Brouwer fixed-point theorem

Template:Short description Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function <math>f</math> mapping a nonempty compact convex set to itself, there is a point <math>x_0</math> such that <math>f(x_0)=x_0</math>. The simplest forms of Brouwer's theorem are for continuous functions <math>f</math> from a closed interval <math>I</math> in the real numbers to itself or from a closed disk <math>D</math> to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset <math>K </math> of Euclidean space to itself.

Among hundreds of fixed-point theorems,<ref>E.g. F & V Bayart Théorèmes du point fixe on [email protected] Template:Webarchive</ref> 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, the invariance of dimension and the Borsuk–Ulam theorem.<ref>See page 15 of: D. Leborgne Calcul différentiel et géométrie Puf (1982) Template:ISBN</ref> This gives it a place among the fundamental theorems of topology.<ref>More exactly, according to Encyclopédie Universalis: Il en a démontré l'un des plus beaux théorèmes, le théorème du point fixe, dont les applications et généralisations, de la théorie des jeux aux équations différentielles, se sont révélées fondamentales. Luizen Brouwer by G. Sabbagh</ref> 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 Kenneth Arrow and Gérard Debreu.

The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile 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 case of differentiable mappings of the Template:Mvar-dimensional closed ball was first proved in 1910 by Jacques Hadamard<ref name="hadamard-1910">Jacques Hadamard: Note sur quelques applications de l'indice de Kronecker in Jules Tannery: Introduction à la théorie des fonctions d'une variable (Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French)</ref> and the general case for continuous mappings by Brouwer in 1911.<ref name="brouwer-1910">Template:Cite journal</ref>

Statement

The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:

In the plane

Every continuous function from a closed disk to itself has at least one fixed point.<ref>D. Violette Applications du lemme de Sperner pour les triangles Bulletin AMQ, V. XLVI N° 4, (2006) p 17. Template:Webarchive</ref>

This can be generalized to an arbitrary finite dimension:

In Euclidean space

Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.<ref>Page 15 of: D. Leborgne Calcul différentiel et géométrie Puf (1982) Template:ISBN.</ref>

A slightly more general version is as follows:<ref>This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see Template:Cite book</ref>

Convex compact set

Every continuous function from a nonempty convex compact subset K of a Euclidean space to K itself has a fixed point.<ref>V. & F. Bayart Point fixe, et théorèmes du point fixe on Bibmath.net. Template:Webarchive</ref>

An even more general form is better known under a different name:

Schauder fixed point theorem

Every continuous function from a nonempty convex compact subset K of a Banach space to K itself has a fixed point.<ref>C. Minazzo K. Rider Théorèmes du Point Fixe et Applications aux Equations Différentielles Template:Webarchive Université de Nice-Sophia Antipolis.</ref>

Importance of the pre-conditions

The theorem holds only for functions that are endomorphisms (functions that have the same set as the domain and codomain) and for nonempty sets that are compact (thus, in particular, bounded and closed) and convex (or homeomorphic to convex). The following examples show why the pre-conditions are important.

The function f as an endomorphism

Consider the function

<math>f(x) = x+1</math>

with domain [-1,1]. The range of the function is [0,2]. Thus, f is not an endomorphism.

Boundedness

Consider the function

<math>f(x) = x+1,</math>

which is a continuous function from <math>\mathbb{R}</math> to itself. As it shifts every point to the right, it cannot have a fixed point. The space <math>\mathbb{R}</math> is convex and closed, but not bounded.

Closedness

Consider the function

<math>f(x) = \frac{x+1}{2},</math>

which is a continuous function from the open interval <math>(-1,1)</math> to itself. Since the point <math>x=1</math> is not part of the interval, there is no point in the domain such that <math>f(x) = x</math>. The set <math>(-1,1)</math> is convex and bounded, but not closed. On the other hand, the function <math>f</math> does have a fixed point in the closed interval <math>[-1,1]</math>, namely <math>x=1</math>. The closed interval <math>[-1,1]</math> is compact, the open interval <math>(-1,1)</math> is not.

Convexity

Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball <math>D^n</math>. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.).

The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function <math>f(x)=-x</math>, which is a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point. The analogous example works for the n-dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex) . The function f Template:Em have a fixed point for the unit disc, since it takes the origin to itself.

A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from the Lefschetz fixed-point theorem.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Notes

The continuous function in this theorem is not required to be bijective or surjective.

