BooksirmaCollected Volumepp. 557–588

Presentation of the fundamental groups of complements of shadows

  • Masaharu Ishikawa

    Keio University, Japan

  • Yuya Koda

    Hiroshima University, Japan

  • Hironobu Naoe

    Chuo University, Japan
Presentation of the fundamental groups of complements of shadows cover
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Abstract

A shadowed polyhedron is a simple polyhedron equipped with half integers on regions, called gleams, which represents a compact, oriented, smooth -manifold. The polyhedron is embedded in the -manifold and it is called a shadow of that manifold. A subpolyhedron of a shadow represents a possibly singular subsurface in the -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 -manifold is always a -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 -ball. The method is very similar to the Wirtinger presentation of links in knot theory.