Thurston norm via Fox calculus

Friedl, Stefan¹ ; Schreve, Kevin² ; Tillmann, Stephan³

¹ Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany
² Department of Mathematics, University of Michigan, Ann Arbor, MI, United States
³ School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, Australia

Geometry & topology, Tome 21 (2017) no. 6, pp. 3759-3784.

Résumé

In 1976 Thurston associated to a $3$–manifold $N$ a marked polytope in $H_1\left(N;ℝ\right)$, which measures the minimal complexity of surfaces representing homology classes and determines all fibered classes in $H^1\left(N;ℝ\right)$. Recently the first and third authors associated to a presentation $\pi$ with two generators and one relator a marked polytope in $H_1\left(\pi ;ℝ\right)$ and showed that it determines the Bieri–Neumann–Strebel invariant of $\pi$. We show that if the fundamental group of a $3$–manifold $N$ admits such a presentation $\pi$, then the corresponding marked polytopes in $H_1\left(N;ℝ\right)=H_1\left(\pi ;ℝ\right)$ agree.

Détail

DOI : 10.2140/gt.2017.21.3759

Classification : 20J05, 57M05, 57M27, 57R19
Keywords: Thurston norm, $3$–manifold, Novikov ring, Fox calculus

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