Presentation Notes



The first time I saw this proof of the Brouwer Fixed Point Theorem was in a talk by Francis Su in Trivial Notions, the graduate student seminar in the Harvard math program.  For this reason, I have opened the talk with a plug for Francis’s Math Fun Facts, which is a wonderful source of mathematical gems for students and teachers alike.  While the site does not contain the proof given below, as it is too long to fit the site’s mandate, there are Fun Facts on Sperner’s Lemma and on the crumpled paper problem.


This talk begins with a demonstration.  I show a piece of regular 8.5x11 paper and mark its position on a tabletop.  I ask a randomly chosen member of the audience to crumple the sheet of paper, taking care not to rip it in the process.  Another member of the audience is enlisted to place the paper inside the marks on the tabletop.


I then present the audience with the following wager:  I bet that there is a point on the page that lies directly above its former position.  The audience votes on my odds of winning: 50%, 75%, 90%, 100%.  People who have not seen this illustration or the theorem will usually lower their hands before I reach 100%.


At this point, I get someone in the audience (a prearranged plant) to actually take the bet.  For the rest of the talk, I am arguing that I get to keep the money.  This is where the slides start.

This talk is brought to you by


Francis Su’s Math Fun Facts.



The Brouwer Fixed Point Theorem

If f: DD is a continuous function, then there is a point a in D such that f(a) = a.






Definition:  D = {(x,y) : x2 + y2 ≤ 1}


D is the closed unit disk.


Continuity in R:


Definition:  A function f: RR is continuous at x if for every ε > 0, there is a δ > 0 such that

if |xx′| ≤ δ,

then |f(x) – f(x′)| ≤ ε.

The Brouwer Fixed Point Theorem:

If f: DD is a continuous function, then there is a point x in D such that f(x) = x.


Continuity in R2:


Definition:  A function f: R2R2 is continuous at x if for every ε > 0, there is a δ > 0 such that

if |xx′| ≤ δ,

then |f(x) – f(x′)| ≤ ε.



Intuition: The plane is made of infinitely stretchable, infinitely compressible material.  f rearranges the plane without tearing it.

The One-Dimensional BFPT

If f:[0,1]→[0,1] is a continuous function, then it has a fixed point.


Proof:  Look at the function g(x) = f(x) – x.


g(0) = f(0) ≥ 0

g(1) = f(1) – 1 ≤ 0



By the Intermediate Value Theorem, somewhere between 0 and 1 is an a such that g(a) = 0.  Thus f(a) = a.


Presentation Notes


When I present the one-dimensional Brouwer Fixed Point Theorem, a common calculus exercise, I mention that there is also an n-dimensional version of the theorem.  The proof presented here works in n dimensions, and I challenge the more ambitious audience members to figure out how to extend the argument to higher dimensions.


I also point out that while the proof above will not work in two dimensions, since there is no analogue of the Intermediate Value Theorem for functions on the plane, our proof will use a similar technique for building a function that compares f (x) to x.


In the next slide, we consider the essential properties of D by looking at sets which admit continuous maps with no fixed points.  We begin by observing that the shape of D had better not be one of these properties if I want to apply the theorem to a rectangular piece of paper.


Food for Thought: What’s special about D?


1.    D is bounded.

2.    D is closed.

3.    D is connected.


Exercise: What property is missing?

Definition:  If S is a set in R2 and R is a subset of S, a retraction of S onto R is a continuous function g: SR such that every point of R is fixed.



S = the square {(a,b) : a,b are in [0,1]}

R = the bottom edge {(a,0) : a is in [0,1]}

g(x,y) = (x,0) is a retraction from S to R.




Let C = {(a,b) : a2 + b2 = 1}

= the boundary of D.


The No Retractions Theorem (NRT)

There is no retraction from D onto C.


Claim: BFPT is true ↔ NRT is true.


Equivalently, BFPT is false ↔ NRT is false.


Proof: (→)

Suppose BPFT is false, and let

f: DD be a continuous function with no fixed points.  We form a retraction g: DC as follows.




g is continuous and g fixes each point of C.



