Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/ium/s12/laudenbach.pdf Дата изменения: Fri Mar 30 21:47:22 2012 Дата индексирования: Tue Oct 2 07:26:37 2012 Кодировка: Поисковые слова: comet

Mini-course proposed by Francois Laudenbach: ё Open books and twisted open books. Application to foliations and contact structures in dimension 3 The concept of open bo ok decomposition go es back to H. Winkelnkemper (1973) and, according to old works by J.W. Alexander in the twenties, every closed orientable 3-manifold admits such a decomposition. In his 1991 paper, E. Giroux gave the material for a Morse theoretical approach to open bo ok decompositions. As a consequence, every closed 3-manifold, orientable or not, has such a decomposition. In an appropriate sense, an open bo ok decomposition of the 3-manifold M carries a co dimension-one foliation and also, when M is orientable, a contact structure (Thurston­Winkelnkemper, 1975). But, these foliations and contact structures are co-orientable, since the normal bundle to a an open bo ok (which makes sense despite the binding set) must be trivial. Ten years ago, E. Giroux showed that every coorientable contact structure on M 3 is carried by some open bo ok. For overcoming the trivialness of the normal bundle, in a joint work with G. Meigniez, we intro duced the concept of twisted open bo ok and we proved a result, similar to the one of Giroux, for concordance classes of co dimension-one foliations with singularities (1 -structures of Haefliger); here the singularities which are meant are singularities of functions: Every 1 -structure on M 3 , whose normal bund le embeds into the tangent bund le M , is concordant to a non-singular foliation carried by a (twisted) open book. This statement is the 3-dimensional part of a famous theorem due to W. Thurston (1976). In this course, this topic will be discussed in a rather detailed manner. The starting point will be a dynamical approach to twisted openbo oks by considering suitable pseudo-gradients vector fields of Morse 1 -structures. That it will lead to a simplified pro of of Giroux's theorem is still a hope.