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 al- gebraic 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 co- homology 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). © Nikolai V. Ivanov, 2009, 2019. Neither the work reported in the present paper, nor its preparation were supported by any corporate entity. 1 Both these alternatives to algebraic topology turn out to be not homological, but cohomolog- ical 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 sim- plicial 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 Å 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 Å 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. Å Geometric interpretation. The labelings in Sperner’s lemma admit a geometric interpreta- tion. Namely, if a vertex v is labeled by i , let us set '(v ) v . This defines a map ' Æ i from the set of vertices of S to the set of vertices of ¢. Since ¢ is a simplex and there- fore every set of vertices of ¢ is a set of vertices of a subsimplex, ' defines a simplicial map S ¢, which we will also denote by '. Clearly, an n-dimensional simplex has ¡! { 1, 2, ... , n 1 } as the set of labels of its vertices if and only if ' maps it onto ¢. In this Å language the assumption of Sperner’s lemma means that ' maps simplices of S contained in a face ¢i of ¢ into ¢i , and the first part of the conclusion means that ' 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 interpre- tation. 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 Æ È us consider one of the faces of ¢, say, the face ¢n 1 . It is an (n 1)-dimensional simplex. Å ¡ The subdivision S defines a subdivision S n 1 of ¢n 1 and the restriction of ' to S n 1 Å Å Å is a simplicial map S n 1 ¢. By the assumptions of Sperner’s lemma the image of this Å ¡! map is contained in ¢n 1 and hence ' defines a simplicial map S n 1 ¢n 1 . The as- Å Å ¡! Å sumptions of Sperner’s lemma obviously hold for this map, and by the inductive assumption 3 ' maps an odd number of simplices of S n 1 onto ¢n 1 . It remains to deduce from this Å Å that ' 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 '(σ) ¢n 1 , then σ is con- Æ Å tained in ¢n 1 . Indeed, otherwise '(σ) would be contained in some other face by the Å assumptions of Sperner’s Lemma. Let σ1 ,..., σh be the full list of such simplices. All of them are contained in ¢n 1 . Let ¿1 ,..., ¿ g be the full list of other (n 1)-dimensional Å ¡ simplices of S mapped by ' onto ¢n 1 . They are contained in the interior of ¢. Å Let us consider n-dimensional simplices of S. We are interested only in n-dimensional sim- plices σ such that the image '(σ) is equal either to ¢ or to ¢n 1 . Clearly, on such sim- Å plices have a face mapped by ' onto ¢n 1 . Let ½1 ,..., ½e be the full list of n-dimensional Å simplices of S having ¢ as the image under '.