Illustrations

The theorem has several "real world" illustrations. Here are some examples.

1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its 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 immediately beneath it.
2. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.
3. In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a delicious cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktail shaken, not stirred defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state. Template:Citation needed

Intuitive approach

Explanations attributed to Brouwer

The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee.<ref>The interest of this anecdote rests in its intuitive and didactic character, but its accuracy is dubious. As the history section shows, the origin of the theorem is not Brouwer's work. More than 20 years earlier Henri Poincaré had proved an equivalent result, and 5 years before Brouwer P. Bohl had proved the three-dimensional case.</ref> If one stirs to dissolve a lump of sugar, it appears there is always a point without motion. He drew the conclusion that at any moment, there is a point on the surface that is not moving.<ref name=Arte>This citation comes originally from a television broadcast: Archimède, Arte, 21 septembre 1999</ref> The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit. The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.

Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet."<ref name=Arte /> Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as Stefan Banach's, that guarantee uniqueness.

One-dimensional case

File:Théorème-de-Brouwer-dim-1.svg

In one dimension, the result is intuitive and easy to prove. The continuous function f is defined on a closed interval [a, b] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval [a, b] which maps x to x (light green).

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function g which maps x to f(x) − x. It is ≥ 0 on a and ≤ 0 on b. By the intermediate value theorem, g has a zero in [a, b]; this zero is a fixed point.

Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."<ref name=Arte />

History

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 first was proved by Piers Bohl in 1904 (published in Journal für die reine und angewandte Mathematik).<ref name=Bohl1904>Template:Cite journal</ref> It was later proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910,<ref name="hadamard-1910" /> and Brouwer found a different proof in the same year.<ref name="brouwer-1910" /> Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals. Although the existence of a fixed point is not constructive in the sense of constructivism in mathematics, methods to approximate fixed points guaranteed by Brouwer's theorem are now known.<ref name=Karamardian1977>Template:Cite book</ref><ref name=Istratescu1981>Template:Cite book</ref>

Before discovery

File:Théorème-de-Brouwer-(cond-1).jpg

For flows in an unbounded area, or in an area with a "hole", the theorem is not applicable.

File:Théorème-de-Brouwer-(cond-2).jpg

The theorem applies to any disk-shaped area, where it guarantees the existence of a fixed point.

At the end of the 19th century, the old problem<ref>See F. Brechenmacher L'identité algébrique d'une pratique portée par la discussion sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des planètes CNRS Fédération de Recherche Mathématique du Nord-Pas-de-Calais</ref> of the stability of the solar system returned into the focus of the mathematical community.<ref>Henri Poincaré won the King of Sweden's mathematical competition in 1889 for his work on the related three-body problem: Jacques Tits Célébrations nationales 2004 Site du Ministère Culture et Communication</ref> Its solution required new methods. As noted by Henri Poincaré, who worked on the three-body problem, there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge."<ref name=methodes>Henri Poincaré Les méthodes nouvelles de la mécanique céleste T Gauthier-Villars, Vol 3 p 389 (1892) new edition Paris: Blanchard, 1987.</ref> He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision".<ref>Quotation from Henri Poincaré taken from: P. A. Miquel La catégorie de désordre Template:Webarchive, on the website of l'Association roumaine des chercheurs francophones en sciences humaines</ref>

He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant flow?<ref>This question was studied in: Template:Cite journal</ref> Poincaré discovered that the answer can be found in what we now call the topological properties in the area containing the trajectory. If this area is compact, i.e. both closed and bounded, then the trajectory either becomes stationary, or it approaches a limit cycle.<ref>This follows from the Poincaré–Bendixson theorem.</ref> Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval t. If the area is a circular band, or if it is not closed,<ref>Multiplication by Template:Sfrac on ]0, 1[² has no fixed point.</ref> then this is not necessarily the case.

