Twisted Alexander modules of hyperplane arrangement complements

Intersection Homology and Alexander Modules of Hypersurface Complements

Let V be a degree d, reduced hypersurface in CP, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP−V ∪H, and let U be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules Hi(U ;Q) of the hypersurface complement and show t...

Ja n 20 02 HYPERSURFACE COMPLEMENTS , ALEXANDER MODULES AND MONODROMY

Complex Hyperbolic Hyperplane Complements

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n−1)-submanifold S . The main result is that the fundamental group of M \S is relatively hyperbolic, relative to fundamental groups of the ends of M \S , and M \S admits a complete finite volume A -regular Riemannian metric of negative sectional curvature. It fo...

Twisted Alexander Invariants of Twisted Links

Let L = 1∪· · ·∪ d+1 be an oriented link in S3, and let L(q) be the d-component link 1 ∪· · ·∪ d regarded in the homology 3-sphere that results from performing 1/q-surgery on d+1. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.

Differential Equations on Hyperplane Complements

The goal of the next two talks is to (1) discuss a class of integrable connections associated to root systems (2) describe their monodromy in terms of quantum groups These connections come in two forms: • Rational form leading to representations of braid groups (this week) • Trigonometric form leading to representations of affine braid groups (next week) The relevance of these connections is th...