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What are the constraints to extend an immersion of the sphere $S^2$ into $\mathbb{R^3}$ to an immersion of the closed unit ball $B(0,1)$ to $\mathbb{R}^3$?

Suppose, I get an immersion of $S^2$ into $\mathbb{R^3}$ which is close to $z\mapsto z^3$, where $\hat{\mathbb{C}}$ is identified with $S^2$. Is it possible that it comes from the restriction to $S^2$ of an immersion from the closed unit ball $B(0,1)$ to $\mathbb{R}^3$?

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$z \mapsto z^3$ has a branch point and I think that all nearby maps (into $R^3$) do, as well. So these maps are not immersions. I couldn't find anything obvious that addresses your first question. There are two papers that may be related to what you want: Smale's famous paper "A classification of immersions of the two-sphere" and Wells' "Cobordism groups of immersions". –  Sam Nead Nov 27 '11 at 16:48
    
You can approximate $z^3$ with an immersion. I will have a look to your references, thx. –  Paul Nov 27 '11 at 17:14
    
Just came across this question from 2011; have you ever found an answer? –  user53153 Feb 5 '13 at 23:06

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