Non-injective antipodal preserving map between spheres My question is if $n\leq m$ then is there a non-injective antipodal preserving (continuous) map
$f:S^n\to S^m$ ? I wonder if such map exists. Can anyone give me a concrete example?
 A: Yes, it exists.
Wlog $m=n$. Consider the natural decomposition
$$S^n = A \sqcup S^{n-1} \sqcup (-A)$$
where $A \approx \operatorname{int} D^n$ and $A \sqcup S^{n-1} \approx S^{n-1} \sqcup (-A) \approx D^n$. Now take any non-injective function $g : D^n \to D^n$ such that $g \restriction S^{n-1} = \mathrm{id}$. I will not give an explicit formula, but it should be obvious that such a function exists (just fold a little close to the center).
We define $f : S^n \to S^n$ so that (via the mentioned identification)
$$f(x) = \begin{cases} g(x) & \text{if } x \in A \sqcup S^{n-1} \\ -g(-x) & \text{if } x \in S^{n-1} \sqcup (-A) \end{cases}$$
Note that the function is well defined because when $x \in S^{n-1}$, the two formulas agree:
$$g(x) = x = -(-x) = -g(-x).$$
Also it is clear from definition that $f$ is antipodal preserving, as if $x \in A \sqcup S^{n-1}$, then $-x \in S^{n-1} \sqcup (-A)$ and vice versa. Finally $f$ is continuous on both $A \sqcup S^{n-1}$ and $S^{n-1} \sqcup (-A)$, which makes it continuous everywhere.
