Concordance of decompositions given by defining sequences

TL;DR

This paper investigates the relationships between certain complex decompositions of 3-manifolds, specifically those arising from defining sequences, and their invariants, revealing a rich structure with uncountably many concordance classes.

Contribution

It establishes the existence of uncountably many concordance classes of decompositions associated with defining sequences in the 3-sphere and relates them to invariants of toroidal decompositions.

Findings

Uncountably many concordance classes of decompositions in the 3-sphere.

Connections between decompositions, invariants, and cobordism of homology manifolds.

Decompositions often involve wild Cantor sets from nested knotted tori.

Abstract

We study the concordance and bordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere.