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Brouwer's fixed-point theorem is a
fixed-point theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors clai ...
in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
to itself. Among hundreds of
fixed-point theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors clai ...
s, 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 In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterior ...
, the
hairy ball theorem The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, if ...
, the invariance of dimension and the
Borsuk–Ulam theorem In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are i ...
. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s and is covered in most introductory courses on
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
. It appears in unlikely fields such as
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
. In economics, Brouwer's fixed-point theorem and its extension, the
Kakutani fixed-point theorem In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed po ...
, play a central role in the proof of existence of
general equilibrium In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
in market economies as developed in the 1950s by economics Nobel prize winners
Kenneth Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972. In economics ...
and
Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize ...
. The theorem was first studied in view of work on differential equations by the French mathematicians around
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
and
Charles Émile Picard Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was ...
. Proving results such as the
Poincaré–Bendixson theorem In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an o ...
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 -dimensional closed ball was first proved in 1910 by
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a tea ...
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a tea ...
:
Note sur quelques applications de l’indice de Kronecker
' in
Jules Tannery Jules Tannery (24 March 1848 – 11 December 1910) was a French mathematician, who notably studied under Charles Hermite and was the PhD advisor of Jacques Hadamard. Tannery's theorem on interchange of limits and series is named after him. He ...
: ''Introduction à la théorie des fonctions d’une variable'' (Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French)
and the general case for continuous mappings by Brouwer in 1911.


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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
disk to itself has at least one fixed point. This can be generalized to an arbitrary finite dimension: :;In Euclidean space:Every continuous function from a
closed ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
into itself has a fixed point. A slightly more general version is as follows: :;Convex compact set:Every continuous function from a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polyto ...
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
subset ''K'' of a Euclidean space to ''K'' itself has a fixed point. An even more general form is better known under a different name: :; Schauder fixed point theorem:Every continuous function from a convex compact subset ''K'' of a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
to ''K'' itself has a fixed point.


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 sets that are ''compact'' (thus, in particular, bounded and closed) and ''convex'' (or
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
to convex). The following examples show why the pre-conditions are important.


The function ''f'' as an endomorphism

Consider the function :f(x) = x+1 with domain 1,1 The range of the function is ,2 Thus, f is not an endomorphism.


Boundedness

Consider the function :f(x) = x+1, which is a continuous function from \mathbb to itself. As it shifts every point to the right, it cannot have a fixed point. The space \mathbb is convex and closed, but not bounded.


Closedness

Consider the function :f(x) = \frac, which is a continuous function from the open interval (−1,1) to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point. The space (−1,1) is convex and bounded, but not closed. The function ''f'' have a fixed point for the closed interval ��1,1 namely ''f''(1) = 1.


Convexity

Convexity is not strictly necessary for BFPT. Because the properties involved (continuity, being a fixed point) are invariant under
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
s, BFPT is equivalent to forms in which the domain is required to be a closed unit ball D^n. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
, bounded,
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, without holes, etc.). The following example shows that BFPT does not work for domains with holes. Consider the function f(x)=-x, 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'' have a fixed point for the unit disc, since it takes the origin to itself. A formal generalization of BFPT for "hole-free" domains can be derived from the
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
.


Notes

The continuous function in this theorem is not required to be
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
or even
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
.


Illustrations

The theorem has several "real world" illustrations. Here are some examples. # 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. # 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. # 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 "Shaken, not stirred" is how Ian Fleming's fictional British Secret Service agent James Bond prefers his martini cocktail. The catchphrase first appears in the novel '' Diamonds Are Forever'' (1956), though Bond himself does not actually say ...
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.


Intuitive approach


Explanations attributed to Brouwer

The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee. 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.This citation comes originally from a television broadcast:
Archimède
',
Arte Arte (; (), sometimes stylized in lowercase or uppercase in its logo) is a European public service channel dedicated to culture. It is made up of three separate companies: the Strasbourg-based European Economic Interest Grouping ARTE, pl ...
, 21 septembre 1999
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." 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 Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
's, that guarantee uniqueness.


One-dimensional case

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 In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two impor ...
, ''g'' has a
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
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."


History

The Brouwer fixed point theorem was one of the early achievements of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, and is the basis of more general
fixed point theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cl ...
s which are important in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on ...
. The case ''n'' = 3 first was proved by
Piers Bohl Piers Bohl (23 October 1865 – 25 December 1921) was a Latvian mathematician, who worked in differential equations, topology and quasi-periodic functions. He was born in 1865 in Walk, Livonia, in the family of a poor Baltic German merchant. I ...
in 1904 (published in ''
Journal für die reine und angewandte Mathematik ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by Augus ...
''). It was later proved by
L. E. J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and c ...
in 1909.
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a tea ...
proved the general case in 1910, and Brouwer found a different proof in the same year. Since these early proofs were all
non-constructive In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existen ...
indirect proofs, they ran contrary to Brouwer's
intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
ideals. Although the existence of a fixed point is not constructive in the sense of constructivism in mathematics, methods to
approximate An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
fixed points guaranteed by Brouwer's theorem are now known.


