Selecta Mathematica

, Volume 22, Issue 2, pp 561–594

Combinatorial covers and vanishing of cohomology

  • Graham Denham
  • Alexander I. Suciu
  • Sergey Yuzvinsky
Article
  • 124 Downloads

Abstract

We use a Mayer–Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen–Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients and obtain new generalizations.

Keywords

Combinatorial cover Cohomology with local coefficients Spectral sequence Hyperplane arrangement Elliptic arrangement Toric complex Cohen–Macaulay property 

Mathematics Subject Classification

Primary 55T99 Secondary 14F05 16E65 20J05 32S22 55N25 

References

  1. 1.
    Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: The polyhedral product functor: a method of computation for moment–angle complexes, arrangements and related spaces. Adv. Math. 225(3), 1634–1668 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bezrukavnikov, R.: Koszul DG-algebras arising from configuration spaces. Geom. Funct. Anal. 4(2), 119–135 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bibby, C.: Cohomology of abelian arrangements. Proc. Am. Math. Soc. arXiv:1310.4866v3
  4. 4.
    Borel, A., et al.: Intersection Cohomology, Progress in Mathematics, vol. 50. Birkhäuser, Boston (1984)CrossRefGoogle Scholar
  5. 5.
    Bott, R., Loring, T.: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)CrossRefMATHGoogle Scholar
  6. 6.
    Brady, N., Meier, J.: Connectivity at infinity for right angled Artin groups. Trans. Am. Math. Soc. 353(1), 117–132 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Crapo, H.: A higher invariant for matroids. J. Comb. Theory 2, 406–417 (1967)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Davis, M.W.: Right-angularity, flag complexes, asphericity. Geom. Dedicata 159(1), 239–262 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Davis, M.W., Leary, I., Januszkiewicz, T., Okun, B.: Cohomology of hyperplane complements with group ring coefficients. Int. Math. Res. Not. 9, 2110–2116 (2011)MathSciNetMATHGoogle Scholar
  10. 10.
    Davis, M.W., Okun, B.: Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products. Groups Geom. Dyn. 6(3), 485–531 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Davis, M.W., Settepanella, S.: Vanishing results for the cohomology of complex toric hyperplane complements. Publ. Mat. 57(2), 379–392 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    De Concini, C., Procesi, C.: Wonderful models of subspace arrangements. Sel. Math. (N.S.) 1(3), 459–494 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Deheuvels, R.: Homologie des ensembles ordonnés et des espaces topologiques. Bull. Soc. Math. Fr. 90, 261–321 (1962)MathSciNetMATHGoogle Scholar
  14. 14.
    Denham, G.: Toric and tropical compactifications of hyperplane complements. Ann. Fac. Sci. Toulouse Math. 23(2), 297–333 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Denham, G., Suciu, A.I.: Moment–angle complexes, monomial ideals, and Massey products. Pure Appl. Math. Q. 3(1), 25–60 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Denham, G., Suciu, A.I.: Multinets, parallel connections, and Milnor fibrations of arrangements. Proc. Lond. Math. Soc. 108(6), 1435–1470 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Denham, G., Suciu, A.I., Yuzvinsky, S.: Abelian duality and propagation of resonance, preprint (2015)Google Scholar
  18. 18.
    Dimca, A.: Sheaves in Topology, Universitext. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  19. 19.
    Dimca, A., Papadima, S., Suciu, A.I.: Topology and geometry of cohomology jump loci. Duke Math. J. 148(3), 405–457 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Eisenbud, D., Popescu, S., Yuzvinsky, S.: Hyperplane arrangement cohomology and monomials in the exterior algebra. Trans. Am. Math. Soc. 355(11), 4365–4383 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Esnault, H., Schechtman, V., Viehweg, E.: Cohomology of local systems on the complement of hyperplanes. Invent. Math. 109, 557–561 (1992). Erratum, ibid. 112(2), 447 (1993)Google Scholar
  22. 22.
    Feichtner, E.M., Yuzvinsky, S.: Chow rings of toric varieties defined by atomic lattices. Invent. Math. 155(3), 515–536 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Folkman, J.: The homology groups of a lattice. J. Math. Mech. 15, 631–636 (1966)MathSciNetMATHGoogle Scholar
  24. 24.
    Fritzsche, K., Grauert, H.: From Holomorphic Functions to Complex Manifolds, Graduate Texts in Mathematics, vol. 213. Springer, New York (2002)CrossRefMATHGoogle Scholar
  25. 25.
    Godement, R.: Topologie algébrique et théorie des faisceaux, Actualités Sci. Ind. no. 1252, Publ. Math. Univ. Strasbourg, no. 13, Hermann, Paris (1958)Google Scholar
  26. 26.
    Harer, J.: The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84(1), 157–176 (1986)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Iversen, B.: Cohomology of Sheaves, Universitext. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  28. 28.
    Jensen, C., Meier, J.: The cohomology of right-angled Artin groups with group ring coefficients. Bull. Lond. Math. Soc. 37(5), 711–718 (2005)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292. Springer, Berlin (1990)CrossRefGoogle Scholar
  30. 30.
    Kohno, T.: Homology of a local system on the complement of hyperplanes. Proc. Jpn. Acad. Ser. A Math. Sci. 62(4), 144–147 (1986)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Levin, A., Varchenko, A.: Cohomology of the complement to an elliptic arrangement. In: Configuration Spaces: Geometry, Combinatorics and Topology, vol. 14, pp. 373–388. CRM Series, Ed. Norm., Pisa (2012)Google Scholar
  32. 32.
    Papadima, S., Suciu, A.I.: Toric complexes and Artin kernels. Adv. Math. 220(2), 441–477 (2009)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Quillen, D.: Homotopy properties of the poset of nontrivial \(p\)-subgroups of a group. Adv. Math. 28(2), 101–128 (1978)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Schechtman, V., Terao, H., Varchenko, A.: Local systems over complements of hyperplanes and the Kac–Kazhdan condition for singular vectors. J. Pure Appl. Algebra 100(1–3), 93–102 (1995)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Schürmann, J.: Topology of Singular Spaces and Constructible Sheaves, Mathematics Institute of the Polish Academy of Sciences, Mathematical Monographs, vol. 63. Birkhäuser, Basel (2003)CrossRefGoogle Scholar
  36. 36.
    Smale, S.: A Vietoris mapping theorem for homotopy. Proc. Am. Math. Soc. 8, 604–610 (1957)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Graham Denham
    • 1
  • Alexander I. Suciu
    • 2
  • Sergey Yuzvinsky
    • 3
  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Department of MathematicsNortheastern UniversityBostonUSA
  3. 3.Department of MathematicsUniversity of OregonEugeneUSA

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