A Glimpse into Continuous Combinatorics of Posets, Polytopes, and Matroids

Abstract

We advocate a systematic study of continuous analogs of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples reviewed are an Euler formula for a class of “continuous convex polytopes” (conjectured by Kalai and Wigderson), a duality result for a class of “continuous matroids,” a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of G.-C. Rota), an exposition of the “homotopy complementation formula” for topological posets and its relation to the results of S. Kallel and R. Karoui about “weighted barycenter spaces,” and a conjecture of Vassiliev about simplicial resolutions of singularities. We also include an extension of the index inequality (Sarkaria’s inequality) based on interpreting diagrams of spaces as continuous posets.

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References

  1. 1.

    L. Anderson and E. Delucchi, “Foundations for a theory of complex matroids,” arXiv:1005. 3560v2[math.CO], Discrete Comput. Geom., 48, 807–846 (2012).

  2. 2.

    A. Bahri and J. M. Coron, “On a non-linear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain,” Commun. Pure Appl. Math., 41, 253–294 (1988).

    Article  Google Scholar 

  3. 3.

    A. Björner and J. W. Walker, “A homotopy complementation formula for partially ordered sets,” Eur. J. Combin., 4, 11–19 (1983).

    MathSciNet  Article  Google Scholar 

  4. 4.

    R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley (1994).

  5. 5.

    D. Jojić, S. Vrećica, and R. Živaljević, “Symmetric multiple chessboard complexes and a new theorem of Tverberg type,” arXiv:1502.05290v2[math.CO], J. Algebraic Combin., 46, No. 1, 15–31 (2017).

  6. 6.

    G. Kalai and A. Wigderson, “Neighborly embedded manifolds,” Discrete Comput. Geom., 40, No. 3, 319–324 (2008).

    MathSciNet  Article  Google Scholar 

  7. 7.

    S. Kallel and R. Karoui, “Symmetric joins and weighted barycenters,” arXiv:math/0602283v3[math. AT], Adv. Nonlinear Stud., 11, 117–143 (2011).

  8. 8.

    D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Lezioni Lincee, Cambridge Univ. Press (1997).

    Google Scholar 

  9. 9.

    D. Kozlov, Combinatorial Algebraic Topology, Algorithms Comput. Math., Vol. 21, Springer (2008).

  10. 10.

    J. Matoušek, Using the Borsuk–Ulam Theorem, Lect. Topol. Methods Combin. Geom., Springer, Berlin (2003).

    Google Scholar 

  11. 11.

    R. T. Rockafellar, Convex Analysis, Princeton Univ. Press (1972).

    Google Scholar 

  12. 12.

    G.-C. Rota, “Ten Mathematics Problems I will never solve,” DMV Mitteilungen, 2, 45–52 (1998).

    MathSciNet  MATH  Google Scholar 

  13. 13.

    R. Schneider, “On Steiner points of convex bodies,” Israel J. Math., 9, 241–249 (1971).

    MathSciNet  Article  Google Scholar 

  14. 14.

    R. Schneider, “Boundary structure and curvature of convex bodies,” in: J. Tölke and J. M. Wills, eds., Contributions to Geometry: Proc. of the Geometry-Symposium held in Singen June 28, 1979 to July 1, 1978, Springer (1979).

  15. 15.

    V. A. Vassiliev, “Geometric realization of the homology of classical Lie groups and complexes, S-dual to flag manifolds,” St.-Petersburg Math. J., 3, No. 4, 108–115 (1991).

    MathSciNet  Google Scholar 

  16. 16.

    V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications. Revised Edition, Transl. Math. Monographs, Vol. 98, Amer. Math. Soc., Providence (1992).

  17. 17.

    V. A. Vassiliev, “Invariants of knots and complements of discriminants,” in: V. I. Arnold and M. Monastyrsky, eds., Developments in Mathematics, the Moscow School, Chapman & Hall (1993), pp. 194–250.

  18. 18.

    V. A. Vassiliev. Topology of Complements of Discriminants, Phasis, Moscow (1997).

    Google Scholar 

  19. 19.

    V. A. Vassiliev, “Topological order complexes and resolutions of discriminant sets,” Publ. Inst. Math., 66 (80), 165–185 (1999).

    MathSciNet  MATH  Google Scholar 

  20. 20.

    V. Welker, G. M. Ziegler, and R. T. Živaljević, “Homotopy colimits—comparison lemmas for combinatorial applications,” J. Reine Angew. Math., 509, 117–149 (1999).

    MathSciNet  MATH  Google Scholar 

  21. 21.

    G. M. Ziegler, Lectures on Polytopes, Grad. Texts Math., Vol. 152, Springer (1995).

  22. 22.

    G. M. Ziegler and R. T. Živaljević, “Homotopy types of subspace arrangements via diagrams of spaces,” Math. Ann., 295, 527–548 (1993).

    MathSciNet  Article  Google Scholar 

  23. 23.

    R. T. Živaljević, “Extremal Minkowski additive selections of compact convex sets,” Proc. Am. Math. Soc., 105, 697–700 (1989).

    MathSciNet  Article  Google Scholar 

  24. 24.

    R. Živaljević, “User’s guide to equivariant methods in combinatorics, I and II,” Publ. Inst. Math. (Beograd) (N. S.), 59 (73), 114–130 (1996), 64 (78), 107–132 (1998).

  25. 25.

    R. T. Živaljević, “Combinatorics of topological posets: Homotopy complementation formulas,” Adv. Appl. Math., 21, No. 4, 547–574 (1998).

    MathSciNet  Article  Google Scholar 

  26. 26.

    R. T. Živaljević, Combinatorics of topological posets. Lect. on the Conference “Geometric Combinatorics,” Satellite Conf. of the Int. Congress of Math. in Berlin 1998; Kotor, Yugoslavia, 28.8—3.9.1998, http://poincare.matf.bg.ac.rs/konferencije/satellite/.

  27. 27.

    R. T. Živaljević, Complex and Quaternionic Relatives of Oriented Matroids (unpublished manuscript).

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Correspondence to R. T. Živaljević.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 143–164, 2016.

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Živaljević, R.T. A Glimpse into Continuous Combinatorics of Posets, Polytopes, and Matroids. J Math Sci 248, 762–775 (2020). https://doi.org/10.1007/s10958-020-04910-1

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