Prehistory

To understand the prehistory of Brouwer's fixed point theorem one needs to pass through
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. At the end of the 19th century, the old problem of the
stability of the solar system The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ...
returned into the focus of the mathematical community. Its solution required new methods. As noted by
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, who worked on the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
, 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."
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
''Les méthodes nouvelles de la mécanique céleste'' T Gauthier-Villars, Vol 3 p 389 (1892) new edition Paris: Blanchard, 1987.
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". 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? Poincaré discovered that the answer can be found in what we now call the
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
properties in the area containing the trajectory. If this area is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
, i.e. both
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
and bounded, then the trajectory either becomes stationary, or it approaches a
limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
. 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, 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 The ''Encyclopædia Universalis'' is a French-language general encyclopedia published by Encyclopædia Britannica, Inc., a privately held company. The articles of the ''Encyclopædia Universalis'' are aimed at educated adult readers, and writte ...
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". In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem, although the connection with the subject of this article was not yet apparent. A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
or sometimes the Poincaré group. 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 Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at th ...
, a contemporary mathematician who generalized the Cauchy–Lipschitz theorem. Picard's approach is based on a result that would later be formalised by another fixed-point theorem, named after
Banach Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase " son of man", combining the Hebrew word ''ben'' ("son of") and Arameic ''nash ...
. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a
contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...
.


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 Piers Bohl (23 October 1865 – 25 December 1921) was a Latvian mathematician, who worked in differential equations, topology and quasi-periodic functions. He was born in 1865 in Walk, Livonia, in the family of a poor Baltic German merchant. I ...
, a
Latvia Latvia ( or ; lv, Latvija ; ltg, Latveja; liv, Leţmō), officially the Republic of Latvia ( lv, Latvijas Republika, links=no, ltg, Latvejas Republika, links=no, liv, Leţmō Vabāmō, links=no), is a country in the Baltic region of ...
n mathematician, applied topological methods to the study of differential equations. In 1904 he proved the three-dimensional case of our theorem, but his publication was not noticed. 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 Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sy ...
and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
. His initial interest lay in an attempt to solve
Hilbert's fifth problem Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathema ...
. In 1909, during a voyage to Paris, he met
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
,
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a tea ...
, and
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was ...
. 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 The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, if ...
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. These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem. The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
, 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 Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education ...
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." Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension, as well as other key theorems such as the invariance of dimension. In the context of this work, Brouwer also generalized the
Jordan curve theorem In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterior ...
to arbitrary dimension and established the properties connected with the
degree of a continuous mapping In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree ...
. This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.


Reception

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. Brouwer's theorem is probably the most important. It is also among the foundational theorems on the topology of
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
s and is often used to prove other important results such as the
Jordan curve theorem In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterior ...
. Besides the fixed-point theorems for more or less
contracting A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tran ...
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 In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are i ...
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 In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s. This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to
multivalued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
s. 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 In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
. The theorem can also be found in existence proofs for the solutions of certain
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s. Other areas are also touched. In
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
, John Nash used the theorem to prove that in the game of Hex there is a winning strategy for white. 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 Hotelling's law is an observation in economics that in many markets it is rational for producers to make their products as similar as possible. This is also referred to as the principle of minimum differentiation as well as Hotelling's linear ci ...
), financial equilibria and incomplete markets. Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are not constructive, 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 In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
, which at the time made a stand against
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
. Brouwer disavowed his original proof of the fixed-point theorem. The first algorithm to approximate a fixed point was proposed by
Herbert Scarf Herbert Eli "Herb" Scarf (July 25, 1930 – November 15, 2015) was an American mathematical economist and Sterling Professor of Economics at Yale University. Education and career Scarf was born in Philadelphia, the son of Jewish emigrants from ...
. A subtle aspect of Scarf's algorithm is that it finds a point that is by a function ''f'', but in general cannot find a point that is close to an actual fixed point. In mathematical language, if is chosen to be very small, Scarf's algorithm can be used to find a point ''x'' such that ''f''(''x'') is very close to ''x'', i.e., d(f(x),x) < \varepsilon . But Scarf's algorithm cannot be used to find a point ''x'' such that ''x'' is very close to a fixed point: we cannot guarantee d(x,y) < \varepsilon, where f(y)=y. Often this latter condition is what is meant by the informal phrase "approximating a fixed point".


