Q:

A question on eversion of (odd) spheres

At  the right  column of the  page  654  of  the  paper, 
R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice  AMS it  is  written "There can be no   eversion of a circle (or  any   odd dimensional spheres)".
In that paper the  eversion is  defined  as a  regular homotopy  between identity  and the  antipodal map.
I  wonder  what is  my  mistake  to think that $f_t(z)=e^{it}z, \quad t\in [0, \pi]$ is  a  regular homotopy  between the  identity and the  antipodal map on the  circle?

A:

Let me summarize the comments. An "eversion" of $S^n$ is a regular homotopy between, on the one part the "identity" embedding $S^n \to \mathbb{R}^{n+1}$, and on the other part an embedding $S^n \to \mathbb{R}^{n+1}$ that turns the sphere inside out.
For an even sphere, the antipodal map turns the sphere inside out (AKA reverses orientation), thus we speak of an eversion for a regular homotopy between the identity and the antipodal map. But for an odd sphere, the antipodal map preserves orientation, so we must use something else. For example, the map $S^{2n+1} \to \mathbb{R}^{2n+2}$ given by $(x_1,\dots,x_{2n+2} \mapsto (-x_1,\dots,-x_{2n+1},x_{2n+2})$. I suspect that this is essentially a typo in Palais's paper. People aren't used of talking about eversion of odd spheres, because, well, they don't exist...
Let me also point out that an eversion is not a homotopy of self-maps $S^n \to S^n$. So notions such as the degree of a map are not preserved. It is not a contradiction that there exists an eversion for even spheres, even though the degree of the identity is $1$ and of the antipodal map $-1$.
However, there exists a different invariant, called the "turning number". This invariant is basically the degree of the self-map of $S^n$ induced by the differential of an embedding $f : S^n \to \mathbb{R}^{n+1}$ (where we used the fact that a regular homotopy is an immersion at each moment, to get a self map of $S^n$: the Gauss map). The turning number of the identity is $1$. The turning number of the "not antipodal" map $S^{2n+1} \to S^{2n+1}$ is $-1$, so there can be no eversion of an odd sphere. However, the turning number of the antipodal map of $S^{2n}$ is $1$, so there is no contradiction.
The legends say – but it was before my birth, so others may confirm/infirm/clarify – that Smale's advisor's, Bott, thought that Smale's proof of the existence of an eversion of the $2$-sphere(1) was wrong; without even looking at the proof, just based on general expectations. Apparently Bott either because he believed an eversion preserved the degree, or (more likely?) because he thought the turning number of the antipodal was $-1$. As we now know, Smale's result was correct: there exists an eversion for $S^2$. It was actually a shock at the time, I gather. It was especially frustrating since the proof was not constructive, and people had to wait years for an explicit eversion.
(1) and of $S^6$, and the proof of inexistence of an eversion for other even spheres – thanks to Mike Miller for this clarification!

