**Brouwer's fixed-point theorem** is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a convex compact subset of Euclidean space to itself.

- Statement
- Importance of the pre-conditions
- The function f as an endomorphism
- Boundedness
- Closedness
- Convexity
- Notes
- Illustrations
- Intuitive approach
- Explanations attributed to Brouwer
- One-dimensional case
- History
- Prehistory
- First proofs
- Reception
- Proof outlines
- A proof using degree
- A proof using the hairy ball theorem
- A proof using homology or cohomology
- A proof using Stokes' theorem
- A combinatorial proof
- A proof by Hirsch
- A proof using oriented area
- A proof using the game Hex
- A proof using the Lefschetz fixed-point theorem
- A proof in a weak logical system
- Generalizations
- Equivalent results
- See also
- Notes 2
- References
- External links

Among hundreds of fixed-point theorems,^{ [1] } Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem.^{ [2] } This gives it a place among the fundamental theorems of topology.^{ [3] } The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.

The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the *n*-dimensional closed ball was first proved in 1910 by Jacques Hadamard ^{ [4] } and the general case for continuous mappings by Brouwer in 1911.^{ [5] }

The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:

- In the plane
- Every continuous function from a closed disk to itself has at least one fixed point.
^{ [6] }

This can be generalized to an arbitrary finite dimension:

- In Euclidean space
- Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.
^{ [7] }

A slightly more general version is as follows:^{ [8] }

- Convex compact set
- Every continuous function from a convex compact subset
*K*of a Euclidean space to*K*itself has a fixed point.^{ [9] }

An even more general form is better known under a different name:

- Schauder fixed point theorem
- Every continuous function from a convex compact subset
*K*of a Banach space to*K*itself has a fixed point.^{ [10] }

The theorem holds only for functions that are *endomorphisms* (functions that have the same set as the domain and range) and for sets that are *compact* (thus, in particular, bounded and closed) and *convex* (or homeomorphic to convex). The following examples show why the pre-conditions are important.

Consider the function

with domain [-1,1]. The range of the function is [0,2]. Thus, f is not an endomorphism.

Consider the function

which is a continuous function from to itself. As it shifts every point to the right, it cannot have a fixed point. The space is convex and closed, but not bounded.

Consider the function

which is a continuous function from the open interval (−1,1) to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point. The space (−1,1) is convex and bounded, but not closed. The function *f**does* have a fixed point for the closed interval [−1,1], namely *f*(1) = 1.

Convexity is not strictly necessary for BFPT. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball . For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.).

The following example shows that BFPT doesn't work for domains with holes. Consider the function , which is a continuous function from the unit circle to itself. Since *-x≠x* holds for any point of the unit circle, *f* has no fixed point. The analogous example works for the *n*-dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex) . The function *f**does* have a fixed point for the unit disc, since it takes the origin to itself.

A formal generalization of BFPT for "hole-free" domains can be derived from the Lefschetz fixed-point theorem.^{ [11] }

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

The theorem has several "real world" illustrations. Here are some examples.

- 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, in 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 directly above 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. - Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.
- In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the 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, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktail shaken, not stirred defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.
^{[ citation needed ]}

The theorem is supposed to have originated from Brouwer's observation of a cup of coffee.^{ [12] } If one stirs to dissolve a lump of sugar, it appears there is always a point without motion. He drew the conclusion that at any moment, there is a point on the surface that is not moving.^{ [13] } The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit. The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.

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."^{ [13] } Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as Stefan Banach's, that guarantee uniqueness.

In one dimension, the result is intuitive and easy to prove. The continuous function *f* is defined on a closed interval [*a*, *b*] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval [*a*, *b*] which maps *x* to *x* (light green).

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function *g* which maps *x* to *f*(*x*) − *x*. It is ≥ 0 on *a* and ≤ 0 on *b*. By the intermediate value theorem, *g* has a zero in [*a*, *b*]; this zero is a fixed point.

Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."^{ [13] }

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 *).^{ [14] } It was later proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910,^{ [4] } and Brouwer found a different proof in the same year.^{ [5] } Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals. Although the existence of a fixed point is not constructive in the sense of constructivism in mathematics, methods to approximate fixed points guaranteed by Brouwer's theorem are now known.^{ [15] }^{ [16] }

To understand the prehistory of Brouwer's fixed point theorem one needs to pass through differential equations. At the end of the 19th century, the old problem^{ [17] } of the stability of the solar system returned into the focus of the mathematical community.^{ [18] } Its solution required new methods. As noted by Henri Poincaré, who worked on the three-body problem, there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge."^{ [19] } He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision".^{ [20] }

He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant flow?^{ [21] } Poincaré discovered that the answer can be found in what we now call the topological properties in the area containing the trajectory. If this area is compact, i.e. both closed and bounded, then the trajectory either becomes stationary, or it approaches a limit cycle.^{ [22] } Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval *t*. If the area is a circular band, or if it is not closed,^{ [23] } then this is not necessarily the case.

To understand differential equations better, a new branch of mathematics was born. Poincaré called it *analysis situs*. The French Encyclopædia Universalis defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing".^{ [24] } In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem,^{ [25] } although the connection with the subject of this article was not yet apparent.^{ [26] } A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the fundamental group or sometimes the Poincaré group.^{ [27] } This method can be used for a very compact proof of the theorem under discussion.

Poincaré's method was analogous to that of Émile Picard, a contemporary mathematician who generalized the Cauchy–Lipschitz theorem.^{ [28] } Picard's approach is based on a result that would later be formalised by another fixed-point theorem, named after Banach. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a contraction.

At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. Piers Bohl, a Latvian mathematician, applied topological methods to the study of differential equations.^{ [29] } In 1904 he proved the three-dimensional case of our theorem,^{ [14] } but his publication was not noticed.^{ [30] }

It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially mathematical logic and topology. His initial interest lay in an attempt to solve Hilbert's fifth problem.^{ [31] } In 1909, during a voyage to Paris, he met Henri Poincaré, Jacques Hadamard, and Émile Borel. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the hairy ball theorem for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point.^{ [32] } These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem.^{ [33] } The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as homotopy, the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. Hans Freudenthal comments on the respective roles as follows: "Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."^{ [34] }

Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension,^{ [5] } as well as other key theorems such as the invariance of dimension.^{ [35] } In the context of this work, Brouwer also generalized the Jordan curve theorem to arbitrary dimension and established the properties connected with the degree of a continuous mapping.^{ [36] } This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became algebraic topology.^{ [37] }

The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory.^{ [38] } Brouwer's theorem is probably the most important.^{ [39] } It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan curve theorem.^{ [40] }

Besides the fixed-point theorems for more or less contracting functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the *n*-dimensional sphere to **R**^{n} has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the Lefschetz fixed-point theorem provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to Banach spaces.^{ [41] } This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to multivalued functions.^{ [42] } One also meets the theorem and its variants outside topology. It can be used to prove the Hartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the Central Limit Theorem. The theorem can also be found in existence proofs for the solutions of certain partial differential equations.^{ [43] }

Other areas are also touched. In game theory, John Nash used the theorem to prove that in the game of Hex there is a winning strategy for white.^{ [44] } In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria (Hotelling's law), financial equilibria and incomplete markets.^{ [45] }

Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are not constructive,^{ [46] } and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of constructivity. He became the originator and zealous defender of a way of formalising mathematics that is known as intuitionism, which at the time made a stand against set theory.^{ [47] } Brouwer disavowed his original proof of the fixed-point theorem. The first algorithm to approximate a fixed point was proposed by Herbert Scarf.^{ [48] } A subtle aspect of Scarf's algorithm is that it finds a point that is *almost fixed* by a function *f*, but in general cannot find a point that is close to an actual fixed point. In mathematical language, if ε is chosen to be very small, Scarf's algorithm can be used to find a point *x* such that *f*(*x*) is very close to *x*, i.e., . But Scarf's algorithm cannot be used to find a point *x* such that *x* is very close to a fixed point: we cannot guarantee where Often this latter condition is what is meant by the informal phrase "approximating a fixed point"^{[ citation needed ]}.

