Tkacz, Przemysław Topologizing Sperner’s lemma. (English) Zbl 07959960 Topol. Methods Nonlinear Anal. 64, No. 1, 1-13 (2024). A statement and proof of Sperner’s lemma can be found on page 250 of [K. Sakai, Geometric aspects of general topology. Tokyo: Springer (2013; Zbl 1280.54001)]. It can be stated as follows. If \(n\in\{0\}\cup\mathbb{N}\), \(\tau\) is an \(n\)-simplex, \(K\) is the natural triangulation of \(\tau\), \(K'\) is a subdivision of \(K\), and \(h:K'\to K\) is a simplicial approximation to the identity map of \(\tau\), then the number of \(n\)-simplexes \(\tau'\) of \(K'\) with \(h(\tau')=\tau\) is odd. This result is equivalent to the Brouwer fixed-point theorem and the no retraction theorem; it is typically used to prove them.The author, in the Introduction, states that Sperner’s lemma “has been generalized in a variety of ways.” Several references are provided as background sources. The applications in the latter are to various types of complexes. Frequently the complexes are provided with labels, that is, a function assigning to each vertex a natural number called its label. In this paper a new type of complex called an \(n\)-Sperner complex (polyhedron) is defined. In Section 2 one can find the definition of an almost \(n\)-pseudomanifold (a certain type of abstract complex), and then in Section 3, an \(n\)-Sperner complex is defined as an \(n\)-pseudomanifold satisfying a list of additional conditions. These definitions are too lengthy to provide in this review.Definition 3.2 provides the notion of a Sperner labeling on an \(n\)-Sperner complex \(\mathcal{K}^n\), and Theorem 3.3 shows that the number of \((n+1)\)-labeled \(n\)-simplexes of \(\mathcal{K}^n\) is odd. Theorem 3.4 presents a topological version of this, i.e., it has the parallel result in the setting of an \(n\)-Sperner polyhedron.In Sections 4 and 5, it is shown that there is no gap between the KKM principle [B. Knaster et al., Fundam. Math. 14, 132–137 (1929; JFM 55.0972.01)], covering (Lebesgue) dimension, and Sperner’s lemma (topological version), in contrast to some of the other results in the literature that the author mentions on page 2. In Section 5 a new type of dimension called the connector dimension is given which is different from covering dimension. Reviewer: Leonard R. Rubin (Norman) MSC: 54F45 Dimension theory in general topology 54H25 Fixed-point and coincidence theorems (topological aspects) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 52B11 \(n\)-dimensional polytopes 05C38 Paths and cycles Keywords:dimension; KKM principle; labeling; simplicial complex; Sperner’s lemma Citations:Zbl 1280.54001; JFM 55.0972.01 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] K.T. Atanasov, On Sperner’s lemma, Studia Sci. Math. Hungar. 32 (1996), 71-74. · Zbl 1012.52501 [2] R.B. 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