A survey 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).Contents 30 9. Conjectures and questions 31 References 33

The minimal genus problem

“…The complexity and Thurston norm defined for 3-manifolds below is very similar to the configuration results described in the previous section. For a survey of the minimal genus problem in 3-manifolds, see Kitayama [18] and Wu [51]. Definition 2.13.…”

“…Given a 3-manifold N , the quantity x 3 (A) is known as the Thurston norm, although it only defines a seminorm on H 1 (N, Z) ≡ H 2 (N, Z), which can however be extended uniquely to H 1 (N, R). This can be viewed as a generalization of the knot genus; see Kitayama [18] for details.…”