Borsuk-Ulam Implies Brouwer: A Direct Construction

The Borsuk-Ulam theorem and the Brouwer fixed point theorem are well-known theorems of topology with a very similar flavor. 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 → S can be homotopic to a constant map. From this one can see that the Brouwer fixed point theorem is a special case, because it can be shown equivalent to the statement that the identity map id : S → S (which is antipode-preserving) is not homotopic to a constant map. However, such an indirect approach is not really necessary, and perhaps a more direct proof would give insight as to how the two theorems are related. The purpose of this note is to provide a completely elementary proof that the Borsuk-Ulam theorem implies the Brouwer theorem by a direct construction, in which the existence of antipodal points in one theorem yields the asserted fixed point in the other.