Borsuk-Ulam Implies Brouwer: A Direct Construction
Francis Edward Su · American Mathematical Monthly · 1997
Introduction The Borsuk-Ulam theorem and the Brouwer xed point theorem are well-known theorems of topology with a very similar avor. Both are non-constructive existence results with somewhat surprising conclusions. Most topology textbooks that cover these theorems (e.g., [4], [5], [6]) do not mention the two are related|although, in fact, the Borsuk-Ulam theorem implies the Brouwer Fixed Point Theorem. The theorems themselves are often proved using the machinery of algebraic topology or the concept of degree of a map. That one theorem implies the other can therefore be established once one understands this machinery, but this requires background. Moreover, such proofs tend to be indirect, relying on the equivalence of these existence theorems with corresponding non-existence theorems. For instance, Dugundji and Granas [3] show that the Borsuk-Ulam theorem is equivalent to the statement that no antipode-preserving, continuous map f : S