Brouwer's original 1911 proof relied on the notion of the degree of a continuous mapping, stemming from ideas in differential topology. Several modern accounts of the proof can be found in the literature, notably Milnor (1965).^{ [49] }^{ [50] }

Let denote the closed unit ball in centered at the origin. Suppose for simplicity that is continuously differentiable. A regular value of is a point such that the Jacobian of is non-singular at every point of the preimage of . In particular, by the inverse function theorem, every point of the preimage of lies in (the interior of ). The degree of at a regular value is defined as the sum of the signs of the Jacobian determinant of over the preimages of under :

The degree is, roughly speaking, the number of "sheets" of the preimage *f* lying over a small open set around *p*, with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of winding number to higher dimensions.

The degree satisfies the property of *homotopy invariance*: let and be two continuously differentiable functions, and for . Suppose that the point is a regular value of for all *t*. Then .

If there is no fixed point of the boundary of , then the function

is well-defined, and

defines a homotopy from the identity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so also has degree one at the origin. As a consequence, the preimage is not empty. The elements of are precisely the fixed points of the original function *f*.

This requires some work to make fully general. The definition of degree must be extended to singular values of *f*, and then to continuous functions. The more modern advent of homology theory simplifies the construction of the degree, and so has become a standard proof in the literature.

The hairy ball theorem states that on the unit sphere *S* in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field **w** on *S*. (The tangency condition means that **w**(**x**) ⋅ **x** = 0 for every unit vector **x**.) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in Milnor (1978).

In fact, suppose first that **w** is * continuously differentiable*. By scaling, it can be assumed that

If **w** is only a * continuous* unit tangent vector on

The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that *n* is odd. If there were a fixed-point-free continuous self-mapping **f** of the closed unit ball *B* of the *n*-dimensional Euclidean space *V*, set

Since **f** has no fixed points, it follows that, for **x** in the interior of *B*, the vector **w**(**x**) is non-zero; and for **x** in *S*, the scalar product **x** ⋅ **w**(**x**) = 1 – **x** ⋅ **f**(**x**) is strictly positive. From the original *n*-dimensional space Euclidean space *V*, construct a new auxiliary

(*n* + 1)-dimensional space *W* = *V* x **R**, with coordinates *y* = (**x**, *t*). Set

By construction **X** is a continuous vector field on the unit sphere of *W*, satisfying the tangency condition **y** ⋅ **X**(**y**) = 0. Moreover, **X**(**y**) is nowhere vanishing (because, if ` x` has norm 1, then

The proof uses the observation that the boundary of the *n*-disk *D*^{n} is *S*^{n−1}, the (*n* − 1)-sphere.

Suppose, for contradiction, that a continuous function *f* : *D*^{n} → *D*^{n} has *no* fixed point. This means that, for every point x in *D*^{n}, the points *x* and *f*(*x*) are distinct. Because they are distinct, for every point x in *D*^{n}, we can construct a unique ray from *f*(*x*) to *x* and follow the ray until it intersects the boundary *S*^{n−1} (see illustration). By calling this intersection point *F*(*x*), we define a function *F* : *D*^{n} → *S*^{n−1} sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever x itself is on the boundary, then the intersection point *F*(*x*) must be *x*.

Consequently, F is a special type of continuous function known as a retraction: every point of the codomain (in this case *S*^{n−1}) is a fixed point of *F*.

