Fundamental group of 2-torus minus a single point

Question

So, if I take one torus and take off one single point, what will be its fundamental group? I think that one single point will not change the topology in this sense. Or will it? If so, how?

Answer 1

If we remove a point the topology will change and the fundamental group will change drastically! The fundamental group of the torus is $$ \pi_1(T)=Z\oplus Z $$ but if you remove a point the fundamental will be even a non abelian group: $$ \pi_1(T-\{p\})= F(\alpha, \beta) $$ where $\alpha$ and $\beta$ are the two cycles of the figure 8, and $F(\alpha, \beta)$ is the free group generated by $\alpha$ and $\beta$.

All theses problems appear because of the existence of the so called "Peano curves" https://en.wikipedia.org/wiki/Space-filling_curve (which are really complicated curves that are surjective). For example following this naive idea that removing a point does not change the topology we could end up with the following WRONG proof for the sphere:

1-"We remove a point to $S^2$ and does not change anything", then $S^2-{p} \simeq R^2$, but $R^2$ is contractible and we get $\pi_1(S^2) = \pi_1(S^2-{p}) = \pi_1(R^2)=0$.

Although the achieved result is true, we have to be careful because this proof is COMPLETELY WRONG! And the reason is the subtle idea that there exist paths filling $S^2$ which clearly will not have a homotopy equivalence class in $\pi_1(S^2-{p})$.