Prove that the Borsuk-Ulam theorem and following ttheorem are equivalent
Borsuk-Ulam theorem: Let $f:S^k \to R^{n+1}$ be a smooth map whos image does not contain the origin, and supposed that $f$ satisfies the symmetry condition $f(-x)=-f(x)$ for all $x\in S^k$ then $W_2(f,0)=1$
Theorem A : if $f:S^k→S^k$ carries antipodal point to antipodal points then $deg_2(f)=1$
So I need to show that Borsuk-Ulam theorem implies theorem A, and the other way around
So assume that the Borsuk theorem is true, let $f:S^k \to S^k$ carries antipodal point to antipodal point. this is where I'm confuse. I know that the antipodal map is mapping $x\to -x$, what do they mean by saying carries antipodal point to antipodal point? Is that mean they use the antipodal map twice?
by definition, the winding number $W(f,0)$ of $ f:S^n\rightarrow R^{n+1}$ is equal to the degree of the function induced $ \frac{f}{||f||}=\overline{f}:S^n\rightarrow S^n$.