Proof outlines


A proof using degree

Brouwer's original 1911 proof relied on the notion of the
degree of a continuous mapping In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree ...
, stemming from ideas in
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
. Several modern accounts of the proof can be found in the literature, notably . Let K=\overline denote the closed unit ball in \mathbb R^n centered at the origin. Suppose for simplicity that f:K\to K is continuously differentiable. A
regular value In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion ...
of f is a point p\in B(0) such that the Jacobian of f is non-singular at every point of the preimage of p. In particular, by the
inverse function theorem In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
, every point of the preimage of f lies in B(0) (the interior of K). The degree of f at a regular value p\in B(0) is defined as the sum of the signs of the
Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
of f over the preimages of p under f: :\operatorname_p(f) = \sum_ \operatorname\,\det (df_x). 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 In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tur ...
to higher dimensions. The degree satisfies the property of ''homotopy invariance'': let f and g be two continuously differentiable functions, and H_t(x)=tf+(1-t)g for 0\le t\le 1. Suppose that the point p is a regular value of H_t for all ''t''. Then \deg_p f = \deg_p g. If there is no fixed point of the boundary of K, then the function :g(x)=\frac is well-defined, and H(t,x) = \frac 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 g also has degree one at the origin. As a consequence, the preimage g^(0) is not empty. The elements of g^(0) 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 In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...
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 The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, if ...
states that on the unit sphere in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field on . (The tangency condition means that = 0 for every unit vector .) 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 . In fact, suppose first that is ''continuously differentiable''. By scaling, it can be assumed that is a continuously differentiable unit tangent vector on . It can be extended radially to a small spherical shell of . For sufficiently small, a routine computation shows that the mapping () = + is a
contraction mapping In mathematics, a contraction mapping, or contraction or contractor, on a metric space (''M'', ''d'') is a function ''f'' from ''M'' to itself, with the property that there is some real number 0 \leq k < 1 such that for all ''x'' and ...
on and that the volume of its image is a polynomial in . On the other hand, as a contraction mapping, must restrict to a homeomorphism of onto (1 + )½ and onto (1 + )½ . This gives a contradiction, because, if the dimension of the Euclidean space is odd, (1 + )/2 is not a polynomial. If is only a ''continuous'' unit tangent vector on , by the
Weierstrass approximation theorem Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, it can be uniformly approximated by a polynomial map of into Euclidean space. The orthogonal projection on to the tangent space is given by () = () - () ⋅ . Thus is polynomial and nowhere vanishing on ; by construction /, , , , is a smooth unit tangent vector field on , a contradiction. The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that is odd. If there were a fixed-point-free continuous self-mapping of the closed unit ball of the -dimensional Euclidean space , set :() = (1 - \cdot ())\, - (1 - \cdot )\, (). Since has no fixed points, it follows that, for in the
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
of , the vector () is non-zero; and for in , the scalar product
⋅ () = 1 – ⋅ () is strictly positive. From the original -dimensional space Euclidean space , construct a new auxiliary
()-dimensional space = x R, with coordinates = (, ). Set :(,t)=(-t\,(), \cdot ()). By construction is a continuous vector field on the unit sphere of , satisfying the tangency condition ⋅ () = 0. Moreover, () is nowhere vanishing (because, if has norm 1, then ⋅ () is non-zero; while if has norm strictly less than 1, then and () are both non-zero). This contradiction proves the fixed point theorem when is odd. For even, one can apply the fixed point theorem to the closed unit ball in dimensions and the mapping (,) = ((),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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.


A proof using homology or cohomology

The proof uses the observation that the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film * Boundary (cricket), the edge of the pl ...
of the ''n''-disk ''D''''n'' is ''S''''n''−1, the (''n'' − 1)-
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
. Suppose, for contradiction, that a continuous function has ''no'' fixed point. This means that, for every point x in ''D''''n'', the points ''x'' and ''f''(''x'') are distinct. Because they are distinct, for every point x in ''D''''n'', we can construct a unique ray from ''f''(''x'') to ''x'' and follow the ray until it intersects the boundary ''S''''n''−1 (see illustration). By calling this intersection point ''F''(''x''), we define a function ''F'' : ''D''''n'' → ''S''''n''−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 mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
(in this case ''S''''n''−1) is a fixed point 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, the impossibility is more basic, because ''S''0 (i.e., the endpoints of the closed interval ''D''1) is not even connected. The case ''n'' = 2 is less obvious, but can be proven by using basic arguments involving the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
s of the respective spaces: the retraction would induce a surjective
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) ...
from the fundamental group of ''D''2 to that of ''S''1, 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 In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topo ...
: the homology ''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 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 In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapt ...
of open subsets of Euclidean space ''E''''n''. For ''n'' ≥ 2, the de Rham cohomology of ''U'' = ''E''''n'' – (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.