Intuitively it seems unlikely that there could be a retraction of *D*^{n} onto *S*^{n−1}, and in the case *n* = 1, the impossibility is more basic, 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 a surjective group homomorphism from the fundamental group of *D*^{2} to that of *S*^{1}, but the latter group is isomorphic to **Z** while the first 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: the homology *H*_{n−1}(*D*^{n}) is trivial, while *H*_{n−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.

The impossibility of a retraction can also be shown using the de Rham cohomology of open subsets of Euclidean space *E*^{n}. For *n* ≥ 2, the de Rham cohomology of *U* = *E*^{n} – (0) is one-dimensional in degree 0 and *n* - 1, and vanishes otherwise. If a retraction existed, then *U* would have to be contractible and its de Rham cohomology in degree *n* - 1 would have to vanish, a contradiction.^{ [52] }

As in the proof of Brouwer's fixed-point theorem for continuous maps using homology, it is reduced to proving that there is no continuous retraction *F* from the ball *B* onto its boundary ∂*B*. In that case it can be assumed that *F* is smooth, since it can be approximated using the Weierstrass approximation theorem or by convolving with non-negative smooth bump functions of sufficiently small support and integral one (i.e. mollifying). If ω is a volume form on the boundary then by Stokes' theorem,

giving a contradiction.^{ [53] }^{ [54] }

More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifold *M* onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the form ω generates the de Rham cohomology group *H*^{n-1}(∂*M*) which is isomorphic to the homology group *H*_{n-1}(∂*M*) by de Rham's theorem.^{ [55] }

The BFPT can be proved using Sperner's lemma. We now give an outline of the proof for the special case in which *f* is a function from the standard *n*-simplex, to itself, where

For every point also Hence the sum of their coordinates is equal:

Hence, by the pigeonhole principle, for every there must be an index such that the th coordinate of is greater than or equal to the th coordinate of its image under *f*:

Moreover, if lies on a *k*-dimensional sub-face of then by the same argument, the index can be selected from among the *k* + 1 coordinates which are not zero on this sub-face.

We now use this fact to construct a Sperner coloring. For every triangulation of the color of every vertex is an index such that

By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is an *n*-dimensional simplex whose vertices are colored with the entire set of *n* + 1 available colors.

Because *f* is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point which satisfies the labeling condition in all coordinates: for all

Because the sum of the coordinates of and must be equal, all these inequalities must actually be equalities. But this means that:

That is, is a fixed point of

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 or by convolving with smooth bump functions. 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.^{ [56] }

R. Bruce Kellogg, Tien-Yien Li, and James A. Yorke turned Hirsch's proof into a computable proof by observing that the retract is in fact defined everywhere except at the fixed points.^{ [57] } For almost any point, *q*, on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from *q* to a fixed point. It is an easy numerical task to follow such a path from *q* to the fixed point so the method is essentially computable.^{ [58] } gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If is a smooth retraction, one considers the smooth deformation and the smooth function

Differentiating under the sign of integral it is not difficult to check that *φ′*(*t*) = 0 for all *t*, so *φ* is a constant function, which is a contradiction because *φ*(0) is the *n*-dimensional volume of the ball, while *φ*(1) is zero. The geometric idea is that *φ*(*t*) is the oriented area of *g*^{t}(*B*) (that is, the Lebesgue measure of the image of the ball via *g*^{t}, taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter *t* passes form 0 to 1 the map *g*^{t} transforms continuously from the identity map of the ball, to the retraction *r*, which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of *r* is necessarily 0, as its image is the boundary of the ball, a set of null measure.^{ [59] }

A quite different proof given by David Gale is based on the game of Hex. The basic theorem regarding Hex, first proven by John Nash, is that no game of Hex can end in a draw; the first player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater). This turns out to be 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.^{ [60] }

The Lefschetz fixed-point theorem says that if a continuous map *f* from a finite simplicial complex *B* to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number

and in particular if the Lefschetz number is nonzero then *f* must have a fixed point. If *B* is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero simplicial homology group is: and *f* acts as the identity on this group, so *f* has a fixed point.^{ [61] }^{ [62] }

In reverse mathematics, Brouwer's theorem can be proved in the system WKL_{0}, and conversely over the base system RCA_{0} Brouwer's theorem for a square implies the weak König's lemma, so this gives a precise description of the strength of Brouwer's theorem.

