In mathematics, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D ⁿ to itself has a fixed point. In this theorem, n is any positive integer, and the closed unit ball is the set of all points in Euclidean n-space Rⁿ which are at distance at most 1 from the origin. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... See also fixed-point arithmetic. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

The theorem has several "real world" illustrations. One common informal version is "you can't comb a hairy ball smooth"; another one works as follows: take two equal size sheets of graph paper with coordinate systems on them, lay one flat on the table and crumple up (but don't rip) the other one and place it any way you like on top of the first. Then there will be at least one point of the crumpled sheet that lies exactly on top of the corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet right beneath it.

The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case n = 3 was proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910, and Brouwer found a different proof in 1912. Since it must have an essentially non-constructive proof, it ran contrary to Brouwer's intuitionist ideals. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ... Jacques Solomon Hadamard (December 8, 1865 - October 17, 1963) was a mathematician best known for his proof of the prime number theorem. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ...

Proof outline

A full proof of the theorem would be too long to reproduce here, but the following paragraph outlines a proof omitting the difficult part. It is hoped that this will at least give some idea why the theorem might be expected to be true. Note that the boundary of D ⁿ is S ^n-1, the (n-1)-sphere. A sphere is, roughly speaking, a ball-shaped object. ...

Suppose f : D ⁿ → D ⁿ is a continuous function that has no fixed point. The idea is to show that this leads to a contradiction. For each x in D ⁿ, consider the straight line that passes through f(x) and x. There is only one such line, because f(x) ≠ x. Following this line from f(x) through x leads to a point on S ^n-1. Call this point g(x). This gives us a continuous function g : D ⁿ → S ^n-1. This is a special type of continuous function known as a retraction: every point of the codomain (in this case S ^n-1) is a fixed point of the function. Given a function , the set B is called the codomain of f. ...

Intuitively it seems unlikely that there could be a retraction of D ⁿ onto S ^n-1, and in the case n = 1 it is obviously impossible because S ⁰ isn't even connected. The case n=2 takes more thought, but can be proven by using basic arguments involving the fundamental groups. For n > 2, however, proving the impossibility of the retraction is considerably more difficult. One way is to make use of homology groups: it can be shown that H_n-1(D ⁿ) is trivial while H_n-1(S ^n-1) is infinite cyclic. This shows that the retraction is impossible, because a retraction cannot increase the size of homology groups. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ...

There is also an almost elementary combinatorial proof. Its main step consists in establishing Sperner's lemma in n dimensions. A combinatorial proof is a method of proving a statement, usually a combinatorics identity, by counting some carefully chosen object in different ways to obtain different expressions in the statement (see also double counting). ... In combinatorial mathematics, Sperners lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. ...

A quite different proof can be given based on the game of Hex. The basic theorem about Hex is that no game can end in a draw. This is equivalent to the Brouwer fixed point theorem for dimension 2. By considering n-dimensional versions of Hex, one can prove that in general that Brouwer's theorem is equivalent to the "no draw" theorem for Hex. Hex is a board game played on a hexagonal grid, usually in the shape of a 10 by 10 or a 11 by 11 rhombus. ...

Generalizations

• Lefschetz fixed-point theorem
• For a number of generalizations of the Brouwer fixed point theorem to infinite dimensions, see fixed point theorems in infinite-dimensional spaces.

In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some... In mathematics, a number of fixed point theorems in infinite-dimensional spaces generalise the Brouwer fixed point theorem. ...

External link

• Brouwer's Fixed Point Theorem for Triangles (http://www.cut-the-knot.org/do_you_know/poincare.shtml#brouwertheorem) (based on Sperner's lemma)