Topological representations of matroids
HTML articles powered by AMS MathViewer
- by E. Swartz
- J. Amer. Math. Soc. 16 (2003), 427-442
- DOI: https://doi.org/10.1090/S0894-0347-02-00413-7
- Published electronically: November 29, 2002
- PDF | Request permission
Abstract:
There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky’s enumeration of the cells of the arrangement still holds. Bounded subcomplexes of an arrangement of homotopy spheres correspond to minimal cellular resolutions of the dual matroid Steiner ideal. As a result, the Betti numbers of the ideal are computed and seen to be equivalent to Stanley’s formula in the special case of face ideals of independence complexes of matroids.References
- Dave Bayer and Bernd Sturmfels, Cellular resolutions of monomial modules, J. Reine Angew. Math. 502 (1998), 123–140. MR 1647559, DOI 10.1515/crll.1998.083
- Margaret Bayer and Bernd Sturmfels, Lawrence polytopes, Canad. J. Math. 42 (1990), no. 1, 62–79. MR 1043511, DOI 10.4153/CJM-1990-004-4
- Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, Oriented matroids, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1999. MR 1744046, DOI 10.1017/CBO9780511586507
- S. Losinsky, Sur le procédé d’interpolation de Fejér, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318–321 (French). MR 0002001
- Charles J. Colbourn and William R. Pulleyblank, Matroid Steiner problems, the Tutte polynomial and network reliability, J. Combin. Theory Ser. B 47 (1989), no. 1, 20–31. MR 1007711, DOI 10.1016/0095-8956(89)90062-2
- Henry H. Crapo, A higher invariant for matroids, J. Combinatorial Theory 2 (1967), 406–417. MR 215744, DOI 10.1016/S0021-9800(67)80051-6
- Jon Folkman, The homology groups of a lattice, J. Math. Mech. 15 (1966), 631–636. MR 0188116
- Jon Folkman and Jim Lawrence, Oriented matroids, J. Combin. Theory Ser. B 25 (1978), no. 2, 199–236. MR 511992, DOI 10.1016/0095-8956(78)90039-4
- Jennifer McNulty, Generalized affine matroids, Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994), 1994, pp. 243–254. MR 1382350 NPS I. Novik, A. Postnikov, and B. Sturmfels. Syzygies of oriented matroids. Duke Math. J., 111(2):287–317, 2002.
- James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112
- Richard P. Stanley, Cohen-Macaulay complexes, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 31, Reidel, Dordrecht-Boston, Mass., 1977, pp. 51–62. MR 0572989
- Michelle L. Wachs and James W. Walker, On geometric semilattices, Order 2 (1986), no. 4, 367–385. MR 838021, DOI 10.1007/BF00367425 W H. Whitney. On the abstract properties of linear dependence. American Journal of Mathematics, 57:509–533, 1935.
- Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. issue 1, 154, vii+102. MR 357135, DOI 10.1090/memo/0154
- Thomas Zaslavsky, A combinatorial analysis of topological dissections, Advances in Math. 25 (1977), no. 3, 267–285. MR 446994, DOI 10.1016/0001-8708(77)90076-7 Za T. Zaslavsky. The Möbius function and the characteristic polynomial. In N.L. White, editor, Combinatorial geometries. Cambridge University Press, 1987.
- Günter M. Ziegler and Rade T. Živaljević, Homotopy types of subspace arrangements via diagrams of spaces, Math. Ann. 295 (1993), no. 3, 527–548. MR 1204836, DOI 10.1007/BF01444901
Bibliographic Information
- E. Swartz
- Affiliation: Malott Hall, Cornell University, Ithaca, New York 14853
- Email: ebs@math.cornell.edu
- Received by editor(s): August 29, 2002
- Received by editor(s) in revised form: November 4, 2002
- Published electronically: November 29, 2002
- Additional Notes: Partially supported by a VIGRE postdoc under NSF grant number 9983660 to Cornell University
- © Copyright 2002 American Mathematical Society
- Journal: J. Amer. Math. Soc. 16 (2003), 427-442
- MSC (2000): Primary 05B35; Secondary 52C40, 13D02, 13F55
- DOI: https://doi.org/10.1090/S0894-0347-02-00413-7
- MathSciNet review: 1949166