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 ℓ^{2} of square-summable real (or complex) sequences, consider the map *f* : ℓ^{2} → ℓ^{2} which sends a sequence (*x*_{n}) from the closed unit ball of ℓ^{2} to the sequence (*y*_{n}) defined by

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 also often an assumption of convexity. See fixed-point theorems in infinite-dimensional spaces for a discussion of these theorems.

There is also finite-dimensional generalization to a larger class of spaces: If is a product of finitely many chainable continua, then every continuous function has a fixed point,^{ [63] } where a chainable continuum is a (usually but in this case not necessarily metric) compact Hausdorff space of which every open cover has a finite open refinement , such that if and only if . Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.

The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in **R**^{n}, but considers upper hemi-continuous set-valued functions (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.

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 *D*^{n}.

There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.^{ [64] }

Algebraic topology | Combinatorics | Set covering |
---|---|---|

Brouwer fixed-point theorem | Sperner's lemma | Knaster–Kuratowski–Mazurkiewicz lemma |

Borsuk–Ulam theorem | Tucker's lemma | Lusternik–Schnirelmann theorem |

- Banach fixed-point theorem
- Infinite compositions of analytic functions
- Nash equilibrium
- Poincaré–Miranda theorem – equivalent to the Brouwer fixed-point theorem
- Topological combinatorics

