Brouwer fixed point theorem

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Brouwer's fixed point theorem is a fixed point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x₀ such that f(x₀) = x₀. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself.

Among hundreds of fixed point theorems,^[1] Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem or the Borsuk–Ulam theorem.^[2] This gives it a place among the fundamental theorems of topology.^[3] The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed point theorem and its extension, the Kakutani fixed point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Gerard Debreu and Kenneth Arrow.

The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Hadamard, and then in 1912 by Luitzen Egbertus Jan Brouwer.

• 1 Statement
  • 1.1 Notes
• 2 Illustrations
• 3 Intuitive approach
  • 3.1 Explanations attributed to Brouwer
  • 3.2 One-dimensional case
• 4 History
  • 4.1 Prehistory
  • 4.2 First proofs
  • 4.3 Reception
• 5 Proof outlines
  • 5.1 A proof using homology
  • 5.2 A proof using Stokes's theorem
  • 5.3 A combinatorial proof
  • 5.4 A proof by Hirsch
  • 5.5 A proof using the game hex
  • 5.6 A proof using the Lefschetz fixed point theorem
  • 5.7 A proof in a weak logical system
• 6 Generalizations
• 7 See also
• 8 Notes
• 9 References
• 10 External links

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