Roots in 3-manifold topology

TL;DR

This paper introduces a general framework for defining roots of objects in 3-manifold topology using simplifying moves, proving existence and uniqueness, and applying it to reprove and extend classical decomposition theorems.

Contribution

It establishes a unified approach to roots in 3-manifold topology, providing new proofs and extensions of prime decomposition theorems for various topological objects.

Findings

Existence and uniqueness of roots under certain conditions

New proof of the Kneser-Milnor prime decomposition theorem

Extensions of prime decomposition to cobordisms, knotted graphs, and orbifolds

Abstract

Let C be some class of objects equipped with a set of simplifying moves. When we apply these to a given object M in C as long as possible, we get a root of M. Our main result is that under certain conditions the root of any object exists and is unique. We apply this result to different situations and get several new results and new proofs of known results. Among them there are a new proof of the Kneser-Milnor prime decomposition theorem for 3-manifolds and different versions of this theorem for cobordisms, knotted graphs, and orbifolds.