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The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent ColinSheila Sandon — 2015

Journal of the European Mathematical Society

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on 2 n × S 1 and P 2 n + 1 . On the other hand we also show by elementary arguments that the...

Reeb vector fields and open book decompositions

Vincent ColinKo Honda — 2013

Journal of the European Mathematical Society

We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux.We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient c equals k = n , where...

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