A Survey of the Thurston Norm

Takahiro Kitayama ()
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Takahiro Kitayama: Graduate School of Mathematical Sciences, The University of Tokyo

Chapter Chapter 5 in In the Tradition of Thurston II, 2022, pp 149-199 from Springer

Abstract: 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.

Keywords: Thurston norm; Knot genus; 3-manifold; Knot; Fibration; Foliation; Alexander polynomial; Reidemeister torsion; Teichmüller polynomial; Seiberg–Witten invariant; Adjunction inequality; Heegaard Floer homology; Knot Floer homology; Twisted Alexander polynomial; Higher-order Alexander polynomial; L2-torsion; Normal surface; Complexity of a 3-manifold; Profinite rigidity (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-97560-9_5

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DOI: 10.1007/978-3-030-97560-9_5

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