To understand differential equations better, a new branch of mathematics was born. Poincaré called it analysis situs. The French Encyclopædia Universalis defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing".<ref>"concerne les propriétés invariantes d'une figure lorsqu'on la déforme de manière continue quelconque, sans déchirure (par exemple, dans le cas de la déformation de la sphère, les propriétés corrélatives des objets tracés sur sa surface". From C. Houzel M. Paty Poincaré, Henri (1854–1912) Template:Webarchive Encyclopædia Universalis Albin Michel, Paris, 1999, p. 696–706</ref> In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem,<ref>Poincaré's theorem is stated in: V. I. Istratescu Fixed Point Theory an Introduction Kluwer Academic Publishers (réédition de 2001) p 113 Template:Isbn</ref> although the connection with the subject of this article was not yet apparent.<ref>Template:SpringerEOM</ref> A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the fundamental group or sometimes the Poincaré group.<ref>Template:Cite book</ref> This method can be used for a very compact proof of the theorem under discussion.

Poincaré's method was analogous to that of Émile Picard, a contemporary mathematician who generalized the Cauchy–Lipschitz theorem.<ref>See for example: Émile Picard Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires Template:Webarchive Journal de Mathématiques p 217 (1893)</ref> Picard's approach is based on a result that would later be formalised by another fixed-point theorem, named after Banach. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a contraction.

First proofs

At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. Piers Bohl, a Latvian mathematician, applied topological methods to the study of differential equations.<ref>J. J. O'Connor E. F. Robertson Piers Bohl</ref> In 1904 he proved the three-dimensional case of our theorem,<ref name="Bohl1904" /> but his publication was not noticed.<ref>Template:Cite journal</ref>

It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially mathematical logic and topology. His initial interest lay in an attempt to solve Hilbert's fifth problem.<ref>J. J. O'Connor E. F. Robertson Luitzen Egbertus Jan Brouwer</ref> In 1909, during a voyage to Paris, he met Henri Poincaré, Jacques Hadamard, and Émile Borel. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the hairy ball theorem for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point.<ref>Template:Cite journal</ref> These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem.<ref>Template:Cite journal</ref> The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as homotopy, the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. Hans Freudenthal comments on the respective roles as follows: "Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."<ref>Template:Cite journal</ref>

Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension,<ref name="brouwer-1910" /> as well as other key theorems such as the invariance of dimension.<ref>If an open subset of a manifold is homeomorphic to an open subset of a Euclidean space of dimension n, and if p is a positive integer other than n, then the open set is never homeomorphic to an open subset of a Euclidean space of dimension p.</ref> In the context of this work, Brouwer also generalized the Jordan curve theorem to arbitrary dimension and established the properties connected with the degree of a continuous mapping.<ref>J. J. O'Connor E. F. Robertson Luitzen Egbertus Jan Brouwer.</ref> This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became algebraic topology.<ref>The term algebraic topology first appeared 1931 under the pen of David van Dantzig: J. Miller Topological algebra on the site Earliest Known Uses of Some of the Words of Mathematics (2007)</ref>

Reception

File:John f nash 20061102 2.jpg

John Nash used the theorem in game theory to prove the existence of an equilibrium strategy profile.

The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory.<ref>V. I. Istratescu Fixed Point Theory. An Introduction Kluwer Academic Publishers (new edition 2001) Template:Isbn.</ref> Brouwer's theorem is probably the most important.<ref>"... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu Fixed Point Theory an Introduction Kluwer Academic Publishers (new edition 2001) Template:Isbn.</ref> It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan curve theorem.<ref>E.g.: S. Greenwood J. Cao Brouwer's Fixed Point Theorem and the Jordan Curve Theorem University of Auckland, New Zealand.</ref>

Besides the fixed-point theorems for more or less contracting functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the n-dimensional sphere to Rⁿ has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the Lefschetz fixed-point theorem provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to Banach spaces.<ref>Template:Cite journal</ref> This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to set-valued functions.<ref>Template:Cite journal</ref> One also meets the theorem and its variants outside topology. It can be used to prove the Hartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the Central Limit Theorem. The theorem can also be found in existence proofs for the solutions of certain partial differential equations.<ref>These examples are taken from: F. Boyer Théorèmes de point fixe et applications CMI Université Paul Cézanne (2008–2009) Archived copy at WebCite (August 1, 2010).</ref>

Other areas are also touched. In game theory, John Nash used the theorem to prove that in the game of Hex there is a winning strategy for white.<ref>For context and references see the article Hex (board game).</ref> In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria (Hotelling's law), financial equilibria and incomplete markets.<ref>P. Bich Une extension discontinue du théorème du point fixe de Schauder, et quelques applications en économie Template:Webarchive Institut Henri Poincaré, Paris (2007)</ref>

Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are not constructive,<ref>For a long explanation, see: Template:Cite journal</ref> and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of constructivity. He became the originator and zealous defender of a way of formalising mathematics that is known as intuitionism, which at the time made a stand against set theory.<ref>Later it would be shown that the formalism that was combatted by Brouwer can also serve to formalise intuitionism, with some modifications. For further details see constructive set theory.</ref> Brouwer disavowed his original proof of the fixed-point theorem.

