The Geometry and Fundamental Groups of Solenoid Complements

TL;DR

This paper explores the geometric properties and fundamental groups of solenoid complements in three-dimensional space, revealing how different embeddings lead to diverse topological structures and fundamental groups.

Contribution

It demonstrates that solenoid complements have varied fundamental groups depending on their embeddings, and establishes the existence of uncountably many non-homeomorphic complements.

Findings

Different solenoid embeddings yield distinct fundamental groups.

Unknotted embeddings produce Abelian fundamental groups.

There are uncountably many non-homeomorphic solenoid complements.

Abstract

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements.