Abelianizing the real permutation action via blowups

Proceedings of the American Mathematical Society

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 Z 4 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 R n , the Z 4 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 Z 4 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.

Kodai Mathematical Journal, 1994

Oberwolfach Reports (to appear)

A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the I-adic completion of the Alexander invariant B(G) also decomposes as a direct sum of "local" invariants and the Chen ranks of G are the sums of the local contributions. Moreover, if B(G) is separated, then the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.

2000

For a real oriented hyperplane arrangement, we show that the corresponding Salvetti complex is homotopy equivalent to the complement of the complexified arrangement. This result was originally proved by M. Salvetti. Our proof follows the framework of a proof given by L. Paris and relies heavily on the notation of oriented matroids. We also show that homotopy equivalence is preserved when we quotient by the action of the corresponding reflection group. In particular, the Salvetti complex of the braid arrangement in $\ell$ dimensions modulo the action of the symmetric group is a cell complex which is homotopy equivalent to the space of unlabelled configurations of $\ell$ distinct points. Lastly, we describe a construction of the orbit complex from the dual complex for all finite reflection arrangements in dimension 2. This description yields an easy derivation of the so-called "braid relations" in the case of braid arrangement.

A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras, and all the nilpotent quotients of G are combinatorially determined. Under some additional hypothesis, we show that the Alexander invariant of G also decomposes as a direct sum of "local" invariants. Consequently, the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.

A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the I-adic completion of the Alexander invariant B(G) also decomposes as a direct sum of ``local" invariants and the Chen ranks of G are the sums of the local contributions. Moreover, if B(G) is separated, then the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.

Algebraic & Geometric Topology

In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all finite Coxeter groups. This in turn follows from Deligne's seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a K(π, 1)-arrangement. In this paper we study the K(π, 1)-property for a certain class of subarrangements of Weyl arrangements, the so called arrangements of ideal type A I. These stem from ideals I in the set of positive roots of a reduced root system. We show that the K(π, 1)-property holds for all arrangements A I if the underlying Weyl group is classical and that it extends to most of the A I if the underlying Weyl group is of exceptional type. Conjecturally this holds for all A I. In general, the A I are neither simplicial, nor is their complexification fiber-type.

Journal of the American Mathematical Society, 1992

We present a method for discretizing complex hyperplane arrangements by encoding their topology into a finite partially ordered set of “sign vectors.” This is used in the following ways: (1) A general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement. (2) For the case of complexified arrangements this specializes to the construction of Salvetti [S]. We study the combinatorial structure of complexified arrangements and the Salvetti complex in some detail. (3) This general method simultaneously produces cell decompositions of the singularity link. This link is shown to have the homotopy type of a wedge of spheres for arrangements in C d , d ≥ 4 {\mathbb {C}^d},\;d \geq 4 . (4) The homology of the link and the cohomology of the complement are computed in terms of explicit bases, which are matched by Alexander duality. This gives a new, more elementary, and more generally valid proof for the Brieskorn-Orlik-Solomon theo...

2000

In "On the homotopy theory of arrangements," published in 1986, the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems.

Mathematical theory and modeling, 2016

The first main objective of the work was to create a combinatorial answer to an essential question; "How Terao generalization of the class of supersolvable arrangements preserved the tensor factorization of the O-S algebra?", by finding a relation among several bases of the O-S algebra. This was achieved in two parts. First, the class of factored arrangements was classified in two subclasses, the subclass of completely factored arrangements and the subclass of factored arrangement that not completely factored. Our classification criteria was, "the existence of an ordering on the hyperplanes of a factored arrangement such that the set of all monomials that related to the sections of a factorization on forms an NBC basis of the O-S algebra as a free module". The second part was, a comparison among the structures of the O-S complex, the NBC complex and the partition complex. In spite of, our classification criteria was failed of the second subclass of factored a...