Proof outlines

A proof using degree

Brouwer's original 1911 proof relied on the notion of the degree of a continuous mapping, stemming from ideas in differential topology. Several modern accounts of the proof can be found in the literature, notably Template:Harvtxt.<ref name="Milnor">Template:Harvnb</ref><ref>Template:Cite book</ref>

Let <math>K=\overline{B(0)}</math> denote the closed unit ball in <math>\mathbb R^n</math> centered at the origin. Suppose for simplicity that <math>f:K\to K</math> is continuously differentiable. A regular value of <math>f</math> is a point <math>p\in B(0)</math> such that the Jacobian of <math>f</math> is non-singular at every point of the preimage of <math>p</math>. In particular, by the inverse function theorem, every point of the preimage of <math>f</math> lies in <math>B(0)</math> (the interior of <math>K</math>). The degree of <math>f</math> at a regular value <math>p\in B(0)</math> is defined as the sum of the signs of the Jacobian determinant of <math>f</math> over the preimages of <math>p</math> under <math>f</math>:

<math>\operatorname{deg}_p(f) = \sum_{x\in f^{-1}(p)} \operatorname{sign}\,\det (df_x).</math>

The degree is, roughly speaking, the number of "sheets" of the preimage f lying over a small open set around p, with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of winding number to higher dimensions.

The degree satisfies the property of homotopy invariance: let <math>f</math> and <math>g</math> be two continuously differentiable functions, and <math>H_t(x)=tf+(1-t)g</math> for <math>0\le t\le 1</math>. Suppose that the point <math>p</math> is a regular value of <math>H_t</math> for all t. Then <math>\deg_p f = \deg_p g</math>.

If there is no fixed point of the boundary of <math>K</math>, then the function

<math>g(x)=\frac{x-f(x)}{\sup_{y\in K}\left|y-f(y)\right|}</math>

is well-defined, and

<math>H(t,x) = \frac{x-tf(x)}{\sup_{y\in K}\left|y-tf(y)\right|}</math>

defines a homotopy from the identity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so <math>g</math> also has degree one at the origin. As a consequence, the preimage <math>g^{-1}(0)</math> is not empty. The elements of <math>g^{-1}(0)</math> are precisely the fixed points of the original function f.

This requires some work to make fully general. The definition of degree must be extended to singular values of f, and then to continuous functions. The more modern advent of homology theory simplifies the construction of the degree, and so has become a standard proof in the literature.

A proof using the hairy ball theorem

The hairy ball theorem states that on the unit sphere Template:Mvar in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field Template:Mvar on Template:Mvar. (The tangency condition means that Template:Mvar = 0 for every unit vector Template:Mvar.) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in Template:Harvtxt.

In fact, suppose first that Template:Mvar is continuously differentiable. By scaling, it can be assumed that Template:Mvar is a continuously differentiable unit tangent vector on Template:Mvar. It can be extended radially to a small spherical shell Template:Mvar of Template:Mvar. For Template:Mvar sufficiently small, a routine computation shows that the mapping Template:Mvar(Template:Mvar) = Template:Mvar + Template:Mvar is a contraction mapping on Template:Mvar and that the volume of its image is a polynomial in Template:Mvar. On the other hand, as a contraction mapping, Template:Mvar must restrict to a homeomorphism of Template:Mvar onto (1 + Template:Mvar)^{Template:Sfrac} Template:Mvar and Template:Mvar onto (1 + Template:Mvar)^{Template:Sfrac} Template:Mvar. This gives a contradiction, because, if the dimension Template:Mvar of the Euclidean space is odd, (1 + Template:Mvar)^{Template:Mvar/2} is not a polynomial.

