Sperner's Lemma, Brouwer's Fixed-Point Theorem, and Cohomology

Sperner’s lemma, Brouwer’s ﬁxed-point theorem, and cohomology

Nikolai V. Ivanov

1. Introduction

The proof of Brouwer’s ﬁxed-point theorem based on Sperner’s lemma [S] is often pre- sented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. A natural question is to what extent this proof is independent from the ideas of algebraic topology, and, in particular, to what extent this proof is independent from Brouwer’s own proof. The latter is based on the degree theory, i.e. on a fragment of algebraic topology developed by Brouwer. After the author discovered [I] that the famous analytic proof of Brouwer’s theorem due to Dunford and Schwartz is nothing else but the usual topological proof in disguise, it was only natural to suspect that the same is true for the proof based on Sperner’s lemma. This suspicion turned out to be correct, and the goal of this note is to uncover the standard topology hidden in this proof.

In fact, the two situations are very similar. Dunford–Schwartz proof can be considered as a cochain-level version of the standard proof based on de Rham cohomology theory (in the de Rham theory cochains are nothing else but differential forms), written in the language of elementary multivariable calculus. Similarly, the combinatorial proof of Sperner’s lemma can be considered as a cochain-level version of a standard argument based on simplicial cohomology theory, written in a combinatorial language. Of course, in this version one needs to use simplicial cochains instead of de Rahm cochains (i.e., of differential forms).

1 Both these alternatives to algebraic topology turn out to be not homological, but cohomological proofs in disguise. This is somewhat surprising; usually homology theory is considered to be more intuitive than cohomology theory. The most likely reason is that cohomology theory is a contravariant functor: preimages under maps behave better than images.

Contrary to a widespread belief, Sperner did not proved his lemma in order to provide an elementary proof of Brouwer’s fixed point theorem. He was interested in other theorems of Brouwer (the invariance of dimension and the invariance of domains), and Brouwer’s fixed point theorem is not even mentioned in his paper [S]. The standard deduction of Brouwer’s theorem from Sperner’s lemma is based on an argument of Knaster–Kuratowski– Mazurkiewich [KKM]. At the first sight this argument is quite different from arguments used in proofs of Brouwer’s fixed point theorem based on algebraic topology.

These proofs of Brouwer’s theorem are usually based on a construction of a retraction of a disc onto its boundary starting from a fixed point free self-map of a disc. There are no such retractions by the no-retraction theorem and Brouwer’s theorem follows. It turns out that this construction of a retraction from a fixed point free map underlies also the Knaster– Kuratowski–Mazurkiewich argument. The no-retraction theorem itself can be proved by a modification of this argument suggested by the notion of simplicial approximations.

Outline of the paper. The rest of the paper is divided into two parts. Section 2 is devoted to Sperner’s lemma. We start with Sperner’s own proof phrased in a geometric language and then present a cohomological proof. In order to compare these two proofs we rewrite the cohomological proof in terms of cochains. The readers not familiar with the cohomology theory may skip the cohomological proof and proceed directly to this cochain-level proof. The latter may be thought of as a geometric realization of Sperner’s arguments.

The version of cohomology theory most suitable for discussing Sperner’s proof is the simplicial cohomology theory. Some familiarity with the cohomology theory on the part of the reader will help, but the author hopes that the cochain-level proof will be accessible to all readers comfortable with Sperner’s lemma and linear algebra methods in combinatorics.

Section 3 is devoted to Brouwer’s ﬁxed-point theorem. We present Knaster–Kuratowski– Mazurkiewich deduction of Brouwer’s theorem from Sperner’s lemma and then explain how the no-retraction theorem and its proof are, in fact, hidden in this deduction. In particular, one can use Sperner’s lemma in order to prove the no-retraction theorem and then complete the proof of Brouwer’s theorem by the well known elementary geometric argument.

Acknowledgments. I am grateful to F. Petrov and the late A. Zelevinsky for their interest in cohomological interpretation of Sperner’s lemma, which stimulated me to write this paper. I am also grateful to M. Prokhorova for careful reading of the ﬁrst version of this paper, for pointing out that the standard construction of the above-mentioned retraction may be not well-deﬁned for a simplex, and for continuing interest in my work.

