Brouwer fixed-point theorem: Difference between revisions

Consider the function

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

===Boundedness===

Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant under [[homeomorphism]]s, 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 set|closed]], bounded, [[connected space|connected]], [[simply connected|without holes]], etc.).

A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from the [[Lefschetz fixed-point theorem]].<ref>{{cite web | url=https://math.stackexchange.com/q/323841 | title=Why is convexity a requirement for Brouwer fixed points? | publisher=Math StackExchange | access-date=22 May 2015 | author=Belk, Jim}}</ref>

# 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.

==Intuitive approach==

===First proofs===

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 ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brouwer.html 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>{{cite journal |first=Hans  |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 495] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |doi-access=free }}</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>{{cite journal |first=Hans |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 495] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |quote=... cette dernière propriété, bien que sous des hypothèses plus grossières, ait été démontré par H. Poincaré |doi-access=free }}</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: <!-- NON-LITERAL QUOTATION! translated back from French -->"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>{{cite journal |first=Hans |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 501] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |doi-access=free }}</ref>

Other areas are also touched. In [[game theory]], [[John Forbes Nash|John Nash]] used the theorem to prove that in the game of [[Hex (board game)|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 ''[http://www.ann.jussieu.fr/~plc/code2007/bich.pdf Une extension discontinue du théorème du point fixe de Schauder, et quelques applications en économie] {{webarchive |url=https://web.archive.org/web/20110611140634/http://www.ann.jussieu.fr/~plc/code2007/bich.pdf |date=June 11, 2011 }}'' Institut Henri Poincaré, Paris (2007)</ref>

==Proof outlines==

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

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>.

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

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 ===

In fact, suppose first that {{mvar|'''w'''}} is ''continuously differentiable''. By scaling, it can be assumed that {{mvar|'''w'''}} is a continuously differentiable unit tangent vector on {{mvar|'''S'''}}. It can be extended radially to a small spherical shell {{mvar|''A''}} of {{mvar|''S''}}. For  {{mvar|''t''}} sufficiently small, a routine computation shows that the mapping {{mvar|'''f'''_''t''}}({{mvar|'''x'''}}) = {{mvar|'''x'''}} + {{mvar|''t'' '''w'''('''x''')}} is a [[contraction mapping]] on {{mvar|''A''}} and that the volume of its image is a polynomial in {{mvar|''t''}}. On the other hand, as a contraction mapping, {{mvar|'''f'''_''t''}} must restrict to a homeomorphism of {{mvar|''S''}} onto (1 + {{mvar|''t''²}})^{{{sfrac|1|2}}} {{mvar|''S''}} and  {{mvar|''A''}} onto (1 + {{mvar|''t''²}})^{{{sfrac|1|2}}} {{mvar|''A''}}. This gives a contradiction, because, if the dimension {{mvar|''n''}} of the Euclidean space is odd, (1 + {{mvar|''t''²}})^{{{mvar|''n''}}/2} is not a polynomial.  

The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that {{mvar|''n''}} is even. If there were a fixed-point-free continuous self-mapping {{mvar|'''f'''}} of the closed unit ball {{mvar|''B''}} of the {{mvar|''n''}}-dimensional Euclidean space {{mvar|''V''}}, set

===A combinatorial proof===

:<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>

*{{cite journal|author=Gale, D. |year=1979|title=The Game of Hex and Brouwer Fixed-Point Theorem | journal=The American Mathematical Monthly | volume=86 | pages=818–827|doi=10.2307/2320146|jstor=2320146|issue=10}}

*{{cite book |first=Morris W. |last=Hirsch | author-link=Morris Hirsch| title=Differential Topology |location=New York |publisher=Springer |year=1988 |isbn=978-0-387-90148-0 }} (see p.&nbsp;72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)

Homology theory: An introduction to algebraic topology|publisher=[[Cambridge University Press]]|location= New York |year=1960|isbn=0521094224}}

*{{cite book |first=Vasile I. |last=Istrăţescu |title=Fixed Point Theory |series=Mathematics and its Applications|volume= 7 |publisher=D. Reidel|location=Dordrecht–Boston, MA |year=1981 |isbn=978-90-277-1224-0|mr=0620639 }}

==External links==

* [http://www.cut-the-knot.org/do_you_know/poincare.shtml#brouwertheorem Brouwer's Fixed Point Theorem for Triangles] at [[cut-the-knot]]

* [http://www.mathpages.com/home/kmath262/kmath262.htm Reconstructing Brouwer] at MathPages

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