Presentation of the fundamental groups of complements of shadows

TL;DR

This paper presents a method to compute the fundamental groups of complements of certain shadows in 4-balls, extending techniques similar to Wirtinger presentations used in knot theory.

Contribution

It introduces a new approach to present fundamental groups of complements of contractible shadows derived from disks with attached annuli, applicable to Milnor fibers and complex arrangements.

Findings

Provides explicit group presentations for complements of shadows in 4-balls.

Extends Wirtinger-like methods to 4-dimensional topology.

Applicable to Milnor fibers and complex line arrangements.

Abstract

A shadowed polyhedron is a simple polyhedron equipped with half integers on regions, called gleams, which represents a compact, oriented, smooth 4-manifold. The polyhedron is embedded in the 4-manifold and it is called a shadow of that manifold. A subpolyhedron of a shadow represents a possibly singular subsurface in the 4-manifold. In this paper, we focus on contractible shadows obtained from the unit disk by attaching annuli along generically immersed closed curves on the disk. In this case, the 4-manifold is always a 4-ball. Milnor fibers of plane curve singularities and complexified real line arrangements can be represented in this way. We give a presentation of the fundamental group of the complement of a subpolyhedron of such a shadow in the 4-ball. The method is very similar to the Wirtinger presentation of links in knot theory.