A survey of the Thurston norm

TL;DR

This survey comprehensively reviews the Thurston norm in 3-manifold topology, exploring its properties, relationships with other invariants, and open questions, serving as a foundational overview for researchers.

Contribution

It consolidates fundamental properties, connections with various invariants, and open problems related to the Thurston norm in a comprehensive survey.

Findings

Thurston norm relates to taut foliations and embedded surfaces.

Connections established between Thurston norm and invariants like Alexander polynomial, Reidemeister torsion, and Seiberg-Witten invariant.

Highlights open questions and conjectures in the study of the Thurston norm.

Abstract

We present an overview of the study of the Thurston norm, introduced by W. P. Thurston in the seminal paper "A norm for the homology of 3-manifolds" (written in 1976 and published in 1986). We first review fundamental properties of the Thurston norm of a 3-manifold, including a construction of codimension-1 taut foliations from norm-minimizing embedded surfaces, established by D. Gabai. In the main part we describe relationships between the Thurston norm and other topological invariants of a 3-manifold: the Alexander polynomial and its various generalizations, Reidemeister torsion, the Seiberg-Witten invariant, Heegaard Floer homology, the complexity of triangulations and the profinite completion of the fundamental group. Some conjectures and questions on related topics are also collected. The final version of this paper will appear as a chapter in the book "In the tradition of Thurston, II", edited by K. Ohshika and A. Papadopoulos (Springer, 2022).