Famous Theorems of Mathematics/Brouwer fixed-point theorem

The Brouwer fixed point theorem is an important fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several general fixed point theorems. It is named after Dutch mathematician L. E. J. Brouwer.

Statement

The theorem states that every continuous function from the closed unit ball Bⁿ to itself has at least one fixed point. In this theorem, n is any positive integer, and the closed unit ball Bⁿ is the set of all points in Euclidean n-space Rⁿ which are at distance at most 1 from the origin. A fixed point of a function f : Bⁿ → Bⁿ is a point x in Bⁿ such that f(x) = x.

Notes

The function f in this theorem is not required to be bijective or even surjective.

Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (and therefore also closed, bounded, connected, without holes, etc.).

The statement of the theorem is false if formulated for the open unit disk, the set of points with distance strictly less than 1 from the origin. Consider for example the function

$f(x,y)=\textstyle ({\frac {1}{2}}(x+{\sqrt {1-y^{2}}}),y)$

which maps every point of the open unit disk in R² to another point of the open unit disk slightly to the right of the given one.

Illustrations

The theorem has several "real world" illustrations. For example: 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 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 exactly on top of 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.

In three dimensions the consequence of the Brouwer fixed point theorem is that no matter how much you stir or shake a cocktail in a glass some point in the liquid will remain in the exact 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, and that the liquid after stirring or shaking is contained within the space originally taken up by it.

Another consequence of the case n = 3 is given by an informational display of a map in, for example, an airport terminal. The function that sends points of the terminal to their image on the map is continuous and therefore has a fixed point, usually indicated by a cross or arrow with the text You are here. A similar display outside the terminal would violate the condition that the function is "to itself" and fail to have a fixed point. For this example, the existence of a fixed point is also a consequence of the Banach fixed point theorem, since the function mapping points in space to the display is a contraction mapping.

History

The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case n = 3 first was proved by Piers Bohl in 1904 (published in Journal für die reine und angewandte Mathematik). It was later proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910, and Brouwer found a different proof in 1912. Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals. Methods to construct (approximations to) fixed points guaranteed by Brouwer's theorem are now known, however; see for example (Karamadian 1977) and (Istrăţescu 1981).

Proof

Assume that there does exist a map from $f:B^{n}\to B^{n}$ with no fixed point. Then let $g(x)$ be the following map: Start at $f(x)$, draw the ray going through $x$ and then let $g(x)$ be the first intersection of that line with the sphere. This map is continuous and well defined only because $f$ fixes no point. Also, it is not hard to see that it must be the identity on the boundary sphere. Thus we have a map $g:B^{n}\to S^{n-1}$, which is the identity on $S^{n-1}=\partial B^{n}$, that is, a retraction. Now, if $i:S^{n-1}\to B^{n}$ is the inclusion map, $g\circ i=\mathrm {id} _{S^{n-1}}$. Applying the reduced homology functor, we find that $g_{*}\circ i_{*}=\mathrm {id} _{{\tilde {H}}_{n-1}(S^{n-1})}$, where $_{*}$ indicates the induced map on homology.

But, it is a well-known fact that ${\tilde {H}}_{n-1}(B^{n})=0$ (since $B^{n}$ is contractible), and that ${\tilde {H}}_{n-1}(S^{n-1})=\mathbb {Z}$. Thus we have an isomorphism of a non-zero group onto itself factoring through a trivial group, which is clearly impossible. Thus we have a contradiction, and no such map $f$ exists.