Contact structures, σ-confoliations and contaminations in 3-manifolds

Ulrich Oertel, Jacek Światkowski

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called σ-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the σ-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact structure up to isotopy. After defining tightness for all pure contaminations in a natural way, generalizing the definition of tightness for contact structures, we obtain some conditions on (the embedding of) a branched surface in a 3-manifold sufficient to guarantee that any pure contamination carried by the branched surface is tight. We also find conditions sufficient to prove that a branched surface carries only overtwisted (non-tight) contact structures. Our long-term goal in developing these methods is twofold: Not only do we want to study tight contact structures and pure contaminations, but we also wish to use them as tools for studying 3-manifold topology.

Original languageEnglish (US)
Pages (from-to)201-264
Number of pages64
JournalCommunications in Contemporary Mathematics
Volume11
Issue number2
DOIs
StatePublished - Apr 2009

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Branched surface
  • Confoliation
  • Contact structure
  • Contamination
  • Tightness

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