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 $0$?