Higher-order Alexander Invariants of Hypersurface Complements

Abstract: We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and higher-order degrees $\delta_{n,i}(\mathcal{U})$ which are invariants of a complex hypersurface complement $\mathcal{U}$. These invariants come from the module structure of the homology of certain solvable covers of the hypersurface complement. Such invariants were originally developed by T. Cochran in [1] and S. Harvey in [8], and were used to study knots and 3-manifolds. In this paper, I generalize the result proved by C. Leidy and L. Maxim [22] from the plane curve complements to higher-dimensional hypersurface complements. Zariski observed that the position of singularities on a singular complex plane curve affects the topology of the curve. My results on higher-order degrees of hypersurface complements also show that global topology is controlled by the local topologies. In particular, the higher-order degrees of the hypersurface complement are bounded by a linear combination of the higher-order degrees of the local link pairs.