Intersection Norms and One-faced Collection of Curves
Résumé
Intersection norms are integer norms on the first homology group of a surface. In this article, we prove that there are some polytopes which are not dual unit balls of such norms. By the way, we prove that, up to diffeomorphism, there are four collections of curves on Σ 2 whose complement is a disk. Intersection norms on surfaces were first quickly introduced by Turaev [6] (page 143), and studied by M. Cossarini and P. Dehornoy [1]. They use intersection norms to classify, up to isotopy all surfaces transverse to the geodesic flow on the complement of special links in the unit tangent bundle of a closed oriented surface. Their result makes explicit Thurston's fiber faced theory for Thurston norms on compact oriented 3-manifolds. It tells us that the intersection norm on a surface (respectively the Thurston norm on a 3-manifold) encodes the open book decompositions of the unit tangent bundle of that surface (respectively the topology of a fibered 3-manifold). 1 Our purpose in this article is to study intersection norms for their own. Let Σ g be a closed oriented surface of genus g ≥ 1, and Γ a collection of closed curves on Σ g. We assume that Γ has only double intersection points. Let α be loop on Σ g , we define the number i Γ (α) as follows : i Γ (α) = inf{#{α ∩ Γ}; α ∼ α; α Γ}; where the symbol ∼ (respectively) is the free homotopy relation (respectively transversality).
Domaines
Topologie géométrique [math.GT]
Origine : Fichiers produits par l'(les) auteur(s)
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