If Template:Mvar is only a continuous unit tangent vector on Template:Mvar, by the Weierstrass approximation theorem, it can be uniformly approximated by a polynomial map Template:Mvar of Template:Mvar into Euclidean space. The orthogonal projection on to the tangent space is given by Template:Mvar(Template:Mvar) = Template:Mvar(Template:Mvar) - Template:Mvar(Template:Mvar) ⋅ Template:Mvar. Thus Template:Mvar is polynomial and nowhere vanishing on Template:Mvar; by construction Template:Mvar/||Template:Mvar|| is a smooth unit tangent vector field on Template:Mvar, a contradiction.

The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that Template:Mvar is even. If there were a fixed-point-free continuous self-mapping Template:Mvar of the closed unit ball Template:Mvar of the Template:Mvar-dimensional Euclidean space Template:Mvar, set

<math>{\mathbf w}({\mathbf x}) = (1 - {\mathbf x}\cdot {\mathbf f}({\mathbf x}))\, {\mathbf x} - (1 - {\mathbf x}\cdot {\mathbf x})\, {\mathbf f}({\mathbf x}).</math>

Since Template:Mvar has no fixed points, it follows that, for Template:Mvar in the interior of Template:Mvar, the vector Template:Mvar(Template:Mvar) is non-zero; and for Template:Mvar in Template:Mvar, the scalar product
Template:Mvar ⋅ Template:Mvar(Template:Mvar) = 1 – Template:Mvar ⋅ Template:Mvar(Template:Mvar) is strictly positive. From the original Template:Mvar-dimensional space Euclidean space Template:Mvar, construct a new auxiliary
(Template:Mvar)-dimensional space Template:Mvar = Template:Mvar x R, with coordinates Template:Mvar = (Template:Mvar, Template:Mvar). Set

<math>{\mathbf X}({\mathbf x},t)=(-t\,{\mathbf w}({\mathbf x}), {\mathbf x}\cdot {\mathbf w}({\mathbf x})).</math>

By construction Template:Mvar is a continuous vector field on the unit sphere of Template:Mvar, satisfying the tangency condition Template:Mvar ⋅ Template:Mvar(Template:Mvar) = 0. Moreover, Template:Mvar(Template:Mvar) is nowhere vanishing (because, if Template:Var has norm 1, then Template:Mvar ⋅ Template:Mvar(Template:Mvar) is non-zero; while if Template:Mvar has norm strictly less than 1, then Template:Mvar and Template:Mvar(Template:Mvar) are both non-zero). This contradiction proves the fixed point theorem when Template:Mvar is even. For Template:Mvar odd, one can apply the fixed point theorem to the closed unit ball Template:Mvar in Template:Mvar dimensions and the mapping Template:Mvar(Template:Mvar,Template:Mvar) = (Template:Mvar(Template:Mvar),0). The advantage of this proof is that it uses only elementary techniques; more general results like the Borsuk-Ulam theorem require tools from algebraic topology.<ref name="Milnor78">Template:Harvnb</ref>

A proof using homology or cohomology

The proof uses the observation that the boundary of the n-disk Dⁿ is Sⁿ⁻¹, the (n − 1)-sphere.

File:Brouwer fixed point theorem retraction.svg

Illustration of the retraction F

Suppose, for contradiction, that a continuous function Template:Nowrap has no fixed point. This means that, for every point x in Dⁿ, the points x and f(x) are distinct. Because they are distinct, for every point x in Dⁿ, we can construct a unique ray from f(x) to x and follow the ray until it intersects the boundary Sⁿ⁻¹ (see illustration). By calling this intersection point F(x), we define a function F : Dⁿ → Sⁿ⁻¹ sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever x itself is on the boundary, then the intersection point F(x) must be x.

Consequently, F is a special type of continuous function known as a retraction: every point of the codomain (in this case Sⁿ⁻¹) is a fixed point of F.

Intuitively it seems unlikely that there could be a retraction of Dⁿ onto Sⁿ⁻¹, and in the case n = 1, the impossibility is more basic, because S⁰ (i.e., the endpoints of the closed interval D¹) is not even connected. The case n = 2 is less obvious, but can be proven by using basic arguments involving the fundamental groups of the respective spaces: the retraction would induce a surjective group homomorphism from the fundamental group of D² to that of S¹, but the latter group is isomorphic to Z while the first group is trivial, so this is impossible. The case n = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields.

For n > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of homology groups: the homology H_n−1(Dⁿ) is trivial, while H_n−1(Sⁿ⁻¹) is infinite cyclic. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.