2 2. Sperner’s lemma and cohomology

Let ∆ be an n-dimensional simplex with the vertices v0 , v1,..., vn . Let ∆i be the face of ∆ opposite to the vertex vi . Its vertices are all the vertices v0 , v1,...,vn except vi . The boundary ∂∆ is equal to the union of the n 1 faces ∆ . Suppose that the simplex ∆ is + i subdivided into smaller simplices forming a simplicial complex S.

Sperner’s lemma. Suppose that vertices of S are labeled by the numbers 1, 2, ... , n 1 in + such a way that if a vertex v belongs to a face ∆i , then the label of v is not equal to i . Then the number of n-dimensional simplices of S such that the set of labels of their vertices is equal to { 1, 2, ... , n 1 } is odd. In particular, there is at least one such simplex. +

Geometric interpretation. The labelings in Sperner’s lemma admit a geometric interpretation. Namely, if a vertex v is labeled by i , let us set ϕ(v ) v . This defines a map ϕ = i from the set of vertices of S to the set of vertices of ∆. Since ∆ is a simplex and therefore every set of vertices of ∆ is a set of vertices of a subsimplex, ϕ defines a simplicial map S ∆, which we will also denote by ϕ. Clearly, an n-dimensional simplex has −→ { 1, 2, ... , n 1 } as the set of labels of its vertices if and only if ϕ maps it onto ∆. In this + language the assumption of Sperner’s lemma means that ϕ maps simplices of S contained in a face ∆i of ∆ into ∆i , and the first part of the conclusion means that ϕ maps an odd number of simplices onto ∆. The second part is deduced from the first one by applying one of the most fundamental principles of mathematics: an odd number is not equal to zero.

In their classical treatise [AH] Alexandroff and Hopf state Sperner’s lemma in this language and use algebraic topology to prove it without even mentioning the combinatorial approach. For quite a while the author naively thought that he rediscovered this geometric interpretation. But nowadays the idea of turning sets into simplicial complexes is well-established, and one can trace its roots to the notion of the nerve of a system of sets, introduced by Alexandroff [A]. The “rediscovery” was, albeit indirectly, based on the original discovery.

Combinatorial proof of Sperner’s lemma. Now we will present a translation of Sperner’s proof in the geometric language. Along the way we introduce some notations used later. The numbers e, f , g , h introduced in this proof have exactly the same meaning as in Sperner’s paper [S] and play a crucial role also in the cochains-based proof later.

Let us use induction by n. The result is trivially true for n 0. Suppose that n 0. Let = > us consider one of the faces of ∆, say, the face ∆n 1 . It is an (n 1)-dimensional simplex. + − The subdivision S deﬁnes a subdivision S n 1 of ∆n 1 and the restriction of ϕ to S n 1 + + + is a simplicial map S n 1 ∆. By the assumptions of Sperner’s lemma the image of this + −→ map is contained in ∆n 1 and hence ϕ deﬁnes a simplicial map S n 1 ∆n 1 . The as- + + −→ + sumptions of Sperner’s lemma obviously hold for this map, and by the inductive assumption

3 ϕ maps an odd number of simplices of S n 1 onto ∆n 1 . It remains to deduce from this + + that ϕ maps an odd number of simplices of S onto ∆. This is the main part of the proof.

If σ is a simplex of S contained in the boundary ∂∆ and ϕ(σ) ∆n 1 , then σ is con- = + tained in ∆n 1 . Indeed, otherwise ϕ(σ) would be contained in some other face by the + assumptions of Sperner’s Lemma. Let σ1 ,..., σh be the full list of such simplices. All of them are contained in ∆n 1 . Let τ1 ,..., τ g be the full list of other (n 1)-dimensional + − simplices of S mapped by ϕ onto ∆n 1 . They are contained in the interior of ∆. + Let us consider n-dimensional simplices of S. We are interested only in n-dimensional simplices σ such that the image ϕ(σ) is equal either to ∆ or to ∆n 1 . Clearly, on such sim- + plices have a face mapped by ϕ onto ∆n 1 . Let ρ1 ,..., ρe be the full list of n-dimensional + simplices of S having ∆ as the image under ϕ. Each of these simplices has exactly one face mapped by ϕ onto ∆n 1 . +

