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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Anderson and E. Delucchi, “Foundations for a theory of complex matroids,” arXiv:1005. 3560v2[math.CO], Discrete Comput. Geom., 48, 807–846 (2012).
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).
A. Björner and J. W. Walker, “A homotopy complementation formula for partially ordered sets,” Eur. J. Combin., 4, 11–19 (1983).
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley (1994).
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).
G. Kalai and A. Wigderson, “Neighborly embedded manifolds,” Discrete Comput. Geom., 40, No. 3, 319–324 (2008).
S. Kallel and R. Karoui, “Symmetric joins and weighted barycenters,” arXiv:math/0602283v3[math. AT], Adv. Nonlinear Stud., 11, 117–143 (2011).
D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Lezioni Lincee, Cambridge Univ. Press (1997).
D. Kozlov, Combinatorial Algebraic Topology, Algorithms Comput. Math., Vol. 21, Springer (2008).
J. Matoušek, Using the Borsuk–Ulam Theorem, Lect. Topol. Methods Combin. Geom., Springer, Berlin (2003).
R. T. Rockafellar, Convex Analysis, Princeton Univ. Press (1972).
G.-C. Rota, “Ten Mathematics Problems I will never solve,” DMV Mitteilungen, 2, 45–52 (1998).
R. Schneider, “On Steiner points of convex bodies,” Israel J. Math., 9, 241–249 (1971).
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).
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).
V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications. Revised Edition, Transl. Math. Monographs, Vol. 98, Amer. Math. Soc., Providence (1992).
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.
V. A. Vassiliev. Topology of Complements of Discriminants, Phasis, Moscow (1997).
V. A. Vassiliev, “Topological order complexes and resolutions of discriminant sets,” Publ. Inst. Math., 66 (80), 165–185 (1999).
V. Welker, G. M. Ziegler, and R. T. Živaljević, “Homotopy colimits—comparison lemmas for combinatorial applications,” J. Reine Angew. Math., 509, 117–149 (1999).
G. M. Ziegler, Lectures on Polytopes, Grad. Texts Math., Vol. 152, Springer (1995).
G. M. Ziegler and R. T. Živaljević, “Homotopy types of subspace arrangements via diagrams of spaces,” Math. Ann., 295, 527–548 (1993).
R. T. Živaljević, “Extremal Minkowski additive selections of compact convex sets,” Proc. Am. Math. Soc., 105, 697–700 (1989).
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).
R. T. Živaljević, “Combinatorics of topological posets: Homotopy complementation formulas,” Adv. Appl. Math., 21, No. 4, 547–574 (1998).
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/.
R. T. Živaljević, Complex and Quaternionic Relatives of Oriented Matroids (unpublished manuscript).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 143–164, 2016.
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04910-1