TITLE: Surface Eversions: Generalizing from Sphere and Torus Eversions
QUESTION [10 upvotes]: In 1958, Smale proved that a $2$-sphere can be "turned inside out", and throughout the 60s, 70s, and 80s, explicit constructions such as Thurston Corrugations, and Minimax eversions were developed to visualize this sphere eversion. I just encountered a recent youtube video: Torus Eversion, which depicts the eversion of a torus. My general question is then this,

Is there a characterization, by Euler characteristic, genus, or some other topological invariant, for which surfaces have eversions?

If not a characterization of all such surfaces which have eversions, is there a finite list of surfaces which can be everted?

REPLY [10 votes]: I'll expand on my comment above. Consider an eversion of a sphere, namely a path of immersions $e_t: S^2\to \mathbb{R}^3$ such that $e_0$ and $e_1$ are embeddings, and which exchanges the two sides. By possibly reparameterizing the maps $e_t$, we may assume that there exists $D^2\subset S^2$ such that  $e_{t|D^2}: D^2\to \mathbb{R}^3$ is an embedding for all $t$. The map $e_{t|D^2}$ may be extended to an ambient isotopy $f_t$ of $\mathbb{R}^3$, which we may also assume takes a  ball $B_0$ intersecting $e_0(S^2)$ in $e_0(D^2)$ to a ball $B_1$ intersecting $e_1(S^2)$ in $e_1(D^2)$. Add some small trivial handles to $D_0$ inside $B_0$, to make a (smooth) embedded surface $\Sigma_g\subset \mathbb{R}^3$ of genus $g$. Then one may perform the regular homotopy $\sigma_t: \Sigma_g \to \mathbb{R}^3$ where $\sigma_{t | S^2-D^2} = e_{t | S^2-D^2}$, and $\sigma_{t| \Sigma_g-(S^2-D^2)}=f_t(\Sigma_g\cap B_0)$. Then $\sigma_1$ will be an embedding which exchanges the two sides of $\Sigma_g$. 
One may also use the torus eversion shown in the link to evert all surfaces of genus $g>0$ in a simpler fashion. 
For the 2-sphere, I like the eversion given a couple of years ago by Aitchison.