A group-theoretic framework for low-dimensional topology

TL;DR

This paper unifies and generalizes group-theoretic frameworks connecting low-dimensional topologies, including 3-manifolds, 4-manifolds, and knotted surfaces, revealing deep algebraic similarities across these fields.

Contribution

It extends existing correspondences between manifolds and surface group surjections to include links and knotted surfaces, unifying multiple low-dimensional topology perspectives.

Findings

Unified algebraic framework for 3- and 4-manifolds and knotted surfaces

Generalization of Heegaard and trisection correspondences

Enhanced understanding of algebraic similarities in low-dimensional topology

Abstract

A correspondence, by way of Heegaard splittings, between closed oriented 3-manifolds and pairs of surjections from a surface group to a free group has been studied by Stallings, Jaco, and Hempel. This correspondence, by way of trisections, was recently extended by Abrams, Gay, and Kirby to the case of smooth, closed, connected, oriented 4-manifolds. We unify these perspectives and generalize this correspondence to the case of links in closed oriented 3-manifolds and links of knotted surfaces in smooth, closed, connected, oriented 4-manifolds. The algebraic manifestations of these four subfields of low-dimensional topology (3-manifolds, 4-manifolds, knot theory, and knotted surface theory) are all strikingly similar, and this correspondence perhaps elucidates some unique character of low-dimensional topology.