Let f be the number of n-dimensional simplices σ such that ϕ(σ) ∆n 1 . We claim = + that every such simplex has exactly two faces mapped by ϕ onto ∆n 1 . Indeed, such sim- + plex σ obviously has a proper face τ mapped by ϕ onto ∆n 1 . If v is the vertex of σ not + belonging to this face, then ϕ(v ) v for some i n. Let w be the vertex of τ mapped = i É by ϕ to vi . Clearly, if we replace w by v in τ, we will get another face τ0 of σ mapped by ϕ onto ∆n 1 , and no other face of σ is mapped onto ∆n 1 . This proves our claim. + + Double counting. Now we are ready for the most acclaimed part of the proof, a double counting argument. Let us count in two ways the number of pairs (σ, τ) such that τ is an (n 1)-dimensional face of an n-dimensional simplex σ and ϕ maps τ onto ∆n 1 . If − + we count the simplices τ ﬁrst, we get 2g h as the number of such pairs, because every + (n 1)-dimensional simplex in the interior of ∆ is a face of exactly two n -dimensional sim- − plices, and every (n 1)-dimensional simplex in the boundary ∂∆ is a face of exactly one − n-dimensional simplex. If we count the simplices σ ﬁrst, then only the simplices having as the image either ∆ or ∆n 1 matter and we get e 2 f as the numbers of pairs. Therefore + + (1) 2g h e 2 f , + = + and hence e is odd if h is. But h is odd by the inductive assumption and hence e is also odd. Since e is the number of simplices mapped onto ∆, this completes the proof. ■

Cohomological proof of Sperner’s lemma. The readers uncomfortable with cohomology theory may skip this proof. We will use the simplicial cohomology theory with coefficients in the field F2 . Since only in the distinction between even and odd numbers matters, this is the most natural choice of coefficients. We will consider the boundary ∂∆ as a simplicial complex having ∆1,..., ∆n 1 as the top-dimensional simplices. Let ∂S be the simplicial + complex consisting of simplices of S contained in the (geometric) boundary of ∆. Clearly, ϕ:S ∆ induces a simplicial map ∂S ∂∆, which we will denote by ϕ . As in the −→ −→ ∂

4 combinatorial proof, we will use an induction by n, the case n 0 being trivial. Suppose = that n 0 and consider the following commutative diagram. >

n 1 δ n H − ( ∂∆ ) H ( ∆, ∂∆ )

(2) ϕ∗∂ ϕ∗

n 1 δ n H − ( ∂S) H (S, ∂S).

Here the maps δ are the connecting homomorphisms in the cohomological sequences of pairs (S, ∂S ) and (∆, ∂∆). Since the cohomology groups of both S and ∆ are trivial, the exactness of the cohomological sequences of pairs (S, ∂S ) and (∆, ∂∆) implies that both maps δ are isomorphisms. In fact, all cohomology groups in (2) are isomorphic to F2 .

The map ϕ∗∂ is the multiplication by the mod 2 degree of ϕ∂ . This degree is equal to the number of (n 1)-simplices mapped by ϕ onto any (n 1)-simplex of ∂∆, for example, − ∂ − onto ∆n 1 . In the notations of the combinatorial proof, there are h such simplices and + hence the mod 2 degree of ϕ∂ is equal to h mod 2. Therefore ϕ∗∂ is equal to the multiplication by h. Similarly, ϕ∗ is equal to the multiplication by e , the number of simplices of S mapped onto ∆. Since the maps δ are isomorphisms, the commutativity of (2) implies that e h mod 2. Since h is assumed to be odd, e is odd also. ≡ ■

Simplicial cochains. Our next goal is to present a cochain-level version of the above cohomological proof. Let us begin with introducing the notions involved. Let X be a finite simplicial complex and let A be a subcomplex of X. An n-dimensional cochain of X is defined as a function on the set of simplices of X of dimension n with values in the field F2 . n Such cochains form a vector space over F2 , denoted by C ( X ). A cochain belongs to the subspace of relative cochains C n ( X, A ) if it is equal to 0 on all simplices of A.

To every n-simplex τ corresponds a cochain taking the value 1 on τ and the value 0 on all other simplices. By an abuse of notations, we will denote this cochain also by τ. Such cochains form a basis of C n ( X ) , and the cochains corresponding to n-simplices not contained in A form a basis of C n ( X, A ). Our abuse of notations allows us to consider cochains as formal sums of simplices. From this point of view, a cochain belongs to C n ( X, A ) if and only if it does not involve simplices of A. With these conventions, the coboundary map

n n 1 ∂∗ :C (X) C + (X) −→ can be deﬁned as follows. If τ is an n-simplex considered as a cochain, then ∂∗(τ) is the sum of all (n 1)-dimensional simplices of X having τ as a face. This map ∂∗ is extended + n n n 1 to the whole space C ( X ) by linearity. Clearly, ∂∗ maps C ( X, A ) to C + (X,A).

