We know that a sphere can be turned inside out. A nice video can be found here.
So now I wonder what happens to a genus $g$ surface $\Sigma_g$.
My Try: I basically want to use Smale Hirsch theory to calculate the path components of this space. Now the formal immersions correspond to maps from $\Sigma_g \to SO(3)$. But I am stuck at this point because I am not sure how one would calculate the homotopy classes of maps from $\Sigma_g$ to $SO(3)$. Any help is appreciated. Thanks