Long-time behavior of 3 dimensional Ricci flow -- C: 3-manifold topology and combinatorics of simplicial complexes in 3-manifolds

TL;DR

This paper develops topological tools and constructs simplicial complexes in 3-manifolds to facilitate the analysis of long-term Ricci flow behavior with surgery, focusing on 3-manifold topology and combinatorics.

Contribution

It introduces new topological constructions and intersection properties in 3-manifolds that are crucial for understanding Ricci flow with surgery.

Findings

Constructed simplicial complexes with specific intersection properties

Established topological results relevant for Ricci flow analysis

Focused on 3-manifold topology without Ricci flow applications

Abstract

In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construction of certain simplicial complexes in a given 3-manifold that exhibit useful intersection properties with embedded, incompressible solid tori. This paper is purely topological in nature and Ricci flows will not be used.