Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds

TL;DR

This paper classifies certain 3-manifolds arising from simplified (2, 0)-trisection maps of 4-manifolds, showing their structure as connected sums and the uniqueness of the source 4-manifold, with examples of hyperbolic cases.

Contribution

It provides a classification of vertical 3-manifolds in simplified (2, 0)-trisections and establishes their relation to the source 4-manifold, including the existence of hyperbolic examples.

Findings

Vertical 3-manifolds are connected sums of six specific 3-manifolds.

Each 6-tuple of summands determines the 4-manifold uniquely up to orientation.

Existence of infinitely many hyperbolic vertical 3-manifolds in certain maps.

Abstract

We classify the $3$-manifolds obtained as the preimages of arcs on the plane for simplified $(2, 0)$-trisection maps, which we call vertical $3$-manifolds. Such a $3$-manifold is a connected sum of a $6$-tuple of vertical $3$-manifolds over specific $6$ arcs. Consequently, we show that each of the $6$-tuples determines the source $4$-manifold uniquely up to orientation reversing diffeomorphisms. We also show that, in contrast to the fact that summands of vertical $3$-manifolds of simplified $(2, 0)$-trisection maps are lens spaces, there exist infinitely many simplified $(2, 0)$-$4$-section maps that admit hyperbolic vertical $3$-manifolds.