Relative Group Trisections

TL;DR

This paper extends the concept of group trisections from closed 4-manifolds to non-closed manifolds, establishing a one-to-one correspondence and a functorial relationship between relative trisections and groups.

Contribution

It introduces the notion of relative group trisections, generalizes existing correspondences, and provides a key lemma for handlebody extensions in the non-closed setting.

Findings

Established a one-to-one correspondence between relative trisections and relative group trisections.

Generalized handlebody extension uniqueness to the non-closed case.

Extended functorial relationships to relative trisections of manifolds.

Abstract

Trisections of closed 4-manifolds, first defined and studied by Gay and Kirby, have proved to be a useful tool in the systematic analysis of 4-manifolds via handlebodies. Subsequent work of Abrams, Gay, and Kirby established a connection with the algebraic notion of a group trisection, which strikingly defines a one-to-one correspondence. We generalize the notion of a group trisection to the non-closed case by defining and studying relative group trisections. We establish an analogous one-to-one correspondence between relative trisections and relative group trisections up to equivalence. The key lemma in the construction may be of independent interest, as it generalizes the classical fact that there is a unique handlebody extension of a surface realizing a given surjection. Moreover, we establish a functorial relationship between relative trisections of manifolds and groups, extending work of Klug in the closed case.