The Borsuk–Ulam Theorem
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The Borsuk–Ulam theorem is one of the most useful tools offered by elementary algebraic topology to the outside world. Here are four reasons why this is such a great theorem: There are (1) several different equivalent versions, (2) many different proofs, (3) a host of extensions and generalizations, and (4) numerous interesting applications. As for (1), Borsuk’s original paper [Bor33] already gives three variants. Below we state six different but equivalent versions, all of them very useful, and several more are given in the exercises.
KeywordsSimplicial Complex Antipodal Point Combinatorial Proof Dimensional Simplex Antipodal Mapping
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