SPERNER’S LEMMA AND BROUWER’S FIXED POINT THEOREM

A fixed point of a function f from a set X into itself is a point x0 satisfying f(x0) = x0. Theorems which establish the existence of fixed points are very useful in analysis. One of the most famous is due to Brouwer. Let Dn = {x ∈ R: |x | ≤ 1}. Theorem (Brouwer’s Fixed Point Theorem.). Every continuous func-tion f: Dn → Dn has a fixed point. The purpose of this note is to provide a proof of Brouwer’s Fixed Point Theorem for n = 2 using a combinatorial result know as Sperner’s Lemma, and to explore both Sperner’s Lemma and Brouwer’s Fixed Point Theorem. We begin by examining the case where n = 1. Proof for n = 1. We have f: [−1, 1] → [−1, 1]. Define g(x) = x−f(x). Then g(1) ≥ 0 and g(−1) ≤ 0 so by the Intermediate Value Theorem there must be a point x0 so that g(x0) = 0. We then have f(x0) = x0. Definition. A fixed point domain is a set G ⊂ Rn such that every continuous function from G to itself has a fixed point. So Brouwer’s Fixed Point Theorem asserts that each Dn, n ≥ 1, is a fixed point domain. On the other hand, Dn less a point, [0, 1] ∪ [2, 3] and R are not fixed point domains. (It is easy to construct appropriate functions with no fixed point for these sets.) The fact that Dn is compact is important in Brouwer’s Fixed Point Theorem. However, if G is a fixed point domain, and f: G → H is a continuous bijection with a continuous inverse, then H is a fixed point domain. Proof. If g: H → H is continuous, so is f−1 ◦ g ◦ f. As this is a continuous function from G to G, it has a fixed point x0. Now (