Presentation of the fundamental groups of complements of shadows

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.

[1]  Masahiko Yoshinaga,et al.  Divides with cusps and Kirby diagrams for line arrangements , 2021, Topology and its Applications.

[2]  M. Ishikawa,et al.  Milnor fibration, A’Campo’s divide and Turaev’s shadow , 2018, Singularities — Kagoshima 2017.

[3]  M. Yoshinaga.  Minimal stratifications for line arrangements and positive homogeneous presentations for fundamental groups , 2011, 1105.1857.

[4]  F. Costantino.  Shadows and branched shadows of 3- and 4-manifolds , 2005 .

[5]  B. Martelli.  Links, two-handles, and four-manifolds , 2004, math/0412511.

[6]  Tomomi Kawamura.  Quasipositivity of links of divides and free divides , 2002 .

[7]  M. Ishikawa,et al.  Links and gordian numbers associated with generic immersions of intervals , 2002 .

[8]  M. Ishikawa,et al.  Links of oriented divides and fibrations in link exteriors , 2002 .

[9]  M. Hirasawa.  Visualization of A'Campo's fibered links and unknotting operation , 2002 .

[10]  B. Perron,et al.  REPRESENTATIVE BRAIDS FOR LINKS ASSOCIATED TO PLANE IMMERSED CURVES , 2000 .

[11]  N. A'campo.  Planar trees, slalom curves and hyperbolic knots , 1998, math/9906087.

[12]  N. A'campo.  Generic immersions of curves, knots, monodromy and Gordian number , 1998, math/9803081.

[13]  N. A'campo.  Real deformations and complex topology of plane curve singularities , 1997, alg-geom/9710023.

[14]  V. Turaev.  Quantum Invariants of Knots and 3-Manifolds , 1994, hep-th/9409028.

[15]  R. Randell.  The fundamental group of the complement of a union of complex hyperplanes , 1982 .

[16]  S. M. Husein-Zade.  THE MONODROMY GROUPS OF ISOLATED SINGULARITIES OF HYPERSURFACES , 1977 .

[17]  N. A'campo.  Le groupe de monodromie du déploiement des singularités isolées de courbes planes I , 1975 .

[18]  Frank Wannemaker,et al.  Arrangements Of Hyperplanes , 2016 .

[19]  M. Ishikawa.  Tangent circle bundles admit positive open book decompositions along arbitrary links , 2004 .

[20]  Amruth N. Kumar,et al.  Links , 1999, INTL.

[21]  V. Turaev.  Shadow links and face models of statistical mechanics , 1992 .

[22]  E. Looijenga.  THE MILNOR FIBRATION , 1984 .

[23]  S. Gusein-Zade.  Intersection matrices for certain singularities of functions of two variables , 1974 .

[24]  S. Gusein-Zade.  Dynkin diagrams for singularities of functions of two variables , 1974 .

[25]  J. Milnor.  Singular points of complex hypersurfaces , 1968 .