Talk:Brouwer fixed-point theorem: Difference between revisions

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Courant and Robbins provide an accessible proof.

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== Accessible proof==

According to Lyusternik ''Convex Figures and Polyhedra'', the theorem was first proved by a Lettish mathematician named Bol. No references are provided. Anyone know what this is about?--[[User:192.35.35.36|192.35.35.36]] 00:08, 18 Feb 2005 (UTC)

Courant and Robbins provide an accessible proof.

==Proofs==

<small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:198.144.199.xxx|198.144.199.xxx]] ([[User talk:198.144.199.xxx|talk]] • [[Special:Contributions/198.144.199.xxx|contribs]]) 30 August 2001</span></small><!-- Template:Unsigned -->

== Citation style ==

Why is wikipedia not the place to reroduce a long proof? --anon

This article mixes parenthetical referencing with footnoted references. The parenthetical ones were there first, so according to [[WP:CITEVAR]] we'd have to use that until explicit consensus. However, it would be significantly easier to turn the couple of parenthetical ones into footnotes than about 50 footnotes into parentheticals. Can we form consensus to continued using footnoted references? <span style="font-family: serif; letter-spacing: 0.1em">–&nbsp;[[User:Finnusertop|Finnusertop]]</span> ([[User talk:Finnusertop|talk]] ⋅ [[Special:Contributions/Finnusertop|contribs]]) 19:59, 24 February 2019 (UTC)

:What? [[User:El C|El_C]] 20:10, 22 September 2005 (UTC)

:I'm sure that'd be okay here. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 00:04, 25 February 2019 (UTC)

:: See [[Wikipedia:WikiProject Mathematics/Proofs]] and the [[math style manual]]. [[User:Oleg Alexandrov|Oleg Alexandrov]] 22:44, 22 September 2005 (UTC)

::Great. I've turned the remaining parentheticals into footnotes. <span style="font-family: serif; letter-spacing: 0.1em">–&nbsp;[[User:Finnusertop|Finnusertop]]</span> ([[User talk:Finnusertop|talk]] ⋅ [[Special:Contributions/Finnusertop|contribs]]) 00:10, 25 February 2019 (UTC)

== HexFunction mapping in multipleclosedness dimensions?section ==

It is stated that the function f(x) = (x+1)/2 is a continous function from the open interval (-1,1) to itself.

The article says:

Is it not the case that the function maps from (-1,1) to (0,1)? [[User:Salomonaber|Salomonaber]] ([[User talk:Salomonaber|talk]]) 00:13, 11 March 2020 (UTC)

:A quite different proof can be given based on the game of [[Hex_(board_game)|Hex]]. The basic theorem about Hex is that no game can end in a draw. This is equivalent to the Brouwer fixed point theorem for dimension 2. By considering ''n''-dimensional versions of Hex, one can prove that in general that Brouwer's theorem is equivalent to the "no draw" theorem for Hex.

But how does one play an "''n''-dimensional version of Hex"?

:It doesn't claim (nor is it required) that the function is [[surjective]], so what's there is correct and appropriate. The example could have even arranged for a bijection, but I don't think it matters much either way. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 00:32, 11 March 2020 (UTC)

:Consider the original 2D hexboard as being made from a lattice, where one connects the lower left corner of a square of a lattice to the upper right corner with an edge. These added diagonals make it so you can have 6 neighbors instead of 4 (on the original lattice). It's easy to see how to cut out an n by n Hex board from this modified lattice. In general, to create an n x n .... x n board, consider a lattice in R^m (where m is the number of dimensions of the board) and then in each m-dimensional cube add a diagonal. Then cut out a board as in 2D.

== Highly skeptical that the remarks "said to have [been] added" by Brouwer are actually due to him ==

:Each player has an opposite pair of sides as before, but now some sides belong to neither player. The game is played the same way as before. It doesn't appear to be so interesting to play, but people have devised other higher dimensional versions which are probably more fun and interesting mathematically. In any case, in the version I described, there can never be a draw, and this no-draw result is equivalent to the Brouwer fixed point theorem. --[[User:Chan-Ho Suh|Chan-Ho]][[User talk:Chan-Ho Suh| (Talk)]] 07:03, 15 January 2006 (UTC)

<blockquote>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."</blockquote>

== elemantary proof with stokes' theorem ==

The citation is apparently from a French-language educational TV show (https://archive.is/20130113210953/http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html). The remarks appear to be spoken by a fictional Brouwer trying to explain his result. The web page that this refers to gives no citation.

first, the retraction is given by:

[[Image:Theorem of brouwer-F.png|framed|Illustration of the retraction]]

: <math>F(x):=x + \left( \sqrt{1-|x|^2 + \left\langle x,\frac{x-f(x)}{|x-f(x)|} \right\rangle^2 } - \left\langle x, \frac{x-f(x)}{|x-f(x)|} \right\rangle \right) \frac{x-f(x)}{|x-f(x)|} </math>

and it can be proved using the strokes' theorem for differential forms.

