On the contact homology of the first exotic contact form of J. Gonzalo and F. Varela

Abstract

This paper contains a detailed study of the behavior of the first exotic contact structure of J. Gonzalo and F. Varela on S3 along a remarkable vector field v in its kernel found by Martino (Adv Nonlinear Stud, 2011). All contact forms are assumed to verify the condition that reads as follows: d(θ α)(v,.) is a contact form with the same orientation than α. α is the first exotic contact form of Gonzalo and Varela (Third Schnepfenried Geometry Conference, vol 1, Asterisque no 107–108, pp 163–168. Société Mathématique de France, Paris, 1983). We also prove in this paper that the contact homology (via dual Legendrian curves) is non-zero for a sequence of indexes tending to infinity for the contact forms θα of the first exotic contact structure of J. Gonzalo and F. Varela on S3, under the assumption that they can be connected to the first contact form of this contact structure through a path along which a special pseudo-gradient which we build in this paper is assumed to verify a Fredholm condition (see the Sect. 1 and Bahri in Morse relations and Fredholm deformations of v-convex contact forms, 2014 for the definition of this notion). We do not know whether this assumption is verified for the pseudo-gradient which we use here.