Brouwer fixed-point theorem

Following are some of the terms used in topology which have similar definition or literal English meanings:

Convex set: A subset $U$ of $\mathbb{R}^n$ is called convex¹ , if it contains, along with any pair of its points $x,y$ , also the entire line segement joining the points.

Star-convex set: A subset $U$ of $\mathbb{R}^n$ is called star-convex if there exists a point $p\in U$ such that for each $x\in U$ , the line segment joining $x$ to $p$ lies in $U$ .

Simply connected: A topological space $X$ is called simply connected if it is path-connected² and any loop in $X$ defined by $f : S^1 \to X$ can be contracted³ to a point.

Deformation retract: Let $U$ be a subspace of $X$ . We say is a $X$ deformation retracts to $U$ if there exists a retraction⁴ $r : X \to U$ a retraction such that its composition with the inclusion is homotopic⁵ to the identity map on $X$ .

Various examples to illustrate the interdependence of these terms. Shown here are pentagon, star, sphere, and annulus.

A stronger version of Jordan Curve Theorem, known as Jordan–Schoenflies theorem, implies that the interior of a simple polygon is always a simply-connected subset of the Euclidean plane. This statement becomes false in higher dimensions.

The n-dimensional sphere $S^n$ is simply connected if and only if $n \geq 2$ . Every star-convex subset of $\mathbb{R}^n$ is simply connected. A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the Klein bottle are NOT simply connected.

The boundary of the n-dimensional ball $S^n$ , that is, the $(n-1)$ -sphere, is not a retract of the ball. Using this we can prove the Brouwer fixed-point theorem. However, $\mathbb{R}^n-0$ deformation retracts to a sphere $S^{n-1}$ . Hence, though the sphere shown above doesn’t deformation retract to a point, it is a deformation retraction of $\mathbb{R}^3-0$ .

Footnotes

In general, a convex set is defined for vector spaces. It’s the set of elements from the vector space such that all the points on the straight line line between any two points of the set are also contained in the set. If $a$ and $b$ are points in the vector space, the points on the straight line between $a$ and $b$ are given by $x = \lambda a + (1-\lambda)b$ for all $\lambda$ from 0 to 1.
A path from a point $x$ to a point $y$ in a topological space $X$ is a continuous function $f$ from the unit interval $[0,1]$ to $X$ with $f(0) = x$ and $f(1) = y$ . The space $X$ is said to be path-connected if there is a path joining any two points in $X$ .
There exists a continuous map $F : D^2 \to X$ such that $F$ restricted to $S^1$ is $f$ . Here, $S^1$ and $D^2$ denotes the unit circle and closed unit disk in the Euclidean plane respectively. In general, a space $X$ is contractible if it has the homotopy-type of a point. Intuitively, two spaces $X$ and $Y$ are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations.
Then a continuous map $r: X\to U$ is a retraction if the restriction of $r$ to $U$ is the identity map on $U$ .
A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X \times [0,1] \to Y$ such that, if $x \in X$ then $H(x,0) = f(x)$ and $H(x,1) = g(x)$ . Deformation retraction is a special type of homotopy equivalence, i.e. a deformation retraction is a mapping which captures the idea of continuously shrinking a space into a subspace.

Yesterday, I was fortunate enough to attend a lecture delivered by Dr. Ritwik Mukherjee, one of my professors, to motivate the study of algebraic topology. Instead of using the “soft targets” like Möbius strip etc. he used the following profound theorem for motivation:

If $f: S^n \to \mathbb{R}^n$ is continuous then there exists an $x\in S^n$ such that: $f(-x)=f(x)$ .

This is known as Borsuk-Ulam Theorem. To appreciate this theorem, one need to know a fundamental theorem about continuous functions known as Intermediate Value Theorem:

If a continuous function, $f$ , with an interval, $[a, b]$ , as its domain, takes values $f(a)$ and $f(b)$ at each end of the interval, then it also takes any value between $f(a)$ and $f(b)$ at some point within the interval.

Here is a video by James Grime illustrating Borsuk-Ulam Theorem in 3D:

Though the implications of the theorem itself are beautiful, following corollary known as Ham sandwich theorem is even more interesting. Here is a video by Marc Chamberland explaining this theorem:

Also, yesterday Grant Sanderson uploaded a video exploring the relation of Borsuk-Ulam Theorem with a fair division problem known as Necklace splitting problem:

But, to my amazement, this theorem is related to one of the other most astonishing theorem of algebraic topology called Brouwer fixed-point theorem:

Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

Here is a video by Michael Stevens illustrating Brouwer fixed-point theorem in some interesting cases:

Now the applications of this theorem are numerous, and there is a book dedicated to this theorem: “Fixed Points” by Yu. A. Shashkin. But my favourite application of this fixed point theorem is to the board game called Hex, explained by Marc Chamberland here:

If you come across some other video/article discussing the coolness of “Borsuk-Ulam Theorem” please let me know.

Edit (18 May 2018): Proof of Brouwer’s Fixed Point Theorem by Tai-Danae Bradley:

M	T	W	T	F	S	S
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30	31

Gaurish4Math

For the love of Mathematics

Tag Archives: Brouwer fixed-point theorem

Confusing terms in topology

Footnotes

Borsuk-Ulam Theorem