Surface singularities and planar contact structures
Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1791-1823.

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of A n -singularities. In general, we characterize completely planar links of normal surface singularities (in terms of their resolution graphs); these singularities are precisely the rational singularities with reduced fundamental cycle (also known as minimal singularities). We also establish non-planarity of tight contact structures on certain small Seifert fibered L-spaces and of contact structures arising from the Boothby–Wang construction applied to surfaces of positive genus. Additionally, we prove that every finitely presented group is the fundamental group of a Lefschetz fibration with planar fibers.

Dans cet article on démontre que si une structure de contact sur une variété de dimension trois est portée par un livre ouvert à pages planaires, alors une certaine configuration d’intersections n’apparaît dans l’homologie d’aucun de ses remplissages minimaux. On démontre de plus que les remplissages d’une telle variété de contact ne contiennent pas de surface symplectique de genre positif. En appliquant ces obstructions aux structures de contact canoniques sur les bords des singularités normales de surfaces, on montre que les bords des singularités isolées de surfaces dans l’espace complexe de dimension trois sont planaires seulement pour les singularités de type A n . En général, nous caractérisons complètement les bords planaires des singularités normales de surfaces (par leurs graphes de résolution)  : ces singularités sont précisément les singularités rationnelles avec cycle fondamental réduit (aussi appelées singularités minimales). On montre aussi la non-planarité des structures de contact tendues sur certains petits L-espaces de Seifert ainsi que celle des structures de contact obtenues par la construction de Boothby–Wang appliquée aux surfaces de genre positif. De plus, on démontre que tout groupe de présentation finie est le groupe fondamental d’une fibration de Lefschetz à fibres planaires.

Published online:
DOI: 10.5802/aif.3384
Classification: 57R17, 32S55
Keywords: Contact structure, open book decomposition, isolated singularity.
Ghiggini, Paolo 1; Golla, Marco 1; Plamenevskaya, Olga 2

1 CNRS Laboratoire Jean Leray Université de Nantes Nantes (France)
2 Department of Mathematics Stony Brook University Stony Brook, NY (U.S.A.)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ghiggini, Paolo; Golla, Marco; Plamenevskaya, Olga. Surface singularities and planar contact structures. Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1791-1823. doi : 10.5802/aif.3384.

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