Regular homotopy classes of immersions of 3-disks into $\mathbb R^3$

Question

What are the regular homotopy classes of immersions of disks $B^3 \to \mathbb{R}^3$ ?

Here, $B^3$ is the unit disk in $\mathbb{R}^3,$ i.e., $B^3 = \{ x\in\mathbb{R}^3\,;\, ||x||\leq 1\}.$ For immersion and regular homotopy, I am using the usual definitions, e.g., see https://en.wikipedia.org/wiki/Immersion_(mathematics) .

I suspect that there are exactly 2 classes. The first class contains the identity map, and the second class contains the "mirror" immersion that reverses orientation: $$i_{\textrm{mirror}} : B^3 \to \mathbb{R}^3\\ i_{\textrm{mirror}}:(x_1, x_2, x_3) \mapsto (x_2, x_1, x_3).$$ I do not know how to prove this (if it is even true).

There is the well-known result by Stephen Smale that there is only one class of immersions of the sphere $S^2$ into $\mathbb{R}^{3}$ (Sphere eversion and Smale-Hirsch theorem). My question is different because I am asking for immersions of the 3-disk into $\mathbb{R}^3.$ So, unlike in Smale's sphere eversion problem, my immersions are mapping between manifolds of the same dimension (3).

This question seems natural to pose, but I was not able to find an answer in literature. Maybe it has an easy answer? Has it been answered somewhere in literature?

Answer 1

As you say, there are only two classes, distinguished as follows: one that preserves local orientation; the other that reverses local orientation.

Here's the proof for the case that $f : B^3 \to \mathbb R^3$ preserves local orientation. We know that there exists $r > 0$ such that $f$ restricts to an embedding on the ball $r B^3 = \{x \mid |x| \le r\}$. Define a regular homotopy of $f$: $$H(x,t) = f((1-t+tr)x) $$ At time $t=1$, this becomes a diffeomorphism between $B^3$ and $r B^3$. Now there's still some work to do, to show that any two orientation preserving embeddings of $B^3$ are smoothly isotopic, but that's the gist of it.

Answer 2

In general, the space of immersions $\operatorname{Imm}(D^k,M^n)$ is homotopy equivalent to $Fr_k(TM)$ (the bundle of $k$-frames on $TM$. The proof of this follows basically from differentiation at the origin, and its homotopy inverse is provided by the exponetial map from Differential Geometry.

Regular homotopy classes are given by $\pi_0(Fr_k(TM))$.

When $M=\mathbb R^k$, this shows that $\pi_0 Imm(D^k, \mathbb R^k) \cong \pi_0(\mathbb R^k \times O(k))=\mathbb Z/2$. The two classes are as Lee Mosher says by checking determinant via the derivative map.

This is a kind of fun way of thinking about it because it is a special case of the Smale-Hirsch theorem (that you mention) which is a nonconstructive way to instead study regular homotpy classes of $S^2$ in $\mathbb R^3$ and show that $\pi_0$ of the resulting space is trivial, implying the existence of sphere eversion.

In particular, when $\operatorname{dim} N< \operatorname{dim} {M}$, one can show that immersions are in bijection with "formal immersions" which are vector bundle monomorphisms from $TN$ into $TM$ covering an arbitrary continuous map $f:N \to M$. This result also works when $N$ is non-closed and of the same dimension which is why your question is a special case.