TY - JOUR

T1 - An equivariant discrete model for complexified arrangement complements

AU - Delucchi, Emanuele

AU - Falk, Michael J.

N1 - Publisher Copyright:
© 2016 American Mathematical Society.

PY - 2017

Y1 - 2017

N2 - We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M, e) associated to the pointed (or affine) oriented matroid (M, e). If M is the oriented matroid of an arrangement A of linear hyperplanes in Rn, the Z4 action corresponds to the diagonal action of C on the complement M of the complexification of A: |Q| is equivariantly homotopyequivalent to M under the identification of Z4 with the multiplicative subgroup {±1, ±i} ⊂ C∗, and |Q(M, e)| is homotopy-equivalent to the complement of the decone of 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.

AB - We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M, e) associated to the pointed (or affine) oriented matroid (M, e). If M is the oriented matroid of an arrangement A of linear hyperplanes in Rn, the Z4 action corresponds to the diagonal action of C on the complement M of the complexification of A: |Q| is equivariantly homotopyequivalent to M under the identification of Z4 with the multiplicative subgroup {±1, ±i} ⊂ C∗, and |Q(M, e)| is homotopy-equivalent to the complement of the decone of 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.

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U2 - 10.1090/proc/13328

DO - 10.1090/proc/13328

M3 - Article

AN - SCOPUS:85007393292

SN - 0002-9939

VL - 145

SP - 955

EP - 970

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -