Topologizing Sperner's lemma
DOI:
https://doi.org/10.12775/TMNA.2024.008Keywords
dimension, labeling, KKM principle, simplicial complex, Sperner's lemmaAbstract
The aim of this paper is to extend Sperner's lemma for a new class of complexes, called $n$-Sperner. Next, we consider a topological version of Sperner's lemma, which leads to characterization of the covering dimension and KKM-principle. Finally, for an arbitrary topological space a new dimension is defined.References
K.T. Atanasov, On Sperner’s lemma, Studia Sci. Math. Hungar. 32 (1996), 71–74.
R.B. Bapat, A constructive proof of a permutation-based generalization of Sperner’s lemma, Math. Program. 44 (1989), no. 1, 113–120.
R.B. Bapat, Sperner’s lemma with multiple labels, Modeling, Computation and Optimization (S.K. Neogy, A.K. Das, R.B. Bapat, eds.), World Scientific, 2009, pp. 257–261.
E.D. Bloch, Mod 2 degree and a generalized no retraction theorem, Math. Nachr. 279 (2006), 490–494.
J.A. De Loera, E. Peterson and F.E. Su, A polytopal generalization of Sperner’s lemma, J. Combin. Theory Ser. A 100 (2002), no. 1–26.
R. Engelking, Dimension Theory, North-Holland Pub. Co., Amsterdam, 1978.
V.V. Fedorchuk and J. Van Mill, Dimensionsgrad for locally connected Polish spaces, Fund. Math. 163 (2000), 77–82.
D. Gale, Equilibrium in a discrete exchange economy with money, Internat. J. Game Theory 13 (1984), 61–64.
R. Hochberg, C. Mcdiarmid and M. Saks, On the bandwidth of triangulated triangles, Discrete Math. 138 (1995), 261–265.
A. Idzik, W. Kulpa and P. Maćkowiak, Equivalent forms of the Brouwer fixed point theorem II, Topol. Methods Nonlinear Anal. 57, no. 1 (2021), 57–71.
B. Knaster, K. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929), 132–137.
F. Meunier, Sperner labellings: a combinatorial approach, J. Combin. Theory Ser. A 113 (2006), 1462–1475.
O.R. Musin, Around Sperner’s lemma, preprint, arXiv:1405.7513.
O.R. Musin, Homotopy invariants of covers and KKM-type lemmas, Algebr. Geom. Topol. 16 (2016), no. 3, 1799–1812.
L. R. Rubin, R.M. Schori and J.J. Walsh, New dimension-theory techniques for constructing infinite-dimensional examples, General Topology Appl. 10 (1979), no. 1, 93–102.
E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6 (1928), 265–272.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