- ↑ E.g. F & V Bayart
*Théorèmes du point fixe*on Bibm@th.net Archived December 26, 2008, at the Wayback Machine - ↑ See page 15 of: D. Leborgne
*Calcul différentiel et géométrie*Puf (1982) ISBN 2-13-037495-6 - ↑ More exactly, according to Encyclopédie Universalis:
*Il en a démontré l'un des plus beaux théorèmes, le théorème du point fixe, dont les applications et généralisations, de la théorie des jeux aux équations différentielles, se sont révélées fondamentales.*Luizen Brouwer by G. Sabbagh - 1 2 Jacques Hadamard:
*Note sur quelques applications de l’indice de Kronecker*in Jules Tannery:*Introduction à la théorie des fonctions d’une variable*(Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French) - 1 2 3 Brouwer, L. E. J. (1911). "Über Abbildungen von Mannigfaltigkeiten".
*Mathematische Annalen*(in German).**71**: 97–115. doi:10.1007/BF01456931. S2CID 177796823. - ↑ D. Violette
*Applications du lemme de Sperner pour les triangles*Bulletin AMQ, V. XLVI N° 4, (2006) p 17. Archived June 8, 2011, at the Wayback Machine - ↑ Page 15 of: D. Leborgne
*Calcul différentiel et géométrie*Puf (1982) ISBN 2-13-037495-6. - ↑ This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see Florenzano, Monique (2003).
*General Equilibrium Analysis: Existence and Optimality Properties of Equilibria*. Springer. p. 7. ISBN 9781402075124 . Retrieved 2016-03-08. - ↑ V. & F. Bayart
*Point fixe, et théorèmes du point fixe*on Bibmath.net. Archived December 26, 2008, at the Wayback Machine - ↑ C. Minazzo K. Rider
*Théorèmes du Point Fixe et Applications aux Equations Différentielles Archived 2018-04-04 at the Wayback Machine*Université de Nice-Sophia Antipolis. - ↑ Belk, Jim. "Why is convexity a requirement for Brouwer fixed points?". Math StackExchange. Retrieved 22 May 2015.
- ↑ The interest of this anecdote rests in its intuitive and didactic character, but its accuracy is dubious. As the history section shows, the origin of the theorem is not Brouwer's work. More than 20 years earlier Henri Poincaré had proved an equivalent result, and 5 years before Brouwer P. Bohl had proved the three-dimensional case.
- 1 2 3 This citation comes originally from a television broadcast:
*Archimède*, Arte, 21 septembre 1999 - 1 2 Bohl, P. (1904). "Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage".
*J. Reine Angew. Math*.**127**(3/4): 179–276. - ↑ Karamardian, Stephan (1977).
*Fixed points: algorithms and applications*. New York: Academic Press. ISBN 978-0-12-398050-2. - ↑ Istrăţescu, Vasile (1981).
*Fixed point theory*. Dordrecht-Boston, Mass.: D. Reidel Publishing Co. ISBN 978-90-277-1224-0. - ↑ See F. Brechenmacher
*L'identité algébrique d'une pratique portée par la discussion sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des planètes*CNRS Fédération de Recherche Mathématique du Nord-Pas-de-Calais - ↑ Henri Poincaré won the King of Sweden's mathematical competition in 1889 for his work on the related three-body problem: Jacques Tits
*Célébrations nationales 2004*Site du Ministère Culture et Communication - ↑ Henri Poincaré
*Les méthodes nouvelles de la mécanique céleste*T Gauthier-Villars, Vol 3 p 389 (1892) new edition Paris: Blanchard, 1987. - ↑ Quotation from Henri Poincaré taken from: P. A. Miquel
*La catégorie de désordre Archived 2016-03-03 at the Wayback Machine*, on the website of l'Association roumaine des chercheurs francophones en sciences humaines - ↑ This question was studied in: Poincaré, H. (1886). "Sur les courbes définies par les équations différentielles".
*Journal de Mathématiques Pures et Appliquées*.**2**(4): 167–244. - ↑ This follows from the Poincaré–Bendixson theorem.
- ↑ Multiplication by 1/2 on ]0, 1[
^{2}has no fixed point. - ↑ "concerne les propriétés invariantes d'une figure lorsqu’on la déforme de manière continue quelconque, sans déchirure (par exemple, dans le cas de la déformation de la sphère, les propriétés corrélatives des objets tracés sur sa surface". From C. Houzel M. Paty
*Poincaré, Henri (1854–1912) Archived 2010-10-08 at the Wayback Machine*Encyclopædia Universalis Albin Michel, Paris, 1999, p. 696–706 - ↑ Poincaré's theorem is stated in: V. I. Istratescu
*Fixed Point Theory an Introduction*Kluwer Academic Publishers (réédition de 2001) p 113 ISBN 1-4020-0301-3 - ↑ Voitsekhovskii, M.I. (2001) [1994], "Brouwer theorem",
*Encyclopedia of Mathematics*, EMS Press, ISBN 1-4020-0609-8 - ↑ Dieudonné, Jean (1989).
*A History of Algebraic and Differential Topology, 1900–1960*. Boston: Birkhäuser. pp. 17–24. ISBN 978-0-8176-3388-2. - ↑ See for example: Émile Picard
*Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires Archived 2011-07-16 at the Wayback Machine*Journal de Mathématiques p 217 (1893) - ↑ J. J. O'Connor E. F. Robertson
*Piers Bohl* - ↑ Myskis, A. D.; Rabinovic, I. M. (1955). "Первое доказательство теоремы о неподвижной точке при непрерывном отображении шара в себя, данное латышским математиком П.Г.Болем" [The first proof of a fixed-point theorem for a continuous mapping of a sphere into itself, given by the Latvian mathematician P. G. Bohl].
*Успехи математических наук*(in Russian).**10**(3): 188–192. - ↑ J. J. O'Connor E. F. Robertson
*Luitzen Egbertus Jan Brouwer* - ↑ Freudenthal, Hans (1975). "The cradle of modern topology, according to Brouwer's inedita".
*Historia Mathematica*.**2**(4): 495–502 [p. 495]. doi: 10.1016/0315-0860(75)90111-1 . - ↑ Freudenthal, Hans (1975). "The cradle of modern topology, according to Brouwer's inedita".
*Historia Mathematica*.**2**(4): 495–502 [p. 495]. doi: 10.1016/0315-0860(75)90111-1 .... cette dernière propriété, bien que sous des hypothèses plus grossières, ait été démontré par H. Poincaré

