Betti number of the cut torus

Question

I've read that the cut torus (the torus with a disk chopped off the end) has homotopy type of the figure eight curve.

Why is this and what is the Betti number of the cut torus?

Answer 1

You can represent the torus by identifying the opposite sides of a square. If you remove a disc in the middle, the resulting surface deformation retracts on the identification of the boundary which is the figure eight.

The Betti numbers of the resulting surface correspond to the Betti numbers of the figure eight. They are: $b_0=1, b_1=2, b_2=0$.

Answer 2

For those who want concrete evidence, there is a nice video of the punctured (cut torus) deformation retracting onto the figure eight.

https://www.youtube.com/watch?v=j2HxBUaoaPU