So I have been mulling the following question over in my head for awhile now, and want to see if anyone else might have any ideas.

Begin with $M$ a manifold and suppose that $M$ has an antipodal map $\alpha:M\rightarrow M$, i.e. $\forall m\in M$ one has:

$\alpha(m)\neq m$

$\alpha(\alpha(m)) = m$

Let $A^n$ be the usual antipodal map on $S^n$: $A^n(s) = -s$. What I am wondering is if there is always an embedding $e:M\hookrightarrow S^n$ (for some $n$, not necessarily of minimal dimension), such that $e\circ\alpha = A^n|_{e(M)}$, that is that the antipodal map on the submanifold extends to the entire sphere.

One thing I suspect may be true is that when $M$ has more than one class of antipodal map, the necessary dimension of the sphere may depend on which class of maps I start with.

As an example, if I start with $M = \mathbb{T}^2$ with coordinates $(\theta,\phi)$, then I can easily define two classes of antipodal maps:

$\alpha_1(\theta,\phi) = (\theta+\pi,\phi)$

$\alpha_2(\theta,\phi) = (\theta+\pi,\phi+\pi)$

I have been able to convince myself with mental pictures that I can embed $\mathbb{T}^2$ into $S^3$ such that $\alpha_2$ coincides with $A^3$ (although I could even be mistaken about this), but I cannot convince myself that there is an embedding such that $\alpha_1$ coincides with $A^3$, so perhaps for this class of antipodal map, one must embed in a higher dimensional sphere.

Anyone have any insights or know of any results in this direction?