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 found a criterion to determine if a curve in $\mathbb{R}^2$ bounds an immersed disk. An exposition has been given by Valentin Poenaru, and the criterion has been extended to $S^2$ by Frisch. There is also a bit of discussion in these papers about the higher dimensional problem. Poenaru states that assuming the Poincare conjecture, an immersion of a sphere extends to an immersion of a ball if and only if it is regularly homotopic to an embedded sphere bounding a ball. Thus, the 3-dimensional case seems to be resolved now.
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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 found a criterion to determine if a curve in $\mathbb{R}^2$ bounds an immersed disk. An exposition has been given by Valentin Poenaru, and the criterion has been extended to $S^2$ by Frisch. There is also a bit of discussion in these papers about the higher dimensional problem. Poenaru states that assuming the Poincare conjecture, an immersion of a sphere extends to an immersion of a ball if and only if it is regularly homotopic to an embedded sphere bounding a ball. Thus, the 3-dimensional case seems to be resolved now. |
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