Borsuk - Ulam theorem states in particular that for any continous map $f:S^2\rightarrow \mathbb{R}^{2}$, there exists point $x\in S^{2}$ such that $f(-x)=f(x)$. I call $(x,-x)$ a "good" pair

My question is:

How should $f$ look like if we want to have at least two different good pairs? This is $x,y\in S^2$ such that $f(x)=f(-x)$ and $f(y)=f(-y)$ Is the answer connected to differentiability of $f$?

More generally i am intrested in the structure of set of these antipodal points. When does this set contain continuum of good pairs?



1 Answer 1


Obtaining a structure theorem on the set of good pairs for all such continuous functions is at best impossible; however, some specialized examples can answer the other questions. Trivially, a constant function gives you a continuum of good pairs. Regarding differentiable function with a single good pair, take for example $$f(x,y,z):=(x,y)$$ where $x^2+y^2+z^2=1$ and observe that the only good pair is the north and south poles —that is, $(0,0,1)$ and $(0,0,-1)$.


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