Constructing Combinatorial 4-Manifolds
Abstract
Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and Montesinos to dimension four. Izmestiev and Joswig [Adv. Geom. 3(2):191-225, 2003] gave a combinatorial equivalent of the Hilden and Montesinos result, constructing closed oriented combinatorial 3-manifolds as simplicial branched covers of combinatorial 3-spheres. The construction of Izmestiev and Joswig is generalized and applied to the result of Piergallini, obtaining closed oriented combinatorial 4-manifolds as simplicial branched covers of simplicial 4-spheres.
Cite
@article{arxiv.0707.1415,
title = {Constructing Combinatorial 4-Manifolds},
author = {Nikolaus Witte},
journal= {arXiv preprint arXiv:0707.1415},
year = {2007}
}
Comments
Stronger results and a shorter proof are presented in "Constructing Simplicial Branched Covers" by the author. Nevertheless we present some interesting techniques and a combinatorial analog of the (topological) proof by Piergallini