Topological Methods in Nonlinear Analysis

Combinatorial lemmas for nonoriented pseudomanifolds

Adam Idzik and Konstanty Junosza-Szaniawski

Full-text: Open access

Abstract

Sperner lemma type theorems are proved for nonoriented primoids and pseudomanifolds. A rank function of a primoid is defined. Applications of these theorems to the geometric simplex are given. Also Knaster-Kuratowski-Mazurkiewicz type theorems on covering of the geometric simplex are presented.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 22, Number 2 (2003), 387-398.

Dates
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475266344

Mathematical Reviews number (MathSciNet)
MR2036384

Zentralblatt MATH identifier
1038.05010

Citation

Idzik, Adam; Junosza-Szaniawski, Konstanty. Combinatorial lemmas for nonoriented pseudomanifolds. Topol. Methods Nonlinear Anal. 22 (2003), no. 2, 387--398. https://projecteuclid.org/euclid.tmna/1475266344


Export citation

References

  • P. Alexandroff and B. Pasynkoff, Elementary proof of the essentiality of the identity mapping of a simplex , Uspiehi Mat. Nauk, 12 (1957), 175–179, (Russian) \ref\key 2
  • R. B. Bapat, A generalization of a theorem of Ky Fan on simplicial maps , J. Combin. Theory Ser. A, 29(1978), 32–38 \ref\key 3 ––––, A constructive proof of permutation-based generalization of Sperner's lemma , Math. Programming, 44(1988), 113–120 \ref\key 4
  • L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten , Math. Ann., 71(1912), 97–115 \ref\key 5
  • I. A. Cohen, On the Sperner Lemma , J. Combin. Theory, 2(1967), 585–587 \ref\key 6
  • K. Fan, A generalization of Tucker's combinatorial lemma with topological applications , Ann. of Math., 52(1952), 431–437 \ref\key 7 ––––, Simplicial maps from orientable $n$-pseudomanifold into $S^n$ with octahedral triangulation , J. Combin. Theory, 2(1967), 588–602 \ref\key 8 ––––, A combinatorial property of pseudomanifolds and covering properties of simplexes , J. Math. Anal. Appl., 31(1970), 68–80 \ref\key 9
  • F. J. Gould and J. W. Tolle, A unified approach to complementarity in optimalization , Discrete Math., 7(1974), 225–271 \ref\key 10
  • T. Ichiishi, Alternative version of Shapley's theorem on closed coverings of a simplex , Proc. Amer. Math. Soc., 104 (1988), 759–763 \ref\key 11
  • T. Ichiishi and A. Idzik, Theorems on closed coverings of simplex and their applications to cooperative game theory , J. Math. Anal. Appl., 146(1990), 259–270 \ref\key 12
  • A. Idzik, KKM Theorem and Matroids , ICS PAS Reports, 693(1990) \ref\key 13
  • B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $n$-dimentiosionale Simplexe, , Fund. Math., 14(1964), 132–137 \ref\key 14
  • W. Kry\plsmc äski, Remarks on matroids and Sperner's lemma, , European J. Combin., 11(1990), 485–488 \ref\key 15
  • G. van der Laan, D. Talman and Z. Yang, Existence of balanced simplices on polytopes , J. Combin. Theory Ser. A, 96(2001), 288–302 \ref\key 16
  • L. Lovász, Matroids and Sperner's lemma , European J. Combin., 2(1981), 65–66 \ref\key 17
  • H. Scarf (with the collaboration of T. Hansen), Computation of Economics Equilibria , Yale University Press, New Haven (1973) \ref\key 18
  • L. S. Shapley, On balanced games without side payments (T. C. Hu and S. M. Robinson, Mathematical Programming, eds.), New York, Academic Press (1973), 261–290 \ref\key 19
  • E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebiets , Abh. Math. Sem. Univ. Hamburg, 6 (1928), 265–272 \ref\key 20
  • S. Park, The Knaster–Kuratowski–Mazurkiewicz theorem and almost fixed points , Topol. Methods Nonlinear Anal., 16(2000), 195–200 \ref\key 21
  • M. J. Todd, A generalized complementary pivoting algorithm , Math. Programming, 6(1974), 243–263