Topological stability of contact structures in dimension 3
It has been known for some time that a contact structure \(\xi_1\) on a closed manifold \(V\) that is sufficiently \(C^1\)-close to a contact structure \(\xi_0\) is actually isotopic to it. This follows from the result of J. Gray that a path of contact structures \((\xi_t)_{t\in [0,1]}\) is of the form \(\xi_t = \phi_{t\ast}(\xi_0)\) for some isotopy \((\phi_t)_{t\in [0,1]}\) of \(V\) with \(\phi_0 = \text{id}_V\). In this paper, the author shows that the contact structure \(\xi_1\) is also isotopic to the contact structure \(\xi_0\) when \(V\) is a closed \(3\)-manifold and \(\xi_1\) is only \(C^0\)-close to \(\xi_0\). He also points out that the same result can not be true in dimensions higher than three. In fact, for overtwisted contact structures, the \(3\)-manifold result can be proved from Eliashberg's classification of overtwisted structures on \(3\)-manifolds. As the author notes, his result is therefore akin to a local version of this classification -- a local version that also holds for tight contact structures.
scientific article; zbMATH DE number 1425243