I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology is that if $S$ is a noncompact connected surface, then $\pi_1(S)$ is a free group (possibly trivial or $\mathbb{Z}$). I've had a lot of people ask me for references for this fact. I know of two such references:
1) In section 44A of Ahlfors's book on Riemann surfaces, he gives a very complicated combinatorial proof of this fact.
2) This isn't a reference, but a high-powered 2-line proof. Introducing a conformal structure, the uniformization theorem shows that the universal cover of $S$ is contractible. In other words, $S$ is a $K(\pi,1)$ for $\pi=\pi_1(S)$. Next, since $S$ is a noncompact $2$-manifold, its integral homology groups vanish in dimensions greater than or equal to $2$. We conclude that $\pi_1(S)$ is a group of cohomological dimension $1$, so a deep theorem of Stallings and Swan says that $\pi_1(S)$ is free.
There should be a proof of this that you can present in a first course in topology! Does anyone know a reference for one?
Best Answer
I'm reluctant to advertise, but since no one else has answered yet, I'll mention the proof on pp. 142--144 of my book Classical Topology and Combinatorial Group Theory.