Is there a more elementary (but not trivial) example of a fundamental group than that of $S^1$?

Question

I just watched a video lecture which proved that the fundamental group of the $1-$sphere $S^1$ is the integers $\mathbb Z$ under addition. While I followed most of the proof, it was very long and technical (the video was over one hour), and though the end result was interesting, the proof itself was not particularly motivating to me.

Are there some simpler examples of non-trivial fundamental groups of (path-connected) topological spaces, to help motivate the definition of fundamental group and also to see how one might go about computing the fundamental group of a space?

Answer 1

There does not exist any simpler example. Indeed, proving that any space has nontrivial fundamental group is at least as hard as proving that $S^1$ has nontrivial fundamental group, because if any space has nontrivial fundamental group it follows immediately that $S^1$ must have nontrivial fundamental group. To see this, suppose $\pi_1(S^1,x_0)$ is trivial. Then in particular the identity map of pointed spaces $(S^1,x_0)\to (S^1,x_0)$ is nullhomotopic. This then gives a nullhomotopy of any pointed map $(S^1,x_0)\to (Y,y_0)$ for any pointed space $(Y,y_0)$, by simply composing with a nullhomotopy of the identity map on $(S^1,x_0)$, and so $\pi_1(Y,y_0)$ is trivial.

Answer 2

This is a somewhat incomplete answer, but by and large the computation of $\pi_1(X)$ for a space $X$ is just as difficult as computing $\pi_1(S^1)$, and often actually uses the fact that $\pi_1(S^1) \cong \mathbb{Z}$.

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$S^1$ has the most elementary (non-trivial) fundamental group computation: there is a direct hands-on argument (see for example Massey's "A Basic Course in Algebraic Topology", Chapter II $\S 5$). The gist is that you divide $S^1$ into two halves, consider a loop $\gamma$ in $S^1$, and analyze the way $\gamma$ travels between the two halves, and describe how to homotope $\gamma$ into a loop that goes directly around the circle an integer number of times. The argument is elementary, but it's extremely difficult to do something similar for other spaces.

For a more general technique that can be used on other spaces, there are a few options. There's the Seifert-van Kampen theorem, where you write your space as a union of two open sets (with some assumptions) and your fundamental group is described as an "amalgamated product" of the fundamental groups of the two open sets and their intersection. A typical application of this theorem is to compute the fundamental group of any closed surface in terms of its fundamental polygon, and the usual argument uses the fact that $\pi_1(S^1) \cong \mathbb{Z}$. Unfortunately this theorem does NOT apply to compute $\pi_1 (S^1)$, but there is a more sophisticated version for "fundamental groupoids" which does apply (it may seem very daunting with all the abstraction, see for example this question, but as Eric Wofsey commented it actually essentially boils down to the argument I sketched above).

Another general tool for computing fundamental groups is the theory of Covering Spaces, which also isn't really elementary, though the arguments tend to be very nice. One outcome is that if $G$ is a discrete group acting on a simply-connected space $X$ (in a nice enough way) then the quotient space $X/G$ has fundamental group $G$. This theory does apply to computing $\pi_1(S^1) \cong \mathbb{Z}$, since $\mathbb{Z}$ acts on $\mathbb{R}$ by translation and $\mathbb{R}/\mathbb{Z} \cong S^1$, but the hands-on argument above is more elementary. Another example is the projective space $\mathbb{R}P^n$ for $n \geq 2$: the group $\mathbb{Z}/2$ acts on $S^n$, where the non-trivial element sends a vector $v\in S^n$ to $-v$ and the quotient by this action is by definition the projective space, so $\pi_1(\mathbb{R}P^n)\cong \mathbb{Z}/2$. (More generally there is a theory of Fibrations which can be used to compute $\pi_1$ in some nice cases, but covering spaces are deeply linked to the fundamental group.)