The impossibility of a retraction can also be shown using the de Rham cohomology of open subsets of Euclidean space Eⁿ. For n ≥ 2, the de Rham cohomology of U = Eⁿ – (0) is one-dimensional in degree 0 and n – 1, and vanishes otherwise. If a retraction existed, then U would have to be contractible and its de Rham cohomology in degree n – 1 would have to vanish, a contradiction.<ref>Template:Harvnb</ref>

A proof using Stokes' theorem

As in the proof of Brouwer's fixed-point theorem for continuous maps using homology, it is reduced to proving that there is no continuous retraction Template:Mvar from the ball Template:Mvar onto its boundary ∂Template:Mvar. In that case it can be assumed that Template:Mvar is smooth, since it can be approximated using the Weierstrass approximation theorem or by convolving with non-negative smooth bump functions of sufficiently small support and integral one (i.e. mollifying). If Template:Mvar is a volume form on the boundary then by Stokes' theorem,

<math>0<\int_{\partial B}\omega = \int_{\partial B}F^*(\omega) = \int_BdF^*(\omega)= \int_BF^*(d\omega)=\int_BF^*(0) = 0,</math>

giving a contradiction.<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref>

More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifold Template:Mvar onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the form Template:Mvar generates the de Rham cohomology group Template:Mvar(∂Template:Mvar) which is isomorphic to the homology group Template:Mvar(∂Template:Mvar) by de Rham's theorem.<ref>Template:Harvnb</ref>

A combinatorial proof

The BFPT can be proved using Sperner's lemma. We now give an outline of the proof for the special case in which f is a function from the standard n-simplex, <math>\Delta^n,</math> to itself, where

<math>\Delta^n = \left\{P\in\mathbb{R}^{n+1}\mid\sum_{i = 0}^{n}{P_i} = 1 \text{ and } P_i \ge 0 \text{ for all } i\right\}.</math>

For every point <math>P\in \Delta^n,</math> also <math>f(P)\in \Delta^n.</math> Hence the sum of their coordinates is equal:

<math>\sum_{i = 0}^{n}{P_i} = 1 = \sum_{i = 0}^{n}{f(P)_i}</math>

Hence, by the pigeonhole principle, for every <math>P\in \Delta^n,</math> there must be an index <math>j \in \{0, \ldots, n\}</math> such that the <math>j</math>th coordinate of <math>P</math> is greater than or equal to the <math>j</math>th coordinate of its image under f:

<math>P_j \geq f(P)_j.</math>

Moreover, if <math>P</math> lies on a k-dimensional sub-face of <math>\Delta^n,</math> then by the same argument, the index <math>j</math> can be selected from among the Template:Nowrap coordinates which are not zero on this sub-face.

We now use this fact to construct a Sperner coloring. For every triangulation of <math>\Delta^n,</math> the color of every vertex <math>P</math> is an index <math>j</math> such that <math>f(P)_j \leq P_j.</math>

By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is an n-dimensional simplex whose vertices are colored with the entire set of Template:Nowrap available colors.

Because f is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point <math>P</math> which satisfies the labeling condition in all coordinates: <math>f(P)_j \leq P_j</math> for all <math>j.</math>

Because the sum of the coordinates of <math>P</math> and <math>f(P)</math> must be equal, all these inequalities must actually be equalities. But this means that:

<math>f(P) = P.</math>

That is, <math>P</math> is a fixed point of <math>f.</math>

A proof by Hirsch

There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. Let f denote a continuous map from the unit ball Dⁿ in n-dimensional Euclidean space to itself and assume that f fixes no point. By continuity and the fact that Dⁿ is compact, it follows that for some ε > 0, ∥x - f(x)∥ > ε for all x in Dⁿ. Then the map f can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem or by convolving with smooth bump functions. One then defines a retraction as above by sending each x to the point of ∂Dⁿ where the unique ray from x through f(x) intersects ∂Dⁿ, and this must now be a differentiable mapping. Such a retraction must have a non-singular value p ∈ ∂Dⁿ, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image f ^-1(p) would be a compact 1-manifold with boundary. Such a boundary would have to contain at least two endpoints, and these would have to lie on the boundary of the original ball. This would mean that the inverse image of one point on ∂Dⁿ contains a different point on ∂Dⁿ, contradicting the definition of a retraction Dⁿ → ∂Dⁿ.<ref>Template:Harvnb</ref>

