Which of these - the Borsuk-Ulam theorem, the Brouwer's fixed point theorem (for the ball) and Tucker's lemma implies which? I'm a little confused with this. I suspect they may be equivalent. If that is indeed so, can someone please provide a proof (or a link to a proof) of the equivalence of the three?
Tucker's lemma and the Borsuk-Ulam theorem can be proved almost along the same lines (triangulations + Sperner's lemma or Ki Fan's lemma) and they are essentially equivalent. Brouwer's theorem is a consequence: however, Brouwer's theorem does not imply Borsuk-Ulam (at least, as far as I know). Have a look at this related question, too.