Higher-order Alexander Invariants of Hypersurface Complements

TL;DR

This paper introduces higher-order Alexander invariants for complex hypersurface complements, extending previous work from plane curves to higher dimensions, and shows how local singularity topology influences global topology.

Contribution

It generalizes the concept of higher-order Alexander invariants to higher-dimensional hypersurface complements and relates global invariants to local singularity data.

Findings

Higher-order degrees are bounded by local invariants.

Extension of invariants from knots and 3-manifolds to hypersurfaces.

Global topology controlled by local singularity topology.

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.