Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?


Remark: If the sphere had dimension k smaller than n-1, then such an immersion would exist if and only if the corresponding map from the k-sphere to the Stiefel manifold is 0-homotopic. This is the Hirsch-Smale Theorem and in fact an example of an h-principle. However the case k=n-1 is exactly the exceptional case which does NOT obey an h-principle. Easy examples (Figure 8.1. in the book by Eliashberg-Mishachev) show that there exist immersions of the circle in the plane which have a formal extension but not a genuine extension to the 2-disk. So, is there anything known about sufficient conditions for extendability?



Codimension zero immersions
This is subtle, even for $n=2$. In this case, clearly the problem reduces to $S^2$ or $\mathbb{R}^2$ since every surface has one of these as a universal cover. [Samuel Blank](http://www.ams.org/mathscinet-getitem?mr=2616455) found a criterion to determine if a curve in $\mathbb{R}^2$ bounds an immersed disk. An exposition has been given by [Valentin Poenaru](http://archive.numdam.org/article/SB_1966-1968__10__473_0.pdf), and the criterion has been extended to $S^2$ by [Frisch](http://arxiv.org/abs/1012.4923). There is also a bit of discussion in these papers about the higher dimensional problem. 