- ↑ Freudenthal, Hans (1975). "The cradle of modern topology, according to Brouwer's inedita".
*Historia Mathematica*.**2**(4): 495–502 [p. 501]. doi: 10.1016/0315-0860(75)90111-1 . - ↑ If an open subset of a manifold is homeomorphic to an open subset of a Euclidean space of dimension
*n*, and if*p*is a positive integer other than*n*, then the open set is never homeomorphic to an open subset of a Euclidean space of dimension*p*. - ↑ J. J. O'Connor E. F. Robertson
*Luitzen Egbertus Jan Brouwer*. - ↑ The term
*algebraic topology*first appeared 1931 under the pen of David van Dantzig: J. Miller*Topological algebra*on the site Earliest Known Uses of Some of the Words of Mathematics (2007) - ↑ V. I. Istratescu
*Fixed Point Theory. An Introduction*Kluwer Academic Publishers (new edition 2001) ISBN 1-4020-0301-3. - ↑ "... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu
*Fixed Point Theory an Introduction*Kluwer Academic Publishers (new edition 2001) ISBN 1-4020-0301-3. - ↑ E.g.: S. Greenwood J. Cao
*Brouwer’s Fixed Point Theorem and the Jordan Curve Theorem*University of Auckland, New Zealand. - ↑ Schauder, J. (1930). "Der Fixpunktsatz in Funktionsräumen".
*Studia Mathematica*.**2**: 171–180. doi: 10.4064/sm-2-1-171-180 . - ↑ Kakutani, S. (1941). "A generalization of Brouwer's Fixed Point Theorem".
*Duke Mathematical Journal*.**8**(3): 457–459. doi:10.1215/S0012-7094-41-00838-4. - ↑ These examples are taken from: F. Boyer
*Théorèmes de point fixe et applications*CMI Université Paul Cézanne (2008–2009) Archived copy at WebCite (August 1, 2010). - ↑ For context and references see the article Hex (board game).
- ↑ P. Bich
*Une extension discontinue du théorème du point fixe de Schauder, et quelques applications en économie Archived June 11, 2011, at the Wayback Machine*Institut Henri Poincaré, Paris (2007) - ↑ For a long explanation, see: Dubucs, J. P. (1988). "L. J. E. Brouwer : Topologie et constructivisme".
*Revue d'Histoire des Sciences*.**41**(2): 133–155. doi:10.3406/rhs.1988.4094. - ↑ Later it would be shown that the formalism that was combatted by Brouwer can also serve to formalise intuitionism, with some modifications. For further details see constructive set theory.
- ↑ H. Scarf found the first algorithmic proof: Voitsekhovskii, M.I. (2001) [1994], "Brouwer theorem",
*Encyclopedia of Mathematics*, EMS Press, ISBN 1-4020-0609-8 . - ↑ Milnor 1965 , pp. 1–19
- ↑ Teschl, Gerald (2019). "10. The Brouwer mapping degree".
*Topics in Linear and Nonlinear Functional Analysis*(PDF). Graduate Studies in Mathematics. American Mathematical Society . Retrieved 1 February 2022. - ↑ Milnor 1978
- ↑ Madsen & Tornehave 1997 , pp. 39–48
- ↑ Boothby 1971
- ↑ Boothby 1986
- ↑ Dieudonné 1982
- ↑ Hirsch 1988
- ↑ Kellogg, Li & Yorke 1976.
- ↑ Chow, Mallet-Paret & Yorke 1978.
- ↑ Kulpa 1989
- ↑ David Gale (1979). "The Game of Hex and Brouwer Fixed-Point Theorem".
*The American Mathematical Monthly*.**86**(10): 818–827. doi:10.2307/2320146. JSTOR 2320146. - ↑ Hilton & Wylie 1960
- ↑ Spanier 1966
- ↑ Eldon Dyer (1956). "A fixed point theorem".
*Proceedings of the American Mathematical Society*.**7**(4): 662–672. doi: 10.1090/S0002-9939-1956-0078693-4 . - ↑ Nyman, Kathryn L.; Su, Francis Edward (2013), "A Borsuk–Ulam equivalent that directly implies Sperner's lemma",
*American Mathematical Monthly*,**120**(4): 346–354, doi:10.4169/amer.math.monthly.120.04.346, MR 3035127

