Can an immersion from $D^n$ to $S^n$ be surjective?

Question

Suppose there is an immersion $f$ from the closed disc $D^n$ to the sphere $\mathbb S^n$ (with the standard differential structures), where $n\ge 2$. Then can this map be surjective? I kind of hope it can't be, but have no idea....

Answer 1

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Basis}{\mathbf{e}}\newcommand{\eps}{\varepsilon}$For every $n \geq 2$, there exists a surjective immersion from a "barbell" comprising two $n$-balls joined by a solid cylinder to the $n$-sphere:

Here's a sketch of a construction.

Denote polar coordinates on the $n$-ball by $(r, \Vec{x})$, with $0 \leq r$, $\Vec{x}$ in $S^{n-1}$, and define $P:\Reals^{n} \to S^{n}$ by $$ P(r, \Vec{x}) = (\sin r)\Vec{x} + (\cos r)\Basis_{n+1}. $$ The map $P$ is an immersion in the open ball of radius $\pi$ centered at the origin.

Fix $0 < \eps < \frac{\pi}{2}$, and let $B_{1}$ and $B_{2}$ be the (disjoint) balls of radius $\frac{\pi}{2} + \eps$, with respective centers $\Vec{p}_{1} = \Vec{0}$ and $\Vec{p}_{2} = (2\pi, 0, \dots, 0)$. Map their union to the $n$-sphere by \begin{align*} \Vec{p}_{1} + (r, \Vec{x}) &\mapsto P(r, \Vec{x}), \\ \Vec{p}_{2} + (r, \Vec{x}) &\mapsto -P(-r, \Vec{x}) = (\sin r)\Vec{x} - (\cos r)\Basis_{n+1}. \end{align*} In words, map $B_{1}$ (blue) to cover a neighborhood of the northern hemisphere, and $B_{2}$ (green) to cover a neighborhood of the southern hemisphere.

Now take a smooth curve that starts at the boundary of the image of $B_{2}$ traveling "north", and ends at the boundary of the image of $B_{1}$ coming from the "south". Thicken this (in $S^{n}$) into a tube (purple). Modulo details of patching, this constructs a surjective immersion from the union of two balls and a cylinder to the $n$-sphere.