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Arrangements and cohomology

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Abstract

To a matroidM is associated a graded commutative algebraA=A(M), the Orlik-Solomon algebra ofM. Motivated by its role in the construction of generalized hypergeometric functions, we study the cohomologyH * (A, d ω) ofA(M) with coboundary mapd ω given by multiplication by a fixed element ω ofA 1. Using a description of decomposable relations inA, we construct new examples of “resonant” values of ω, and give a precise calculation ofH 1 (A, d ω) as a function of ω. We describe the setR 1(A={ω|H 1(A(M,d ω)≠}, and use it as a tool in the classification of Orlik-Solomon algebras, with applications to the topology of complex hyperplane complements. We show thatR 1(A) is a complete invariant of the quadratic closure ofA, and show under various hypotheses that one can reconstruct the matroidM, or at least its Tutte polynomial, from the varietyR 1(A). We demonstrate with several examples thatR 1 is easily calculable, may contain nonlocal components, and that combinatorial properties ofR 1(A) are often sufficient to distinguish nonisomorphic rank three Orlik-Solomon algebras.

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References

  1. A. Björner, Topological methods, In: Handbook of Combinatorics, Elsevier, Amsterdam, 1995, pp. 1819–1872.

    Google Scholar 

  2. J.E. Blackburn, H.H. Crapo, and D.A. Higgs, A catalogue of combinatorial geometries, Math. Computation27, (1973) 155–166.

    Article  MathSciNet  Google Scholar 

  3. T. Brylawski, Hyperplane reconstruction of the Tutte polynomial of a geometric lattice, Discrete Math.35 (1981) 25–28.

    Article  MathSciNet  Google Scholar 

  4. D. Cohen and A. Suciu, The Chen groups of the pure braid group, In: Proceedings of the Čech Centennial Homotopy Theory Conference, B. Cenkl and H. Miller, Eds. Contemporary Mathematics, Vol. 181, Providence, Amer. Math. Soc., 1995, pp. 45–64.

  5. C. Eschenbrenner and M. Falk, Tutte polynomials and Orlik-Solomon algebras, preprint.

  6. M. Falk, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc.309 (1988) 543–556.

    Article  MathSciNet  Google Scholar 

  7. M. Falk, The cohomology and fundamental group of a hyperplane complement, In: Singularities, R. Randell, Ed., Contemporary Mathematics, Vol. 90, Providence, Amer. Math. Soc., 1989, pp. 55–72.

    Google Scholar 

  8. M. Falk, On the algebra associated with a geometric lattice, Adv. Math.80 (1989) 152–163.

    Article  MathSciNet  Google Scholar 

  9. M. Falk, Homotopy types of line arrangements, Invent. Math.111 (1993) 139–150.

    Article  MathSciNet  Google Scholar 

  10. M. Falk and R. Randell, On the homotopy theory of arrangements, In: Complex Analytic Singularities, Advanced Studies in Mathematics, Vol. 8, North Holland, 1987, pp. 101–124.

  11. I.M. Gelfand and A.V. Zelevinsky, Algebraic and combinatorial aspects of the general theory of hypergeometric functions, Functional Analysis and Appl.20 (1986) 183–197.

    Article  MathSciNet  Google Scholar 

  12. E. Viehweg, H. Esnault, and V. Schechtman, Cohomology of local systems on the complement of hyperplanes, Invent. Math.109 (1992) 557–561. Erratum: ibid112 (1993) 447.

    Article  MathSciNet  Google Scholar 

  13. P. Orlik and L. Solomon, Topology and combinatorics of complements of hyperplanes, Invent. Math.56 (1980) 167–189.

    Article  MathSciNet  Google Scholar 

  14. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin Heidelberg New York, 1992.

    MATH  Google Scholar 

  15. J.G. Oxley, Matroid Theory, Oxford University Press, Oxford New York Tokyo, 1992.

    MATH  Google Scholar 

  16. G. Rybnikov, Private correspondence.

  17. G. Rybnikov, On the fundamental group of a complex hyperplane arrangement, DIMACS Technical Reports13 (1994).

  18. V.V. Schechtman, H. Terao, and A.N. Varchenko, Cohomology of local systems and Kac-Kazhdan condition for singular vectors. J. Pure Appl. Algebra100 (1995) 93–102.

    Article  MathSciNet  Google Scholar 

  19. V.V. Schechtman and A.N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math.106 (1991) 139–194.

    Article  MathSciNet  Google Scholar 

  20. B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc., to appear.

  21. D.J.A. Welsh, Matroid Theory, Academic Press, London, 1976.

    MATH  Google Scholar 

  22. N. White, Ed. Theory of Matroids, Cambridge University Press, Cambridge, 1986.

    MATH  Google Scholar 

  23. N. White, Ed., Matroid Applications, Cambridge University Press, Cambridge, 1992.

    MATH  Google Scholar 

  24. S. Yuzvinsky, Cohomology of Brieskom-Orlik-Solomon algebras, Comm. Algebra23 (1995) 5339–5354.

    MathSciNet  Google Scholar 

  25. G.M. Ziegler, Private communication, 1988.

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Authors and Affiliations

  1. Department of Mathematics, Northern Arizona University, 86011, Flagstaff, AZ, USA

    Michael Falk

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  1. Michael Falk
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Falk, M. Arrangements and cohomology. Annals of Combinatorics 1, 135–157 (1997). https://doi.org/10.1007/BF02558471

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  • DOI: https://doi.org/10.1007/BF02558471

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