Reference for self-intersections of immersions

Question

I believe the following is a true fact: if $f : S^1 \to \mathbb{R}^2$ is an immersion, there is an arbitrarily small homotopy of f to another immersion $g : S^1 \to \mathbb{R}^2$ such that $g$ only intersects itself finitely many times. This probably has a much more general statement with arbitrary manifolds in place of $S^1$ and $\mathbb{R}^2$. I've been looking and can't seem to find this sort of basic stuff anywhere. Does anyone know where can I find it?

Answer 1

This is a very good question. (I wish I could award bounties to people who ask good questions!) The results stated below are frequently stated as fact in more advanced topology references and never mentioned in the more introductory ones. My bible for these sort of fundamental statements tends to be Hirsch's differential topology book; I highly recommend looking through it.

The correct statement should say that for $M$ closed, immersions with only transverse double self-intersections (the preimage of a point is at most two points) are dense in all maps $M \to N$, where $\dim N \geq 2\dim M$.

This is proved in Hirsch's "Differential topology". First he proves the related statements that immersions (with no restriction on preimages) are dense if $\dim N \geq 2\dim M$ and that embeddings are dense if $\dim N > 2\dim M$; these are theorems 2.2.12 and 2.2.13. He later provides a different proof of the density of immersions at the end of the transversality section (3.2.9) and leaves your question as an exercise, the first of section 3.2. I haven't attempted the exercise but I suspect it should follow from essentially the same techniques as the rest of the section. I'll probably do it tomorrow if you don't get a chance to before then.