Definitions

Brouwer fixed point theorem

[brou-er fikst-point]
In mathematics, 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 Dn to itself has at least one fixed point. In this theorem, n is any positive integer, and the closed unit ball 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 : DnDn is a point x in Dn 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

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

A full proof of the theorem would be too long to reproduce here, but the following outlines a proof omitting the most difficult part. The proof uses the observation that the boundary of Dn is Sn − 1, the (n − 1)-sphere.

The argument proceeds by contradiction, supposing that a continuous function f : Dn → Dn has no fixed point, and then attempting to derive an inconsistency, which proves that the function must in fact have a fixed point. For each x in Dn, there is only one straight line that passes through f(x) and x, because it must be the case that f(x) and x are distinct by hypothesis (recall that f having no fixed points means that f(x) ≠ x). Following this line from f(x) through x leads to a point on Sn − 1, denoted by F(x). This defines a continuous function F : Dn → Sn − 1, which is a special type of continuous function known as a retraction: every point of the codomain (in this case Sn − 1) is a fixed point of the function.

Intuitively it seems unlikely that there could be a retraction of Dn onto Sn − 1, and in the case n = 1 it is obviously impossible because S 0 (i.e., the endpoints of the closed interval D 1) is not even connected. The case n = 2 is less obvious, but can be proven by using basic arguments involving the fundamental groups of the respective spaces: the retraction would induce an injective group homomorphism from the fundamental group of S 1 to that of D 2, but the first group is isomorphic to Z while the latter group is trivial, so this is impossible. The case n = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields.

For n > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of homology groups: it can be shown that the homology Hn − 1(Dn) is trivial, while Hn − 1(S n − 1) is infinite cyclic. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.

There is also an almost elementary combinatorial proof. Its main step consists in establishing Sperner's lemma in n dimensions.

There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that the map f can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem, for example. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.

A quite different proof given by David Gale is based on the game of 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 in general that Brouwer's theorem is equivalent to the determinacy theorem for Hex.

Generalizations

The Brouwer fixed point theorem forms the starting point of a number of more general fixed point theorems.

The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary Hilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space 2 of square-summable real (or complex) sequences, consider the map f : ℓ2 → ℓ2 which sends a sequence (xn) from the closed unit ball of ℓ2 to the sequence (yn) defined by

$y_0=sqrt\left\{1-|x|_2^2\right\}qquadmbox\left\{ and \right\}qquad y_n=x_\left\{n-1\right\}quadmbox\left\{ for \right\}quad ngeq 1.$
It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ 2, but does not have a fixed point.

The generalizations of the Brouwer fixed point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and in addition also often an assumption of convexity. See fixed point theorems in infinite-dimensional spaces for a discussion of these theorems.

The Kakutani fixed point theorem generalizes the Brouwer fixed point theorem in a different direction: it stays in Rn, but considers upper semi-continuous correspondences (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.

By using forcing to collapse cardinals, the Brouwer fixed point theorem can be generalized to arbitrary cardinality: if L is a compact, connected order topology, then any continuous function from Ln to itself has a fixed point. Note that if we require L to be separable, this is precisely the Brouwer fixed point theorem.

The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of Dn.