I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow \mathbb{R}^2$ ? Especially, is a computable invariant for such immersed closed curves ?
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3$\begingroup$ The paper "Extensions of codimension one immersions" by C. Pappas (1996) TAMS covers your question, although I suspect your question was answered earlier than that. The Pappas paper may give the appropriate reference. $\endgroup$Ryan Budney– Ryan Budney2014-09-18 14:13:31 +00:00Commented Sep 18, 2014 at 14:13
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3$\begingroup$ See this answer: mathoverflow.net/a/68021/1345 $\endgroup$Ian Agol– Ian Agol2014-09-18 23:14:15 +00:00Commented Sep 18, 2014 at 23:14
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Here is a cool example (due I believe independently to Eliashberg, Milnor, and
Blank) related to your question. There is an immersion of $S^1$ into $\mathbb{R}^2$ that extends to two different immersions of $D^2$ into $\mathbb{R}^2$. Best done with a picture but I
don't know how to up load one. A reference is page 150 of "Topology of Spaces of S-Immersions" by Eliashberg and Mishachev. Glueing these two together gives rise to a map from $S^2$ to $\mathbb{R}^2$ with only fold singularities which is not homotopic through such maps to the standard quish the 2-sphere onto the plane, which is the topic of the E and M paper.
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$\begingroup$ Thanks Ian! Now can you prove your true powers by making the two different immersions evident ;-). $\endgroup$Tom Mrowka– Tom Mrowka2014-09-23 19:40:22 +00:00Commented Sep 23, 2014 at 19:40