Brouwer Fixed Point Theorem

Algebraic Topology · Axiom Academy

LESSON Brouwer Fixed Point Theorem A fundamental result connecting topology, algebra, and analysis with profound applications Let's start with the 1-dimensional case ( n = 1), where the theorem follows from the Intermediate Value Theorem. Consider D 1 = [0, 1] and a continuous function f : [0, 1] → [0, 1]. Define g ( x ) = f ( x ) - x . We have g (0) = f (0) - 0 ≥ 0 and g (1) = f (1) - 1 ≤ 0. By the Intermediate Value Theorem, there exists c ∈ [0, 1] where g ( c ) = 0, meaning f ( c ) = c . 2. Proof Strategy via Homology For higher dimensions, we use algebraic topology. The key insight is to prove by contradiction: assume no fixed point exists, and show this leads to a retraction from D n to its boundary S n -1 , which is impossible. This would make r a retraction , i.e., a continuous map r : D n → S n -1 such that r restricted to S n -1 is the identity. 3. No Retraction from Ball to Sphere The crucial algebraic topology result is: there is no retraction from D n to S n -1 . 4. Applications in Economics and Game Theory The Brouwer Fixed Point Theorem has remarkable applications beyond pure mathematics:

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