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 from the ball onto its boundary ∂. In that case it can be assumed that is smooth, since it can be approximated using the
Weierstrass approximation theorem Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
or by convolving with non-negative smooth
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bu ...
s of sufficiently small support and integral one (i.e. mollifying). If is a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...
on the boundary then by
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan(j ...
, :0<\int_\omega = \int_F^*(\omega) = \int_BdF^*(\omega)= \int_BF^*(d\omega)=\int_BF^*(0) = 0, giving a contradiction. More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifold onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the form generates the de Rham cohomology group (∂) which is isomorphic to the homology group (∂) by de Rham's theorem.


A combinatorial proof

The BFPT can be proved using
Sperner's lemma In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an ...
. We now give an outline of the proof for the special case in which ''f'' is a function from the standard ''n''-
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, \Delta^n, to itself, where :\Delta^n = \left\. For every point P\in \Delta^n, also f(P)\in \Delta^n. Hence the sum of their coordinates is equal: :\sum_^ = 1 = \sum_^ Hence, by the pigeonhole principle, for every P\in \Delta^n, there must be an index j \in \ such that the jth coordinate of P is greater than or equal to the jth coordinate of its image under ''f'': :P_j \geq f(P)_j. Moreover, if P lies on a ''k''-dimensional sub-face of \Delta^n, then by the same argument, the index j can be selected from among the coordinates which are not zero on this sub-face. We now use this fact to construct a Sperner coloring. For every triangulation of \Delta^n, the color of every vertex P is an index j such that f(P)_j \leq P_j. 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 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 P which satisfies the labeling condition in all coordinates: f(P)_j \leq P_j for all j. Because the sum of the coordinates of P and f(P) must be equal, all these inequalities must actually be equalities. But this means that: :f(P) = P. That is, P is a fixed point of f.


A proof by Hirsch

There is also a quick proof, by
Morris Hirsch Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley. A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of ...
, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that 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 Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
or by convolving with smooth
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bu ...
s. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction. 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. For almost any point, ''q'', on the boundary, (assuming it is not a fixed point) the one 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. 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 r\colon B\to \partial B is a smooth retraction, one considers the smooth deformation g^t(x):=t r(x)+(1-t)x, and the smooth function :\varphi(t):=\int_B \det D g^t(x) \, dx. Differentiating under the sign of integral it is not difficult to check that '(''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 form 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.


A proof using the game Hex

A quite different proof given by
David Gale David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...
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 Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and simil ...
theorem for Hex.


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 :\displaystyle \sum_n(-1)^n\operatorname(f, H_n(B)) 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 In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
group is: H_0(B) and ''f'' acts as the identity on this group, so ''f'' has a fixed point.


A proof in a weak logical system

In
reverse mathematics Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ...
, 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 König's lemma Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ...
, 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 theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors clai ...
s. The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. For example, in the Hilbert space 2 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 :y_0 = \sqrt\quad\text\quad y_n = x_ \text n \geq 1. 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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, ...
. 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 X is a product of finitely many chainable continua, then every continuous function f:X\rightarrow X has a fixed point, where a chainable continuum is a (usually but in this case not necessarily
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
)
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
of which every
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
has a finite open refinement \, such that U_i \cap U_j \neq \emptyset if and only if , i-j, \leq 1. 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 R''n'', but considers upper hemi-continuous
set-valued function A set-valued function (or correspondence) is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including optimizatio ...
s (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 In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of
singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of ''D''''n''.


Equivalent results


See also

*
Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of cert ...
*
Infinite compositions of analytic functions In mathematics, infinite Function composition, compositions of analytic functions (ICAF) offer alternative formulations of Generalized continued fraction, analytic continued fractions, series (mathematics), series, product (mathematics), products a ...
*
Nash equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equ ...
* Poincaré–Miranda theorem – equivalent to the Brouwer fixed-point theorem *
Topological combinatorics The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics. History The discipline of combinatorial topology used combinatorial concepts in top ...


Notes


References

* * * * * * * * (see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction) * * * * * *Leoni, Giovanni (2017).
A First Course in Sobolev Spaces: Second Edition
'.
Graduate Studies in Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General T ...
. 181. American Mathematical Society. pp. 734. * * * * *


External links


Brouwer's Fixed Point Theorem for Triangles
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Brouwer theorem
from
PlanetMath PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...
with attached proof.
Reconstructing Brouwer
at MathPages
Brouwer Fixed Point Theorem
at Math Images. {{DEFAULTSORT:Brouwer Fixed Point Theorem Fixed-point theorems Theory of continuous functions Theorems in topology Theorems in convex geometry