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Stephen Smale famously proved in [Trans. Amer. Math. Soc. 90 (1959), 281-290] that any two $C^2$ immersions $S^2\to\mathbb R^3$ are regularly homotopic. This is how we knew that one can do a sphere eversion before ever constructing such a thing, for example.

Later, Morris Hirsch wrote [Trans. Amer. Math. Soc. 93 (1959), 242-276], where he moves from the problem of immersing spheres to immersing general manifolds. His paper is based on Smale's, he says, in roughly the same way that obstruction theory is based on the theory of homotopy groups.

In his paper Hirsch uses his results to solve various problems in immersion theory, but I cannot find in it, nor after a google search, an answer to this:

What is the classification up to regular homotopy of immersions $M\to\mathbb{R^3}$ from a closed oriented surface?

In particular, is there just a single regular homotopy class of immersions of a closed orientable surface of positive genus? Can one everse a sphere with handles?

Later. Looking at the explicit construction of representatives of regular homotopy classes of immersions given in the paper by Joel Hass and John Hughes that Danny Ruberman mentions in his answer, the following question asks itself. In some sense, it is closer to the eversion problem, as when turning spheres inside-out we start and end with embeddings.

What is the classification up to regular homotopy of embeddings $M\to\mathbb{R^3}$ of a closed oriented surface?

Of course, along the homotopy the embedding may degrade into just an immersion. Corollary 3.3 in the paper states that if $M$ has genus $g$, then of the $4^g$ regular homotopy classes of immersions $M\to M\times[0,1]$ which are homotopic to the the obvious map $\iota:p\in M\mapsto (p,0)\in M\times [0,1]$ only one can be represented by an embedding, the one which contains $\iota$. So the second question above is what happens when we replace $M\times[0,1]$ by $\mathbb{R^3}$.

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(Based on the comment of Mariano Suárez-Álvarez, there was a false assumption in my original answer. This is an attempt to correct it.)

1) Let $M$ be a closed smooth manifold with $k < n$. According to Smale-Hirsch theory, the space of immersions $M^k \to \Bbb R^n$ is homotopy equivalent to the space of tangent bundle monomorphisms $TM\to \Bbb R^n$, by which I mean the space of sections of the fiber bundle over $M$ whose fiber at $x\in M$ is given by the space of linear injections $T_xM\to \Bbb R^n$. If $M$ is oriented, then this last space is identified with the space of linear orientation preserving isomorphisms $T_xM \oplus \Bbb R \to \Bbb R^n$.

2) In (1) there is no loss in assuming $M$ is equipped with a Riemannian metric and that the isomorphisms $T_xM \oplus \Bbb R \to \Bbb R^n$ are linear isometries.

3) Assume $k=2$ and $M$ is oriented. Then the space of immersions $I(M,\Bbb R^3)$ is identified with the space of sections of fiber bundle over $E\to M$ whose fiber at $x\in M$ is the space of orientation preserving linear isometries $T_x M\oplus \Bbb R \to \Bbb R^3$.

4) Any oriented surface $M$ is stably parallelizable since it embeds in $\Bbb R^3$. So a choice of stable parallelization gives a preferred isomorphism $T_x M \oplus \Bbb R\cong\Bbb R^3$. So the bundle in $(2)$ is trivializable: $E \cong M \times SO(3)$. Hence the space of sections coincides with the space of maps $\text{map}(M,SO(3))$.

Hence there is a bijection between the homotopy classes of immersions of $M$ in $\Bbb R^3$ and the set of homotopy classes $[M,SO(3)]$.

5) $SO(3) \cong \Bbb RP^3$ and the map $\Bbb RP^3 \to \Bbb RP^\infty$ is $3$-connected. So $[M,SO(3)] \cong H^1(M,\Bbb Z_2) \cong \oplus_{2g} \Bbb Z_2$, where $g$ is the genus.

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  • $\begingroup$ In (4) you mean $n=3$, I guess. If $M$ is not a torus, then $M$ is not going to be parallelizable. Trying to get to something liker your (5) is where I get stuck for a general surface. $\endgroup$ – Mariano Suárez-Álvarez Jul 9 '19 at 2:53
  • $\begingroup$ Yes, you're right. dumb mistake. I will repair it. $\endgroup$ – John Klein Jul 9 '19 at 4:27
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A paper of Joel Hass and John Hughes (Immersions of surfaces in 3-manifolds. Topology 24 (1985), no. 1, 97–112) gives a similar calculation where the target is an arbitrary 3-manifold. A bonus, which is of interest even in the case when the target is $\mathbb{R}^3$, is an explicit geometric representative for each regular homotopy class of immersions (in a given homotopy class).

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    $\begingroup$ The construction of representives they give is very nice! Thanks. $\endgroup$ – Mariano Suárez-Álvarez Jul 9 '19 at 3:16
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Regarding the classification up to regular homotopy of embeddings, U􏰄. Pinkall􏰀 proves in [Topology 24 (1984), 421–434] that

if $f$, $g:M\to\mathbb R^3$ are two embeddings of a compact orientable surface, then there is a diffeomorphism $h:M\to M$ such that $f$ and $g\circ h$ are regularly homotopic.

This is not quite as good as «one can everse a compact orientable surface», though: that the usual embedding $f:S^2\to\mathbb R^3$ and $g=-f$ are related as in Pinkall's theorem is obvious.

Pinkall notes that it is not true that two embeddings $f$, $g:M\to\mathbb R^3$ are necesarily regularly homotopic, but does not say anything how often they are.

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