5 n Simplicial cohomology. A cochain α C ( X ) is called a cocycle if ∂∗ (α) 0 and is ∈ n 1 n = called a coboundary if it belongs to the image of ∂∗ :C − (X) C ( X ). The relative −→ cocycles and coboundaries are defined similarly. The cohomology group Hn ( X ) is defined as the quotient of the F2-vector space of n-dimensional cocycles by the subspace of n- dimensional coboundaries. The relative cohomology group Hn ( X, A ) is defined similarly. n 1 n 1 So, a cocycle α C − ( X ) defines an element of H − ( X ), the cohomology class of α, ∈ n 1 n 1 and a cocycle α C − ( X, A ) an element of H − ( X, A ). The connecting map ∈

n 1 n δ :H − (A) H (X,A) −→

n 1 is deﬁned as follows. If a H − ( A ) is represented by a cocycle α, one extends α as a ∈ n 1 function on simplices in an arbitrary way to a cochain αe C − ( X ), and then takes the n ∈ coboundary ∂∗( αe ). This coboundary belongs to C ( X, A ). One can check that ∂∗( αe ) is a cocycle and its cohomology class δ(a ) does not depend on the choices of α and αe.

The induced maps. A simplicial map ψ:X Y leads to a map −→

n n ψ∗ :C (Y) C (X) −→ called the induced map and deﬁned as follows. For an n-simplex τ the cochain ψ∗ (τ ) is deﬁned as the formal sum of all n-simplices of X mapped by ψ onto τ. The map ψ∗ is n extended to the whole space C ( Y ) by linearity. The cochain-level map ψ∗ induces a map

n n ψ∗ :H (Y) H (X) −→ of cohomology groups (also denoted also by ψ∗ ). The induced maps commute with the connecting maps in the following sense. Let

ϕ:(X0,A0 ) (X,A) −→ be a simplicial map of pairs, i.e. a simplicial map ϕ :X0 X such that ϕ(A0 ) A, and −→ ⊂ let ϕ :A0 A be the map induced by ϕ. Then the diagram A −→

n 1 δ n H − (A) H (X,A)

(3) ϕ∗A ϕ∗

n 1 δ n H − (A0 ) H (X0,A0 ), is commutative. Clearly, the diagram (2) is a special case of the diagram (3).

6 The commutativity of the diagram (3). Let us outline a proof of this commutativity. Let n 1 n 1 a H − ( A ) be a cohomology class. It can be represented by a cocycle α C − ( A ). Let ∈ n 1 ∈ us extend α to a cochain α C − ( X ) as in the deﬁnition of the connecting map δ. Then e ∈ ϕ∗ ( ∂∗( αe ) ) is a cocycle representing the cohomology class ϕ∗ ( δ(a ) ). On the other hand, the cochain ϕ∗ ( αe ) extends the cochain ϕ∗A (α) and therefore the coboundary ∂∗ ( ϕ∗ ( αe )) represents δ ( ϕ∗A (a ) ). Therefore it is sufﬁcient to prove that ¡ ¢ ¡ ¢ (4) ϕ∗ ∂∗ ( α ) ∂∗ ϕ∗ ( α ) . e = e

In fact, ϕ∗ ∂∗ ∂∗ ϕ∗ , i.e. ◦ = ◦ ¡ ¢ ¡ ¢ (5) ϕ∗ ∂∗ ( β) ∂∗ ϕ∗ ( β) = n 1 for every cochain β C − (X). ∈ By linearity, in order to prove (5) it is sufﬁcient to consider the case when β corresponds to a simplex τ. In this case both sides of (5) are equal to the sum of all n-dimensional simplices σ of X0 such that ϕ(σ) is also n-dimensional and has τ as a face. The only subtle point here is the fact n-dimensional simplices σ of X0 such that ϕ(σ) τ enter into = ∂∗ ( ϕ∗ ( β) ) with the coefﬁcient 2, which is equal to 0 in F2 . In the context of Sperner’s Lemma such simplices are exactly the n-dimensional simplices σ such that ϕ(σ) ∆n 1 , = + and arguments used in the combinatorial proof work in the present situation also.

