A note on three-fold branched covers of $S^4$

TL;DR

This paper demonstrates that certain 4-manifolds with specific trisections are irregular 3-fold covers of the 4-sphere, with branch sets being surfaces that are mostly smooth except for a singular cone point on a link.

Contribution

It establishes a correspondence between 4-manifolds with particular trisections and irregular 3-fold covers of $S^4$, linking handle decompositions to branched cover structures.

Findings

4-manifolds with $(g;k_1,k_2,0)$-trisections are irregular 3-fold covers of $S^4$

Branch sets are surfaces with a single singular cone point

Such manifolds have handle decompositions with no 1-handles

Abstract

We show that any 4-manifold admitting a $(g;k_1,k_2,0)$-trisection is an irregular 3-fold cover of the 4-sphere whose branching set is a surface in $S^4$, smoothly embedded except for one singular point which is the cone on a link. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles; it is conjectured that all simply-connected 4-manifolds have this property.