Constructing Combinatorial 4-Manifolds

TL;DR

This paper extends the combinatorial construction of 3-manifolds as branched covers to 4-manifolds, showing that every closed oriented PL 4-manifold can be represented as a simplicial branched cover of a 4-sphere.

Contribution

It generalizes the combinatorial approach for 3-manifolds to 4-manifolds, providing a new method to construct all closed oriented PL 4-manifolds as branched covers.

Findings

Every closed oriented PL 4-manifold is a simplicial branched cover of a 4-sphere.

The combinatorial construction is a direct generalization of the 3-manifold case.

The approach provides a new perspective on the topology of 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.