In mathematics, the **Borsuk–Ulam theorem** states that every continuous function from an *n*-sphere into Euclidean *n*-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.

In mathematics, a **diffeomorphism** is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.

In the mathematical field of topology, a **homeomorphism**, **topological isomorphism**, or **bicontinuous function** is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called **homeomorphic**, and from a topological viewpoint they are the same. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape or form, introduced to mathematics by Henri Poincaré in 1895.

In mathematical analysis, the **intermediate value theorem** states that if is a continuous function whose domain contains the interval [*a*, *b*], then it takes on any given value between and at some point within the interval.

In mathematics, a **normed vector space** or **normed space** is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

- It is nonnegative, meaning that for every vector
- It is positive on nonzero vectors, that is,
- For every vector and every scalar
- The triangle inequality holds; that is, for every vectors and

In mathematics, physics, and engineering, a **vector space** is a set whose elements, often called *vectors*, may be added together and multiplied ("scaled") by numbers called *scalars*. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called *vector axioms*. The terms **real vector space** and **complex vector space** are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

**Invariance of domain** is a theorem in topology about homeomorphic subsets of Euclidean space . It states:

In topology, the **Jordan curve theorem** asserts that every *Jordan curve* divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. *"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."*. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

The **hairy ball theorem** of algebraic topology states that there is no nonvanishing continuous tangent vector field on even-dimensional *n*-spheres. For the ordinary sphere, or 2‑sphere, if *f* is a continuous function that assigns a vector in **R**^{3} to every point *p* on a sphere such that *f*(*p*) is always tangent to the sphere at *p*, then there is at least one pole, a point where the field vanishes.

The **Arzelà–Ascoli theorem** is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

In mathematics, a **fixed-point theorem** is a result saying that a function *F* will have at least one fixed point, under some conditions on *F* that can be stated in general terms. Some authors claim that results of this kind are amongst the most generally useful in mathematics.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, the **real coordinate space** of dimension n, denoted **R**^{n} or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

In mathematical analysis, the **Kakutani fixed-point theorem** is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

A mathematical object *X* has **the fixed-point property** if every suitably well-behaved mapping from *X* to itself has a fixed point. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point. But another use is in order theory, where a partially ordered set *P* is said to have the fixed point property if every increasing function on *P* has a fixed point.

**Two-dimensional Euclidean space** or simply **two-dimensional space** is a geometric setting in which two values are required to determine the position of an element on the plane. The set of pairs of real numbers with appropriate structure often serves as the canonical example of a **Euclidean plane**, the two-dimensional Euclidean space; for a generalization of the concept, see dimension. Two-dimensional space can be seen as a projection of the physical universe onto a plane. Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.

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- Brouwer's Fixed Point Theorem for Triangles at cut-the-knot
- Brouwer theorem, from PlanetMath with attached proof.
- Reconstructing Brouwer at MathPages
- Brouwer Fixed Point Theorem at Math Images.

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