Some Applications of Number Theory to 3-Manifold Theory

TL;DR

This paper introduces the interplay between number theory and 3-manifold theory, highlighting key results that leverage arithmetic hyperbolic 3-manifolds and automorphic forms to advance understanding in topology.

Contribution

It provides an accessible overview of how number theory techniques are applied to 3-manifold problems, fostering interdisciplinary collaboration.

Findings

Connections between number theory and 3-manifold topology are elucidated.

Key results from Labesse-Schwermer, Calegari-Dunfield, Dunfield-Ramakrishnan are summarized.

The paper encourages collaboration between topologists and number theorists.

Abstract

These are the extended notes of a talk I gave at the Geometric Topology Seminar of the Max Planck Institute for Mathematics in Bonn on January 30th, 2012. My goal was to familiarize the topologists with the basics of arithmetic hyperbolic 3-manifolds and sketch some interesting results in the theory of 3-manifolds (such as Labesse-Schwermer, Calegari-Dunfield, Dunfield-Ramakrishnan) that are obtained by exploiting the connections with number theory and automorphic forms. The overall intention was to stimulate interaction between the number theorists and the topologists present at the Institute.