If a sphere eversion is possible using a half-way model how model is used for $cylinder$ eversion ?
I need to make some premises to be able to frame the true nature of the problem
In sphere eversion a sphere is transformed through homotopy into a Morin surface.
A half-way model is an immersion of the sphere $S^2$ into $R^3$.
$S^2$ is raised into $^2$ because from Smale Theorem the original sphere come from from $(n − 1)$-dimensional hyperplane, so when that 'sphere' (is not a real sphere but for simplicity we imagine that it is so) "it goes down" into a standard $R^3$ we have a embedding situation for hyperplane sphere (such as a group that is a subgroup).
The sphere $S^n$ can be turned inside out (more exactly, the standard embedding of $S^n$ into the Euclidean space $R^ {n+1}$ is regularly homotopic to the composition of the standard embedding and of the reflection with respect to an $(n − 1)$ dimensional hyperplane) if and only if $n ∈ {0, 2, 6}$.
This happens because
classes of immersions of spheres can be realizated using a special group: Regular homotopy group of immersions but with a particular precaution to avoid that $S^2$ sphere in $R^3$ vanishes: the standard embedding and the inside-out one must be regular homotopic.
But this is no sufficient because we need that
their Gauss maps have the same degree/winding number.
But also this is not sufficient because we must have consider homotopy groups of Stiefel manifolds. Why Stiefel manifold ?
In other words real Stiefel manifold of $\mathbb{R}^k$ of canonical embedding is the coset topological space where the action (group action) is via its canonical embedding. See here
I remember that also that
every finite group G can be embedded in a symmetric group
My question: from what manifold we need to start to build an eversion for any-tridimensional object ?