I would like to know who originally came up with the "crumpled paper theorem" explanation of the BFPT. It could have been Brouwer himself, but my guess is it was not. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Natkuhn|Natkuhn]] ([[User talk:Natkuhn#top|talk]] • [[Special:Contributions/Natkuhn|contribs]]) 01:06, 8 May 2020 (UTC)</small> <!--Autosigned by SineBot-->

for <math>\omega^{n-1}:= F^1\, dF^2\wedge\cdots\wedge dF^n </math>, <math>d\omega^{n-1} = 0</math>, so you get:

<math> 0 = \int_{D^n} \mathrm d\omega^{n-1} = \int_{S^{n-1}} \omega^{n-1} = \int_{S^{n-1}} x_1 dx^2 \wedge \cdots \wedge dx^n = vol (D^n) \neq 0 </math>. first = because the jacobian is 0 by theorem of implicit functions.

:Oops, yeah, that's a good catch. This probably deserves some looking into. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 01:19, 8 May 2020 (UTC)

See the german page as example. this could be integrated. ~ibotty

== Did Brouwer offer the first proof for continuous functions? ==

:: Looks a bit messy to me.... [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 16:24, 9 February 2006 (UTC)

In the book ''History of Topology'' by James on pages 273-274: "Bohl's theorem is also equivalent to the Brouwer theorem. Bohl's theorem was published in 1904, with a proof that required that f be differentiable. Brouwer published his fixed point theorem, for continuous functions on the 3-ball, in 1909. When the first proof for the n-ball, with f '''differentiable''', appeared in print a year later, in an appendix by '''J. Hadamard''' to a text by Tannery, the theorem was called the 'Brouwer Fixed Point Theorem', which suggests that the result was already famous by that time. It is not known in what year Brouwer made his discovery and, apparently, communicated it to other mathematicians in an informal manner. The first published proof of the general case, that is, for '''continuous''' functions on the n-ball, was by '''Brouwer''' himself in 1912."

:It's not that messy! Don't be scared by the symbols, Oleg :-) I think with a little reworking it would make a fine addition. This is a pretty famous proof. Unfortunately, I take a bit of an issue with "elementary" describing this proof. The proof, as given, only proves Brouwer's theorem for sufficiently smooth maps f, since one needs to take the exterior derivative. Luckily, we can ''homotope'' f to be smooth while still keeping the map fixed point free (we can pick a straight-line homotopy that moves every point less than epsilon, where epsilon is smaller than the minimal distance between x and f(x)); however, this is not so trivial to show and can be regarded as a technicality that makes the entire proof not as elementary. And of course, we could debate, if we wished, whether this whole business of Stoke's theorem and smoothing maps is really more elementary than some simple homology (or homotopy) theory. Personally, I think the only kind of proof of Brouwer that really qualifies as elementary are the ones involving some form of coloring trick, e.g. Sperner's lemma or Hex. --[[User:Chan-Ho Suh|Chan-Ho]][[User talk:Chan-Ho Suh| (Talk)]] 01:46, 2 April 2006 (UTC)

In the book ''Brouwer Degree. The Core of Nonlinear Analysis'' by Mawhin on page 393: "In 1910, Jules Tannery published the second volume of the second edition of his book ‘Introduction à la théorie des fonctions de variables réelles’, for the time and still now a very modern presentation of analysis, introducing Weierstrass’ rigor in France. This volume two ended with a Note of Jacques Hadamard, connected in the following way to Tannery’s book material: 'The proof, following M. Ames, of Jordan’s theorem on closed curves without double point is based upon the concept of order of a point or, equivalently on the consideration of the variation of the argument. The generalization to the case where the dimension is larger than two is given by the Kronecker index. It is a now classical notion, mainly since the publication of the Traité d’Analyse of Mr. Picard (T. I, p. 123; T. II, p. 193). It has received new applications in various recent works. My aim is to present here some of them. All the following reasonings [...] only use the continuity of the considered functions.'"

:Hmmm...actually I see that since I last closely perused the article, somebody has added another proof of Brouwer for smooth maps. Per my comment right above, this is actually a proof for continuous maps also; I'll add that to the article. --[[User:Chan-Ho Suh|Chan-Ho]][[User talk:Chan-Ho Suh| (Talk)]] 10:42, 2 April 2006 (UTC)

The previous quote from Mawhin's book has a strange omission ([...]) that may indicate that Hadamard did not write down a proof for the continuous case. '''Can someone please check out if Hadamard proved the general case for differentiable functions and not for continuous functions?'''

== Outline of Proof ==

:'''Comments.''' Please sign your posts with <nowiki>~~~~</nowiki> and, per [[WP:TPG]], add comments at the ''end'' of the talk page. Please stop edit-warring to insert your own [[WP:POV|point of view]]. There is already a discussion of the history of the FPT in the section [[Brouwer fixed-point theorem#First proofs]]. Bohl's proof applied to three dimensions. There are several historical accounts of the FPT, notably "A history of algebraic and differential topology, 1900–1960" by [[Jean Dieudonné]]. [[User:Mathsci|Mathsci]] ([[User talk:Mathsci|talk]]) 17:43, 21 February 2022 (UTC)

I believe that there's an error outline of the proof given - in particular the induced transformation from the disk to its boundary need not be a retraction (the points on the boundary need not be fixed by the transformation). At the same time this is only a small error in that it's true that there's no continuous mapping from the disk to its boundary (retraction or not).