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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.

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.

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.

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.

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 give rise to an 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.

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.