Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

An equivariant discrete model for complexified arrangement complements


Authors: Emanuele Delucchi and Michael J. Falk
Journal: Proc. Amer. Math. Soc. 145 (2017), 955-970
MSC (2010): Primary 05B35, 52C35, 52C40, 55U10, 05E45
DOI: https://doi.org/10.1090/proc/13328
Published electronically: November 29, 2016
MathSciNet review: 3589296
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define a partial ordering on the set $ \mathcal {Q}=\mathcal {Q}(\mathsf {M})$ of pairs of topes of an oriented matroid $ \mathsf {M}$, and show the geometric realization $ \vert\mathcal {Q}\vert$ of the order complex of $ \mathcal {Q}$ has the same homotopy type as the Salvetti complex of $ \mathsf {M}$. For any element $ e$ of the ground set, the complex $ \vert\mathcal {Q}_e\vert$ associated to the rank-one oriented matroid on $ \{e\}$ has the homotopy type of the circle. There is a natural free simplicial action of $ \mathbb{Z}_4$ on $ \vert\mathcal {Q}\vert$, with orbit space isomorphic to the order complex of the poset $ \mathcal {Q}(\mathsf {M},e)$ associated to the pointed (or affine) oriented matroid $ (\mathsf {M},e)$. If $ \mathsf {M}$ is the oriented matroid of an arrangement $ \mathscr {A}$ of linear hyperplanes in $ \mathbb{R}^n$, the $ \mathbb{Z}_4$ action corresponds to the diagonal action of $ \mathbb{C}^*$ on the complement $ M$ of the complexification of $ \mathscr {A}$: $ \vert\mathcal {Q}\vert$ is equivariantly homotopy-equivalent to $ M$ under the identification of $ \mathbb{Z}_4$ with the multiplicative subgroup $ \{\pm 1, \pm i\}\subset \mathbb{C}^*$, and $ \vert\mathcal {Q}(\mathsf {M},e)\vert$ is homotopy-equivalent to the complement of the decone of $ \mathscr {A}$ relative to the hyperplane corresponding to $ e$. All constructions and arguments are carried out at the level of the underlying posets.

We also show that the class of fundamental groups of such complexes is strictly larger than the class of fundamental groups of complements of complex hyperplane arrangements. Specifically, the group of the non-Pappus arrangement is not isomorphic to any realizable arrangement group. The argument uses new structural results concerning the degree-one resonance varieties of small matroids.


References [Enhancements On Off] (What's this?)

  • [1] Dragan Acketa, The catalogue of all nonisomorphic matroids on at most 8 elements, Special Issue, vol. 1, University of Novi Sad, Institute of Mathematics, Faculty of Science, Novi Sad, 1983. MR 740223
  • [2] Laura Anderson and Emanuele Delucchi, Foundations for a theory of complex matroids, Discrete Comput. Geom. 48 (2012), no. 4, 807-846. MR 3000567, https://doi.org/10.1007/s00454-012-9458-9
  • [3] Bruno Benedetti, Discrete Morse theory for manifolds with boundary, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6631-6670. MR 2958950, https://doi.org/10.1090/S0002-9947-2012-05614-5
  • [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
  • [5] Anders Björner and Günter M. Ziegler, Combinatorial stratification of complex arrangements, J. Amer. Math. Soc. 5 (1992), no. 1, 105-149. MR 1119198, https://doi.org/10.2307/2152753
  • [6] John E. Blackburn, Henry H. Crapo, and Denis A. Higgs, A catalogue of combinatorial geometries, Math. Comp. 27 (1973), 155-166; addendum, ibid. 27 (1973), no. 121, loose microfiche suppl. A12-G12. MR 0419270
  • [7] Daniel C. Cohen and Alexander I. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), no. 1, 33-53. MR 1692519, https://doi.org/10.1017/S0305004199003576
  • [8] Raul Cordovil and António Guedes de Oliveira, A note on the fundamental group of the Salvetti complex determined by an oriented matroid, European J. Combin. 13 (1992), no. 6, 429-437. MR 1193552, https://doi.org/10.1016/0195-6698(92)90002-H
  • [9] Priyavrat Deshpande, On arrangements of pseudohyperplanes, Proc. Indian Acad. Sci. Math. Sci. 126 (2016), no. 3, 399-420. MR 3531858, https://doi.org/10.1007/s12044-016-0286-3
  • [10] Alexandru Dimca and Sergey Yuzvinsky, Lectures on Orlik-Solomon algebras, Arrangements, local systems and singularities, Progr. Math., vol. 283, Birkhäuser Verlag, Basel, 2010, pp. 83-110. MR 3025861
  • [11] Michael Falk, Arrangements and cohomology, Ann. Comb. 1 (1997), no. 2, 135-157. MR 1629681, https://doi.org/10.1007/BF02558471
  • [12] Michael Falk and Sergey Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143 (2007), no. 4, 1069-1088. MR 2339840, https://doi.org/10.1112/S0010437X07002722
  • [13] I. M. Gelfand and G. L. Rybnikov, Algebraic and topological invariants of oriented matroids, Dokl. Akad. Nauk SSSR 307 (1989), no. 4, 791-795 (Russian); English transl., Soviet Math. Dokl. 40 (1990), no. 1, 148-152. MR 1020668
  • [14] Branko Grünbaum, Arrangements and spreads, American Mathematical Society Providence, R.I., 1972. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 10. MR 0307027
  • [15] Dmitry Kozlov, Combinatorial algebraic topology, Algorithms and Computation in Mathematics, vol. 21, Springer, Berlin, 2008. MR 2361455
  • [16] Anatoly Libgober and Sergey Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), no. 3, 337-361. MR 1761630, https://doi.org/10.1023/A:1001826010964
  • [17] Miguel Ángel Marco Buzunáriz, A description of the resonance variety of a line combinatorics via combinatorial pencils, Graphs Combin. 25 (2009), no. 4, 469-488. MR 2575595, https://doi.org/10.1007/s00373-009-0863-7
  • [18] Daniel Matei and Alexander I. Suciu, Cohomology rings and nilpotent quotients of real and complex arrangements, Arrangements--Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Kinokuniya, Tokyo, 2000, pp. 185-215. MR 1796900
  • [19] Daniel Quillen, Homotopy properties of the poset of nontrivial $ p$-subgroups of a group, Adv. in Math. 28 (1978), no. 2, 101-128. MR 493916, https://doi.org/10.1016/0001-8708(78)90058-0
  • [20] M. Salvetti, Topology of the complement of real hyperplanes in $ {\bf C}^N$, Invent. Math. 88 (1987), no. 3, 603-618. MR 884802, https://doi.org/10.1007/BF01391833
  • [21] R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91-109. MR 510404, https://doi.org/10.1017/S0305004100055535
  • [22] Günter M. Ziegler, What is a complex matroid?, Discrete Comput. Geom. 10 (1993), no. 3, 313-348. MR 1226982, https://doi.org/10.1007/BF02573983

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05B35, 52C35, 52C40, 55U10, 05E45

Retrieve articles in all journals with MSC (2010): 05B35, 52C35, 52C40, 55U10, 05E45


Additional Information

Emanuele Delucchi
Affiliation: Department of Mathematics, University of Fribourg, Chemin du musée 23, 1700 Fribourg, Switzerland

Michael J. Falk
Affiliation: Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, Arizona 86011-5717

DOI: https://doi.org/10.1090/proc/13328
Received by editor(s): October 19, 2015
Received by editor(s) in revised form: March 6, 2016
Published electronically: November 29, 2016
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society