R. Bruce Kellogg, Tien-Yien Li, and James A. Yorke turned Hirsch's proof into a computable proof by observing that the retract is in fact defined everywhere except at the fixed points.Template:Sfn For almost any point q on the boundary — assuming it is not a fixed point — the 1-manifold with boundary mentioned above does exist and the only possibility is that it leads from q to a fixed point. It is an easy numerical task to follow such a path from q to the fixed point so the method is essentially computable.Template:Sfn gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

A proof using oriented area

A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If <math>r\colon B\to \partial B</math> is a smooth retraction, one considers the smooth deformation <math>g^t(x):=t r(x)+(1-t)x,</math> and the smooth function

<math>\varphi(t):=\int_B \det D g^t(x) \, dx.</math>

Differentiating under the sign of integral it is not difficult to check that Template:Prime(t) = 0 for all t, so φ is a constant function, which is a contradiction because φ(0) is the n-dimensional volume of the ball, while φ(1) is zero. The geometric idea is that φ(t) is the oriented area of g^t(B) (that is, the Lebesgue measure of the image of the ball via g^t, taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter t passes from 0 to 1 the map g^t transforms continuously from the identity map of the ball, to the retraction r, which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of r is necessarily 0, as its image is the boundary of the ball, a set of null measure.<ref>Template:Harvnb</ref>

A proof using the game Hex

A quite different proof given by David Gale is based on the game of Hex. The basic theorem regarding Hex, first proven by John Nash, is that no game of Hex can end in a draw; the first player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater). This turns out to be equivalent to the Brouwer fixed-point theorem for dimension 2. By considering n-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the determinacy theorem for Hex.<ref>Template:Cite journal</ref>

A proof using the Lefschetz fixed-point theorem

The Lefschetz fixed-point theorem says that if a continuous map f from a finite simplicial complex B to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number

<math>\displaystyle \sum_n(-1)^n\operatorname{Tr}(f|H_n(B))</math>

and in particular if the Lefschetz number is nonzero then f must have a fixed point. If B is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero simplicial homology group is: <math>H_0(B)</math> and f acts as the identity on this group, so f has a fixed point.<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref>

A proof in a weak logical system

In reverse mathematics, Brouwer's theorem can be proved in the system WKL₀, and conversely over the base system RCA₀ Brouwer's theorem for a square implies the weak Kőnig's lemma, so this gives a precise description of the strength of Brouwer's theorem.

Generalizations

The Brouwer fixed-point theorem forms the starting point of a number of more general fixed-point theorems.

The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary Hilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space ℓ² of square-summable real (or complex) sequences, consider the map f : ℓ² → ℓ² which sends a sequence (x_n) from the closed unit ball of ℓ² to the sequence (y_n) defined by

<math>y_0 = \sqrt{1 - \|x\|_2^2}\quad\text{ and}\quad y_n = x_{n-1} \text{ for } n \geq 1.</math>

It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ², but does not have a fixed point.

The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and also often an assumption of convexity. See fixed-point theorems in infinite-dimensional spaces for a discussion of these theorems.

There is also finite-dimensional generalization to a larger class of spaces: If <math>X</math> is a product of finitely many chainable continua, then every continuous function <math>f:X\rightarrow X</math> has a fixed point,<ref>Template:Cite journal</ref> where a chainable continuum is a (usually but in this case not necessarily metric) compact Hausdorff space of which every open cover has a finite open refinement <math>\{U_1,\ldots,U_m\}</math>, such that <math>U_i \cap U_j \neq \emptyset</math> if and only if <math>|i-j| \leq 1</math>. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.

The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in Rⁿ, but considers upper hemi-continuous set-valued functions (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.

The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of Dⁿ.

See also

• Banach fixed-point theorem
• Fixed-point computation
• Infinite compositions of analytic functions
• Nash equilibrium
• Poincaré–Miranda theorem – equivalent to the Brouwer fixed-point theorem
• Topological combinatorics

References

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• Template:Cite book (see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)
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• Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. Template:ISBN
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External links

• Brouwer's Fixed Point Theorem for Triangles at cut-the-knot
• Brouwer theorem Template:Webarchive, from PlanetMath with attached proof.
• Reconstructing Brouwer at MathPages
• Brouwer Fixed Point Theorem at Math Images.