Topological representations of matroids

Swartz, E.

doi:10.1090/S0894-0347-02-00413-7

Topological representations of matroids
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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

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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

Duke Math. J.

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

American Journal of Mathematics

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

Combinatorial geometries

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