Annals of Combinatorics 4 (2000) 171-182
Lagrangian Matroids and Cohomology
Richard F. Booth1, Alexandre V. Borovik1, Israel M. Gelfand2, and David A. Stone3
1Department of Mathematics, UMIST, PO Box 88, Manchester
M60 1QD, UK
2Department of Mathematics,Rutgers University, New
Brunswick,NJ 08903, USA
3Department of Mathematics, Brooklyn College, 2900 Bedford Avenue, Brooklyn, NY 11210-2889, USA
Received March 15, 1999
AMS Subject Classification: 05B35
Abstract. We prove that -matroids associated with maps on compact closed surfaces are representable, with the space of representation provided by cohomology of the surface with punctured points.
Keywords: -matroid, Lagrangian symplectic matroid, cohomology, -complex
1. T.V. Alekseyevskaya, A.V. Borovik, I.M. Gelfand, and N. White, Matroid homology, In: The Gelfand Mathematical Seminars 1997–1999, I.M. Gelfand and V.S. Retakh, Eds., Birkhäuser-Verlag, 1999, pp. 1–13.
2. E. Artin, Geometric Algebra, Interscience Publishers, 1957.
3. R.F. Booth, Oriented Lagrangian orthogonal matroids, Manchester Centre for Pure Mathematics, 1999/14, preprint.
4. R.F. Booth, A.V. Borovik, and I.M. Gelfand, Lagrangian matroids associated with maps on orientable surfaces, Manchester Centre for Pure Mathematics, 1999/3, preprint.
5. A.V. Borovik and I.M. Gelfand,WP-matroids and thin Schubert cells on Tits systems, Adv. Math. 103 (1994) 162–179.
6. A.V. Borovik, I.M. Gelfand, and D.A. Stone, On the topology of the combinatorial flag varieties, Manchester Centre for Pure Mathematics, 1999/12, preprint.
7. A.V. Borovik, I.M. Gelfand, and N. White, Symplectic matroids, J. Alg. Combin. 8 (1998) 235–252.
8. A.V. Borovik, I.M. Gelfand, and N. White, Combinatorial geometry in characteristic 1, Manchester Centre for Pure Mathematics, 1999/10, preprint .
9. A.V. Borovik, I.M. Gelfand, and N. White, Coxeter Matroids, Birkhäuser, Boston, in preparation.
10. A. Bouchet, Greedy algorithm and symmetric matroids, Mathematical Programming 38 (1987) 147–159.
11. A. Bouchet, Maps and D-matroids, Discrete Math. 78 (1989) 59–71.
12. A. Bouchet, A. Dress, and T. Havel, D-matroids and metroids, Adv. Math. 91 (1992) 136– 142.
13. K.S. Brown, Buildings, Springer-Verlag, 1988.
14. A. Dold, Lectures on Algebraic Topology, 2nd Edition, Springer-Verlag, Berlin, 1980.
15. A.W. Dress and T.F. Havel, Some combinatorial properties of discriminants in metric vector spaces, Adv. Math. 62 (1986) 285–312.
16. P.J. Hilton and S. Wylie, Homology Theory, Cambridge University Press, 1960.
17. M. Kontsevich, Formal (non)-commutative symplectic geometry, In: The Gelfand Mathematical Seminars 1990–1992, Birkh¨auser, 1993, pp. 173–187.
18. M. Kontsevich, Feynman diagrams and low-dimensional topology, In: First European Congress of Mathematics, Paris, July 6–10, 1992, Birkh¨auser, Vol. 2, 1994, pp. 97–122.
19. R. Rado, Note on independence functions, Proc. London Math. Soc. 7 (1957) 300–320.
20. A. Vince and N. White, Orthogonal Matroids, in preparation.
21. D.J.A. Welsh, Matroid Theory, Academic Press, London, 1976.
22. W.Wenzel, Geometric algebra of D-matroids and related combinatorial geometries, Habilitationsschrift, Bielefield, 1991.
23. W. Wenzel, Pfaffian forms and D-matroids, Discrete Math. 115 (1993) 253–266.