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Assume I have a closed curve $\gamma:\mathbb{S}^1\mapsto M$ of a compact $3$-manifold $M$. I want to remove the self-intersections, by looking at a ball (which is convex) in one of the maps, which contains self-intersections. In this ball I want to replace all the curves, by replacing the "pieces that map into this ball" with straight paths, like this:

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And these straights are then easy to rid of self-intersections. I have a problem with the choice of the ball though. I feel like running into a problem, if I have countably many "pieces that map into the ball". First of, is this even a problem? or is it no problem to replace countably many things in the curve? And if it is, can I somehow choose the ball differently, so that I only have finitely many of those straights? And if it isnt possible, is it, if I add, that the manifold has to be differentiable? or smooth?

William Elliot
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PlatinTato
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  • I suggest you look at the Hahn-Marurkiewicz theorem. A corollary is path connected Hausdorff spaces are arc connected. There you' ll discover how a curve can be smooth out so it does not cross itself. It is not a simple proof. General Topology by Stephen Willard has a proof. – William Elliot May 13 '17 at 07:20
  • See Mike Miller's answer here: https://math.stackexchange.com/questions/1448843/reference-for-self-intersections-of-immersions?rq=1 – Moishe Kohan Jan 30 '18 at 03:36

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