Advertisement

Discrete & Computational Geometry

, Volume 10, Issue 3, pp 313–348 | Cite as

What is a complex matroid?

  • Günter M. Ziegler
Article

Abstract

Following an “ansatz” of Björner and Ziegler [BZ], we give an axiomatic development of finite sign vector systems that we callcomplex matroids. This includes, as special cases, the sign vector systems that encode complex arrangements according to [BZ], and the complexified oriented matroids, whose complements were considered by Gel'fand and Rybnikov [GeR].

Our framework makes it possible to study complex hyperplane arrangements as entirely combinatorial objects. By comparing complex matroids with 2-matroids, which model the more general 2-arrangements introduced by Goresky and MacPherson [GoM], the essential combinatorial meaning of a “complex structure” can be isolated.

Our development features a topological representation theorem for 2-matroids and complex matroids, and the computation of the cohomology of the complement of a 2-arrangement, including its multiplicative structure in the complex case. Duality is established in the cases of complexified oriented matroids, and for realizable complex matroids. Complexified oriented matroids are shown to be matroids with coefficients in the sense of Dress and Wenzel [D1], [DW1], but this fails in general.

Keywords

Sign Vector Cohomology Algebra Complex Arrangement Oriented Matroids Geometric Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [B] E. K. Babson: A Combinatorial Flag Space, Ph.D. Thesis, MIT, 1993.Google Scholar
  2. [BKS] L. J. Billera, M. M. Kapranov, and B. Sturmfels: Cellular strings on polytopes,Proc. Amer. Math. Soc., to appear.Google Scholar
  3. [BLS+] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler:Oriented Matroids, Encyclopedia of Mathematics and Its Applications, Vol. 46, Cambridge University Press, Cambridge, 1993.zbMATHGoogle Scholar
  4. [BZ] A. Björner and G. M. Ziegler: Combinatorial stratification of complex arrangements,J. Amer. Math. Soc. 5 (1992), 105–149.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [D1] A. W. M. Dress: Duality theory for finite and infinite matroids with coefficients,Adv. in Math. 59 (1986), 97–123.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [D2] A. W. M. Dress: Personal communication, Djursholm, 1992.Google Scholar
  7. [DW1] A. W. M. Dress and W. Wenzel: Endliche Matroide mit Koeffizienten,Bayreuth. Math. Schr. 26 (1988), 37–98.zbMATHMathSciNetGoogle Scholar
  8. [DW2] A. W. M. Dress and W. Wenzel: Grassmann-Plücker relations and matroids with coefficients,Adv. in Math. 86 (1991), 68–110.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [DW3] A. W. M. Dress and W. Wenzel: Perfect matroids,Adv. in Math. 91 (1992), 158–208.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [EM] J. Edmonds and A. Mandel: Topology of Oriented Matroids, Ph.D. Thesis of A. Mandel, University of Waterloo, 1982.Google Scholar
  11. [FL] J. Folkman and J. Lawrence: Oriented matroids,J. Combin. Theory Ser. B 25 (1978), 199–236.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [GeM] I. M. Gel'fand and R. D. MacPherson: A combinatorial formula for the Pontrjagin classes,Bull. Amer. Math. Soc. 26 (1992), 304–309.MathSciNetCrossRefGoogle Scholar
  13. [GeR] I. M. Gel'fand and G. L. Rybnikov: Algebraic and topological invariants of oriented matroids,Soviet Math. Dokl. 40 (1990), 148–152.zbMATHMathSciNetGoogle Scholar
  14. [GoM] M. Goresky and R. MacPherson:Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, vol. 14, Springer-Verlag, Berlin, 1988.zbMATHCrossRefGoogle Scholar
  15. [K] J. Karlander: A characterization of affine sign vector systems, Preprint, KTH, Stockholm, 1992.Google Scholar
  16. [LV] M. Las Vergnas: Oriented matroids as signed geometries real in corank 2, in:Finite and Infinite Sets (Proc. 6th Hungarian Combinatorial Conf., Eger, 1981), North-Holland, Amsterdam, 1984, pp. 555–565.Google Scholar
  17. [MP] R. D. MacPherson: Combinatorial differential manifolds, Preprint, 1992.Google Scholar
  18. [MZ] N. E. Mnëv and G. M. Ziegler: Combinatorial models for the finite-dimensional Grassmannians,Discrete Comput. Geom., this issue, pp. 241–250.Google Scholar
  19. [OS] P. Orlik and L. Solomon: Combinatorics and topology of complements of hyperplanes,Invent. Math. 56 (1980), 167–189.zbMATHMathSciNetCrossRefGoogle Scholar
  20. [R] G. L. Rybnikov: Personal communication, Luminy, 1991.Google Scholar
  21. [S] M. Salvetti: Topology of the complement of real hyperplanes in ℂN,Invent. Math. 88 (1987), 603–618.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [SW] J. Stoer and Ch. Witzgall:Convexity and Optimization in Finite Dimensions I, Grundlehren der Mathematischen Wissenschaften, vol. 163, Springer-Verlag, Berlin, 1970.CrossRefGoogle Scholar
  23. [SZ] B. Sturmfels and G. M. Ziegler: Extension spaces of oriented matroids,Discrete Comput. Geom. 10 (1993), 23–45.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [Z] G. M. Ziegler: On the difference between real and complex arrangements,Math. Z. 212 (1993), 1–11.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [ZŽ] G. M. Ziegler and R. T. Živaljević: Homotopy types of arrangements via diagrams of spaces,Math. Ann. 295 (1993), 527–548.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Günter M. Ziegler
    • 1
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)Berlin 31Federal Republic of Germany

Personalised recommendations