3-manifolds lying in trisected 4-manifolds

TL;DR

This paper explores how all 3-manifolds can be embedded in the spine of minimal genus trisections of certain 4-manifolds, providing bounds and specific analysis for lens spaces.

Contribution

It introduces the concept of embedding 3-manifolds into the spine of trisected 4-manifolds and establishes that all 3-manifolds can be embedded in such a way within connected sums of $S^2 ilde \times S^2$s.

Findings

Every 3-manifold can be embedded almost in the spine of a minimal genus trisection.

Provided an upper bound on the number of $S^2 \tilde \times S^2$ summands based on a graph distance.

Derived explicit bounds for lens spaces.

Abstract

The spine of a trisected 4-manifold is a singular 3-dimensional set from which the trisection itself can be reconstructed. 3-manifolds embedded in the trisected 4--manifold can often be isotoped to lie almost or entirely in the spine of the trisection. We define this notion and show that in fact every 3-manifold can be embedded to lie almost in the spine of a minimal genus trisection of some connect sum of $S^2 \tilde \times S^2$s. This mirrors the known fact that every 3-manifold can be smoothly embedded in a connect sum of $S^2 \times S^2$s. Our methods additionally give an upper bound for how many copies of $S^2 \tilde \times S^2$ based on a distance calculated in an appropriately defined graph. For the special case of lens spaces we analyze more closely and obtain more explicit bounds.