Cochain-level proof of Sperner’s Lemma. Not surprisingly, the cochain-level proof is based directly on (4) or (5), bypassing the commutativity of the cohomological diagram (3). As before, we will argue by an induction by n, the case n 0 being trivial. Suppose that = n 0 and let us consider the simplex ∆n 1 as a cochain of ∂∆. There are no n-simplices > n + in ∂∆ and hence C ( ∂∆ ) 0. Therefore ∆n 1 is a cocycle. = +

We would like to apply (4) to ∆n 1 in the role of α. In order to do this one needs to extend + ∆n 1 to an (n 1)-dimensional cochain of ∆. Since all (n 1)-simplices of ∆ are contained + − − in ∂∆ , the only possible extension of ∆n 1 is ∆n 1 itself. Hence (4) takes the form + + ¡ ¢ ¡ ¢ (6) ϕ∗ ∂∗ (∆n 1 ) ∂∗ ϕ∗ (∆n 1 ) . + = +

Let us compute the left hand side of the equality (6). Clearly, ∂∗ ( ∆n 1 ) ∆, and hence + = ¡ ¢ ϕ∗ ∂∗ ( ∆n 1 ) ϕ∗ ( ∆ ) . + =

By the deﬁnition, ϕ∗ ( ∆ ) ρ ... ρ , where ρ denote the same simplices as in the = 1 + + e i combinatorial proof. It follows that

¡ ¢ (7) ϕ∗ ∂∗ ( ∆n 1 ) ρ1 ... ρe . + = + +

7 Let us compute now the right hand side of (6). The simplices σ1 ,..., σh and τ1 ,..., τg from the combinatorial proof provide a complete list without repetitions of (n 1)-dimen- − sional simplices of S mapped by ϕ onto ∆n 1 . Therefore +

ϕ∗ ( ∆n 1 ) σ1 ... σh τ1 ... τ g + = + + + + + and we need to compute the coboundary

∂∗ ( σ ... σ τ ... τ ). 1 + + h1 + 1 + + g Every σ is an (n 1)-dimensional simplex contained in ∂S and hence is a face of a unique i − n-dimensional simplex Σ of S. Clearly, ∂∗ (σ ) Σ . It follows that i i = i ¡ ¢ ¡ ¢ (8) ∂∗ ϕ∗( ∆n 1 ) ∂∗ σ1 ... σh τ1 ... τg + = + + + + +

Σ ... Σ ∂∗ (τ ) ... ∂∗ (τ ). = 1 + + h + 1 + + g

By substituting (7) and (8) into (6) we see that

(9) Σ ... Σ ∂∗ (τ ) ... ∂∗ (τ ) ρ ... ρ . 1 + + h + 1 + + g = 1 + + e

Every simplex τk is contained in the interior of S and hence is a face of exactly two n- dimensional simplices of S. Therefore every coboundary ∂∗ (τk ) is a sum of two simplices. It follows that the left hand side of (9) is a sum of h 2g simplices. Clearly, the right hand + side of (9) is a sum of e simplices. This does not implies that h 2g e because we are + = working over F (and h 2g is not equal to e in general). But summing the coefﬁcients 2 + of simplices at both sides of (9) as elements of F2 shows that

(10) h 2g e mod 2 + ≡ and hence h e mod 2. Therefore e is odd if and only if h is odd. ≡ ■

The cochain-level proof and the combinatorial proof. The chain-level proof can be eas- ily modiﬁed to get not only the congruence (10), but also the equality (1), the heart of Sperner’s proof. As we saw, the left hand side of (9) is a sum of h 2g simplices. Some + of them cancel each other. An n-dimensional simplex σ occurs in this sum twice if either σ and τ are faces of σ for some i , k (in this case σ Σ ), or τ and τ are faces i k = i i k of σ for some i k . Such simplices σ are exactly the n-dimensional simplices σ such 6= that ϕ(σ) ∆n 1 , and there are f of them. In other words, there are exactly f cancel- = + lations at the left hand side of (9). Therefore (9) implies that 2g h 2 f e and hence + − = implies (1). One can say that (9) is the cochain-level realization of the equation (1), and the cochains-based proof is a “linearization” of the combinatorial proof.

