I was reading about inversion of sphere. Wikipedia defines it as:
Let $f: S^2\to R^3$ be the standard embedding; then there is a regular homotopy of immersions $f_t\colon S^2\to R^3$ such that $f_0 = f $ and $f_1 = −f.$
I would like to know if such an inversion is impossible via embeddings. Thanks.
(Reposting my comment which I suppose is an answer)
It is impossible. By the generalized Jordan separation theorem, an embedding of the sphere separates $\mathbb{R}^3$ into two regions, and so we can define an orientation of $S^2$ by the field of outward pointing unit normals (i.e. they point towards the unbounded component of the complement). Given a homotopy through embeddings of your original embedding, this orientation will vary continuously and thus will not change. So it's not possible to end up with a reversed orientation.