sphere but with the red coloured model excluded from the sphere. The above yellowed coloured model shows the interior of the sphere.

Because the intention is to understand its topology, I cannot tear or glue the model. I could only stretch and squash the model. The answer given was $1$. However, I could only get $0$ as the answer.

I'm imagining the protruding pole in the middle of the sphere could be expanded to filled up the empty space surrounding it and eventually fills up to become a full sphere again. In this case, the genus is equals to $0$, which means, the topology of this model has no penetrating holes. However, I am wrong because the correct answer should be genus equals to $1$.

How is the genus of this model $1$ when I could easily get $0$?

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1 Answer

In your picture you are only looking at he half of the remains of the ball. The other half is obtained by reflection and glueing the halves togehter. In the middle of the ball is a connection between the two halves (stemming from the hole of the torus you removed, the pole you are referring to) which makes the complement homotopically nontrivial -- let a curve run around it and try to homotop this to a point. Actually, what you have here is just a deformed solid torus.

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Thanks, but I am still a little confuse. I don't get you by letting a curve run around it and homotop this to a point. What do you mean by running a curve and how does homotop changes the model to a solid torus? Sorry because I've only started out on topology and I could only imagine the models being squashed or stretched. –  xenon Aug 18 '12 at 17:13
@xEnOn You do need some imaganition to stretch the object you are referring to in such a way that you see the torus. Regarding the curve: Have a look at the red and green olympic ring, e.g. on this page: de.depositphotos.com/3569095/… If you think of the green one as a torus, the red one as a closed curve in the complement, you get to see what I have in mind. You cannot contract the red ring, cause the green one 'is in the way'. This would not occur if the green one had genus 0. (It is not a proof it is genus one, just an illustration). –  user20266 Aug 18 '12 at 18:04