8 3. Brouwer’s ﬁxed-point theorem

Brouwer’s fixed-point theorem. Every continuous map f : ∆ ∆ has a fixed point. −→ The first proof based on Sperner’s lemma is due to Knaster, Kuratowski, and Mazurkiewich [KKM]. The following proof is a well known simplified version of it. The simplification re- sults from arguing by contradiction. Knaster–Kuratowski–Mazurkiewich construct a labeling satisfying the assumptions of Sperner’s lemma without assuming that f has no fixed points.

n 1 Proof. Let us denote by x the i th coordinate of x R + . We may assume that i ∈ © ¯ ª n 1 ∆ (x 1,..., x n 1 ) ¯ x 1,..., x n 1 0 and x 1 ... x n 1 1 R + = + + Ê + + + = ⊂ and that ∆ is the face of ∆ deﬁned by the equation x 0. i i = Suppose that x, y ∆ and that y x for all i 1, 2, ... , n 1. Since the coordi- ∈ i Ê i = + nates of y are nonnegative and their sum is equal to 1, and the same is true for x , these inequalities imply that y x . It follows that if the map f has no ﬁxed points and x ∆, = ∈ then f (x ) x for some i (because f (x ) x ). At the same time, if x ∆ , then i < i 6= ∈ i f (x ) x because f (x ) is always non-negative and in this case x 0. i Ê i i i =

Suppose that f has no ﬁxed points and choose a sequence of subdivisions S 0,S 1,S 2,... of ∆ in such a way that the maximal diameter of simplices of S tends to 0 when i . i −→ ∞ For example, one can take S 0 ∆ and S k 1 to be the barycentric subdivision of S k . = +

For each subdivision S k and each vertex w of S k let us label w by any i such that f (w ) w . If w ∆ , then the label of w cannot be equal to i . Hence any such label- i < i ∈ i ing satisﬁes the assumptions of Sperner’s lemma. Sperner’s lemma implies that for every k there is a simplex σ of S having { 1, 2, ... , n 1 } as the set of labels of its vertices. Let k k + x (k ) be an arbitrary point of σk . Since ∆ is compact, one can assume (after replacing the sequence S k by a subsequence, if necessary) that the sequence x (k ) converges to some point x ∆. Since the diameters of simplices σ tend to 0, for every i 1, 2, ... , n 1 ∈ k = + there are points w ∆ arbitrarily close to x and such that f (w ) w , for example, the ∈ i < i vertices of the simplices σk labeled by i for sufﬁciently big k . By passing to the limit we conclude that f (x ) x for all i 1, 2, ... , n 1. By the observation at the beginning i É i = + of the proof, this implies that f (x ) x , contrary to the assumption. = ■

Fixed point free maps and retractions. At the first sight this proof completely avoids a key step of almost all proofs of Brouwer fixed-point theorem: the construction of a retraction ∆ ∂∆ from a fixed point free map and then using the no-retraction theorem to the −→ effect that such retractions do not exist. Let us recall this construction.

9 Suppose that f : ∆ ∆ has no ﬁxed points and, in addition, that the following assump- −→ tion holds: (A) for each x ∆ the segment having x and f (x ) as its endpoints is not ∈ contained in a face of ∆. Under these assumptions one can deﬁne a map

r : ∆ ∂∆ −→ by assigning to x ∆ the point r (x ) of intersection of the ray going from f (x ) to x ∈ with ∂∆. This ray is well-defined because f (x ) x . The assumption (A) ensures that 6= the intersection of this ray with ∂∆ consists of only one point and hence r is well-defined. Clearly, r is a retraction ∆ ∂∆. If the map f does not satisfy (A), one can replace f −→ by a map f 0 : ∆ ∆ which does and still has no fixed points. In fact, if f 0 ( ∆ ) ∆ ∂∆, −→ ⊂ à then f 0 satisfies (A). Clearly, such a map f 0 can be chosen to be arbitrarily close to f , and if f 0 is sufficiently close to f , it has no fixed points together with f .

The standard expositions usually deal with maps from a ball to itself. Since a ball is strictly convex in contrast with ∆, the analogue of (A) is automatically satisﬁed. This suggests an- 1 other construction of r . Let h : ∆ B be a homeomorphism. Then h f h − is a con- −→ ◦ ◦ tinuous map B B without ﬁxed points and hence leads to a retraction r B :B ∂B. −→1 −→ The map r h − r h is a retraction ∆ ∂∆. = ◦ B ◦ −→