So NRT is false.



Suppose NRT is false, and let g: DC be a retraction.  Then

f(x,y) = –g(x,y)

(retract and then rotate) is continuous and has no fixed points.


So BFPT is false.

Presentation Notes


As soon as I begin the actual proof of the Brouwer Fixed Point Theorem, I make it clear that this will be a proof in four steps.  Now, at the end of the first step, I note that the audience may be less than satisfied with our progress.  After all, we have just taken one theorem that we don’t know how to prove and shown that it is equivalent to another theorem that we don’t know how to prove.


Nonetheless, I claim that we have gained something.  The Brouwer Fixed Point Theorem is far from obvious—a point that I can reinforce by the observation that some of them bet against it—while the No Retractions Theorem is utterly self-evident.


To illustrate this, I use a cylinder (most recently, an oatmeal container) with a piece of a balloon stretched over it to represent the disk D.  After prying at it for a while in a vain attempt to retract it onto its boundary, I comment that the only way that we can achieve the map is by doing “this,” at which point I pop the balloon.  The noise is very satisfying.


I usually observe that the No Retractions Theorem is so intuitively obvious that perhaps my opponent will consider forfeiting the bet now.  Needless to say, this does not happen.


The rest of the proof involves assuming that we have a retraction of a disk (eventually, a triangle) onto its boundary and finding the tear that we know must be there.



At Step 2, on the next slide, I fill in the sample triangulation with a Sperner labeling as I define the concept. The rules for such a labeling are that the vertices of the big triangle are labeled A, B, and C, the vertices along an outer edge may only be labeled by the two letters at the edge's endpoints, and the middle vertices may be labeled with any of the letters A, B, and C. I allow the audience to call out some of the letters as we go.


At Step 3, when the technique for changing shape is described, I point out that the same technique allows me to apply the Brouwer Fixed Point Theorem to a rectangular region.


Sperner’s Lemma: Locating the tear.


Start with a triangle, and triangulate it.


A Sperner labeling:



Sperner’s Lemma: Any Sperner labeling contains an ABC triangle.


Along the outer edge from vertex A to vertex B, there is an odd number of AB edges.



Score the labeling as follows:

1 point for every external AB edge

2 points for every internal AB edge

The total score is odd.


We get the same score by giving each triangle 1 point for every AB edge it contains.



Since the total score is odd, there is an odd number of ABC triangles, and hence at least one.


Changing shapes.


Let T be a closed triangular region in R2.

We can define a map σ: TD by:




σ is a continuous one-to-one correspondence between T and D which sends the boundary of T to the boundary of D.


If g is a retraction from D to its boundary,




then σ−1° g ° σ is a retraction from T to its boundary.


Thus, to prove NRT, it suffices to show that there is no retraction from T to its boundary.


We need the following.

Definition: f: SS is uniformly continuous on S if for every ε > 0, there is a δ > 0 such that for every x in S,

if |xx′| ≤ δ,

then |f(x) – f(x′)| ≤ ε.


Intuition: There is an upper bound to the stretchiness of f.

Example: f(x) = 1/x is not uniformly continuous on (0, 2).


BUT if S is closed and bounded (like D and T), then any continuous function on S is uniformly continuous.


Advanced Calculus, Gerald B. Folland, pp. 39-40.


Proof of NRT:


Suppose g were a retraction from T to its boundary.


Pick a very small ε, and choose a corresponding δ.  Now, triangulate T with triangles whose sides are smaller than δ.


Label the vertices of T by A, B, and C, and label each vertex v in the triangulation by which of A, B, or C is closest to g(v).


Since g fixes the boundary of T, this is a Sperner labeling.  By Sperner’s Lemma, we have an ABC triangle, whose vertices we will call x, y, and z.



But x, y, and z are within δ of each other, and g(x), g(y), and g(z) are farther apart than ε.




Therefore, NRT is true.


Therefore, BFPT is true.