Geometry and topology of symmetric point arrangements

Complex Arrangements: Algebra, Geometry, Topology

A hyperplane arrangement A is a finite collection of hyperplanes in some fixed (typically real or complex) vector space V. For simplicity, in this overview we work over the complex numbers C. There is a host of beautiful mathematics associated to the complement X = V A. Perhaps the first interesting result in the area was Arnol’d’s computation [2] of the cohomology ring of the complement of the...

Topology of Complex Reflection Arrangements

Let V be a finite dimensional complex vector space and W ⊂ GL(V ) be a finite complex reflection group. Let V reg be the complement in V of the reflecting hyperplanes. A classical conjecture predicts that V reg is a K(π, 1) space. When W is a complexified real reflection group, the conjecture follows from a theorem of Deligne, [20]. Our main result validates the conjecture for duality (or, equi...

Topology of Generic Line Arrangements

Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial f = ∏ i(aix + biy + ci), so that A = (f = 0). We prove that the defining polynomials of two generic line arrangements are, up to a small deformation, topologically equivalent. In higher dimension the related result is that within a ...

Topology of real coordinate arrangements

We prove that if a simplicial complex ∆ is (nonpure) shellable, then the intersection lattice for the corresponding real coordinate subspace arrangement A∆ is homotopy equivalent to the link of the intersection of all facets of ∆. As a consequence, we show that the singularity link of A∆ is homotopy equivalent to a wedge of spheres. We also show that the complement of A∆ is homotopy equivalent ...

The Topology of Hyperplane Arrangements