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](http://www.youtube.com/watch?v=kQcy5DvpvlM), which depicts the eversion of a torus. My general question is then this,



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> Is there a characterization, by Euler characteristic, genus, or some other topological invariant, for which surfaces have eversions?
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If not a characterization of all such surfaces which have eversions, is there a finite list of surfaces which can be everted?



Surface Eversions: Generalizing from Sphere and Torus Eversions
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](https://arxiv.org/abs/1008.0916).


