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.
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). The Borsuk–Ulam Theorem. In: Using the Borsuk–Ulam Theorem. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76649-0_2
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DOI: https://doi.org/10.1007/978-3-540-76649-0_2
Publisher Name: Springer, Berlin, Heidelberg
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