Proving that the fundamental group of Klein bottle is generated by two elements without using covering spaces and van Kampen theorem, etc.

Question

This is a problem in 'Topology and Geometry' by Bredon. I tried hard on this problem, but have no idea what to do. This question was onced asked by someone, but there was no satisfactory answer.

Let $\mathbb{K}^2$ be Klein bottle. The problem asks us to show that $\pi_1 (\mathbb{K}^2 )$ is generated by two elements, say $\alpha$ and $\beta$ obtained from the "longitudinal" and "latitudinal" loops.

The problem is that this problem is given just after defining and proving some elementary facts about fundamental groups. No exact computation of fundamental groups is done until here. Even the fundamental group of the circle. Many useful tools such as covering spaces, path lifting, van Kampen Theorem, properly discontinuous actions are not introduced yet.

However, since this book presents some theory on differentiable manifolds before the fundamental group, we can use some facts about differential manifolds. For example, we have the Smooth Approximation Theorem and Sard's Theorem.

Also, there's a hint given in the book: Hint: Use the fact that a smooth loop must miss a point.

Answer 1

You should convince yourself with a drawing that the Klein bottle $K$ minus a point (not your base point) deformation retracts onto the wedge of circles $S^1\vee S^1$ generated by the two loops $\alpha$ and $\beta$. If $\gamma$ is a loop in $K$, it is based-homotopic to a loop that misses a point $\mathrm{pt}$ (necessarily other than the base point). Use the deformation retraction of $K\setminus\mathrm{pt}$ onto $S^1\vee S^1$ to homotope $c$ to a path drawn in $S^1\vee S^1$. The fundamental group of $S^1\vee S^1$ is generated by $\alpha$ and $\beta$, and so they also generate $\pi_1(K)$.

EDIT I'm afraid you're going to have to prove this the usual way: use the open cover $$\left\lbrace{\Huge\subset},\;{\Huge \mathsf{x}},\;{\Huge\supset}\right\rbrace$$ for $$S^1\vee S^1\equiv{\Huge\infty}$$ Given a based loop $\gamma$ in $S^1\vee S^1$, use the epilson lemma of Lebesgue on the interval $[0,1]$ to get an integer $N$ such $\gamma$ maps every interval $\left[\frac kN,\frac{k+1}N\right]$ into one of the open sets above, and show directly that $\gamma$ is homotopic to a path obtained from concatenations of $\alpha,\beta$ and their inverses. I don't think there's any other way if you can't use the van Kampen theorem and $\pi_1(S^1)\simeq \Bbb Z$.

Answer 2

There is a way around using Van Kampen. In a sense it is a dirty cheat, since the core idea of the argument is essentially the same as the proof of the Van Kampen theorem.

The idea is to use the supplied hint. By Sard's theorem, you can apply a small homotopy leaving the base-point fixed to any map $S^1 \to K^2$, ensuring the new map misses some point in the Klein bottle. Technically if your map $S^1 \to K^2$ is only continuous you first need to apply a smooth approximation, and then apply Sard. So now the map has the form $S^1 \to K^2 \setminus \{*\}$. But $K^2 \setminus \{*\}$ has a wedge of two circles as a deformation-retract, so those loops generate $\pi_1 K^2$.

You can use the same kind of argument to compute $\pi_1 S^1$ precisely, although in this case your map $S^1 \to S^1$ will likely hit every point in $S^1$ many times. Transversality could be applied to get some regularity, and define the degree of the map.