The hidden retraction. In fact, the retraction r is implicitly used in Knaster-Kuratowski- Mazurkiewich proof. To simplify the discussion, let us assume that f satisﬁes the assumption (A). Let w be a vertex. The vector w f (w ) is parallel to the hyperplane −

x 1 ... x n 1 1 + + + = and hence the ray from f (w ) to w is contained in this hyperplane and intersects ∂∆. The vertex w may be labeled by i if and only if w f (w ) , i.e. if and only if w f (w ) i > i − has positive i th coordinate. By the construction of the retraction r this condition is equiv- alent to r (w ) w having positive i th coordinate. Hence w may be labeled by i if and − only if r (w ) is contained in the intersection of ∂∆ with the open half-space

This intersection is contained in ∂∆ ∆ because the face ∆ is deﬁned by x 0. It is à i i i = tempting to allow as a label of w any i such that r (w ) ∂∆ ∆ . This condition has the ∈ à i advantage of ∆ ∆ being independent of w , but after passing to the limit the sets ∆ ∆ à i à i has to be replaced by their closures, which, in contrast with the sets ∆ ∆ themselves, à i have non-empty intersection (for example, every vertex of ∆ belongs to their intersection). But one can avoid passing to the limit by using compactness in a less direct manner and strengthening the condition r (w ) ∂∆ ∆ in a standard in combinatorial topology way. ∈ à i This leads to a proof of the no-retraction theorem based on Sperner’s lemma.

10 No-retraction theorem. There exists no retraction ∆ ∂∆. −→

Proof. Suppose that r is a such retraction. Recall that the open star st (w ) of a vertex w of a simplicial complex is the union of all simplices having w as a vertex with the faces opposite of w removed. In particular, ∂∆ ∆ st (v ), where the star is taken in ∂∆. à i = i By the well known Lebesgue lemma applied to the open covering of ∆ by the preimages 1 r − ( st (vi ) ) every subset of ∆ having sufﬁciently small diameter is contained in one of these preimages. Applying Lebesgue lemma is simply another way to use the compactness of ∆. Clearly, the diameter of a star of a vertex of a simplicial complex is no bigger than twice the maximum of the diameters of simplices of this complex. Therefore, if diameters of simplices of a subdivision S of ∆ are small enough, then for every vertex w of S the image r ( st (w ) ) is contained in some star st (vi ). Let us label each vertex w by any i such that

r ( st(w )) ∂∆ ∆ st (v ). ⊂ à i = i By Sperner’s lemma there is a simplex σ of S having { 1, 2, ... , n 1 } as the set of labels + of its vertices. The interior int σ of σ is contained in st (w ) for every vertex w of σ. Since for each i 1, 2, ... , n 1 some vertex w of σ is labeled by i , it follows that = + r ( int σ ) st (v ) ⊂ i for every i . But the sets st (v ) ∂∆ ∆ obviously have empty intersection. The con- i = à i tradiction between the last two statements completes the proof. ■

Simplicial approximations. The labeling constructed in the above proof leads a simplicial map ϕ :S ∂∆ which is a simplicial approximation of the contunuous map r . In fact, −→ the best way to deﬁne simplicial approximations is to require that the condition

r ( st (w )) st ( ϕ(w )) ⊂ holds for every vertex w . Nowadays usually another deﬁnition is adopted, followed by a proof that this condition is necessary and sufﬁcient for ϕ to be a simplicial approximation of the continuous map r . The strengthening of the condition r (w ) ∂∆ ∆ to ∈ à i r ( st(w )) ∂∆ ∆ ⊂ à i was motivated exactly by this property of simplicial approximations. The use of Lebesgue lemma above is nothing else but the standard way to establish the existence of simplicial approximations. Sperner’s lemma implies that the existence of simplicial approximations of a retraction ∆ ∂∆ leads to a contradiction. −→

11 References

[A] P. Alexandroff, Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, Mathematische Annalen, V. 98 (1928), 617–635.

[AH] P. Alexandroff, H. Hopf, Topologie, Springer, 1935, xiii, 636 pp.

[I] N.V. Ivanov, A topologist’s view of the Dunford-Schwartz proof of the Brouwer ﬁxed- point theorem, The Mathematical Intelligencer, V. 22, No. 3 (2000), 55–57.

[KKM] B. Knaster, C. Kuratowski, S. Mazurkiewich, Ein Beweis des Fizpunktzatzes für n- dimensional Simplexe, Fundamenta Mathematicae, V. 14 (1929), 132–137.

[S] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Semin. Hamburg. Univ., Bd. 6 (1928), 265–272.

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