Abstract. Given a real hyperplane arrangement A, the complement M(A) of the complexification of A admits an action of the group Z2 by complex conjugation. We define the equivariant Orlik-Solomon algebra of A to be the Z2-equivariant cohomology ring of M(A) with coefficients in the field F2. We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orli...

— There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the h...

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the B...

background and objectives: complement proteins are some of the most important plasma proteins of the innate immune system. impaired immune function is reported in subjects who are iron deficient, and there are documents that these patients are prone to infection. this study was conducted to show whether serum c3 and c4 complement change in adult nonpregnant female with iron deficient anemia or ...

In a recent paper, Dimca and N emethi pose the problem of nding a homogeneous polynomial f such that the homology of the complement of the hypersurface de ned by f is torsion-free, but the homology of the Milnor ber of f has torsion. We prove that this is indeed possible, and show by construction that, for each prime p , there is a polynomial with p-torsion in the homology of the Milnor ber. Th...

We study the relations between the fundamental group and the homological operations on integer homology. For the “rational” fundamental group (Malcev completion) see [2, 3, 4, 11, 16, 17]. This work is an attempt to understand the invariant of the fundamental group of the complement of a complex hyperplane arrangement that was used in [13]. Note that this invariant necessarily vanishes over Q (...

We produce a one-parameter family of hyperplane arrangements that are counterexamples to the conjecture of Saito that the complexified complement of a free arrangement is K(n, 1 ). These arrangements are the restriction of a one-parameter family of arrangements that arose in the study of tilings of certain centrally symmetric octagons. This other family is discussed as well. I. Definitions and ...

The Orlik-Solomon algebra A(G) of a matroid G is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing G. On the other hand, some features of the matroid G are reflected in the algebraic structure of A(G). In this mostly expository article,...

Let G be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of G. This resolution is used to define representations of groups which act compatibly on G, generalizing classical constructions of Magnus, Burau, and Gassner. Our construction also yields algorithms for comp...

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev (i.e. unipotent) completion of its fundamental group π1(X,x0) and the cohomology groups H (X,Lρ) with coefficients in rank one local systems. In this paper, we develop a tool that unifies these two approaches. This tool is the Malcev completion of π1(X,x0) relative to a hom...