Fundamental group of cylinder minus a point

Question

I am trying to find the fundamental group of the closed cylinder minus a point $S_1\times[a,b]\setminus \{p\}$. My work so far: the space can be retracted to a graph like this:

I am not sure if this is correct, either.

Answer 1

Your deformation retraction, let us call it $X$, is correct. Since your space is path connected the fundamental group is independent of the choice of a base point.

As Thomas said the space is homotopy equivalent to $S^1\vee S^1$. You might recognize, that $\pi_1(X)=\pi_1(S^1\vee S^1)$, because if you take $x_0$ to be the midpoint of the line in $X$, then you have two loops generating all others. They both go along the line to one of the circles, then they go once around the circle and go back along the line to $x_0$, respectively. Each of this two loops $\alpha$ and $\beta$ generates non trivial loops going around one of the circles $n$ or $-n$ times (direction backwards). Every other non-trivial loop (e.g. it goes around the first circle n times and around the other m times, then -s times around the first one) can be constructed by composing such loops. But such loops $\alpha$ and $\beta$ are homotopic to the generating loops of the figure $8$. Hence it gives the same fundamental group.

This is somehow handwaving, but it might give you an intuition. A more formal approach is using the homotopy equivalence. Then by Seifert - van Kampen you can calculate $\pi_1(S^1\vee S^1)=\pi_1(S^1)\ast \pi_1(S^1)=\mathbb{Z}\ast\mathbb{Z}$ as we saw in the foregoing section intuitively.