Torus minus a point homeomorphic to sphere with three points?

Question

I was able to show they are both homotopy equivalent to $S^1 \wedge S^1$. I cannot find any reason why they would not be homeomorphic, but I also cannot see any way to construct an explicit map. Any hints would be appreciated.

Answer 1

When you take a loop $\gamma$ which goes once around the torus $\Bbb T^2$ (and doesn't go on the point $p$ which was removed) then the complement of $\gamma$ in $\Bbb T^2-p$ is still connected.

The complement of any loop $\gamma$ in $\Bbb S^2-\{p_1,p_2,p_3\}$ is disconnected (because of Jordan curve theorem basically). Therefore they can't be homeomorphic.