Almost contact 5-manifolds are contact
"The existence of a contact structure is proved in any homotopy class of almost contact structures on a closed 5-dimensional manifold." The work, a substantial 47 pages long, underwent a long and rigorous refereeing process, and is regarded as a major milestone. Dishant Pancholi was with ICTP as a postdoc during 2008 and 2009, and is now Associate Professor at the Chennai Mathematical Institute. He is also a Simons Associate of ICTP.
We invited Dishant to tell us more, and here is what he wrote:
A contact structure on a manifold M is a maximally non-integrable codimension one distribution. (A distribution is a sub-bundle of the tangent bundle; a distribution is integrable if given any two vector fields that are sections of the sub-bundle, their Lie bracket is also a section of the sub-bundle. We will not define the notion of "maximally nonintegrable".)
Contact geometry is closely related to symplectic geometry. The study of contact structures is at the forefront of current research in geometric topology because of the intimate relationship with subjects like complex and Kaehler geometry. In 1966, S.S. Chern posed the question: under which condition(s) does a manifold M admit a contact structure? A necessary condition is that the manifold should admit a codimension one distribution that has a structure of a complex vector bundle. (In particular the dimension of M must be odd.)
Such manifolds are now known as almost contact manifolds. M. Gromov showed (using his h-principle philosophy) that any open (that is, non-compact) almost contact manifold is in fact contact.
As for closed (that is, compact and without boundary) manifolds, the first breakthrough came in the seventies and was due to R. Lutz and J. Martinet. They established that any orientable three manifold admits a contact structure. At around the same time, D. Bennequin discovered invariants that distinguished contact structures.
In late eighties and early nineties Y. Eliashberg greatly generalized these constructions to show that every homotopy class of codimension one distribution on a closed 3-manifold admits a unique "overtwisted" contact structure.
Despite the complete answer to Chern's question in dimension three, not much was known in higher dimensions. The most notable advance was due to H. Geiges who showed that a simply connected manifold admits a contact structure if and only if the Stieffel-Whitney class W_3 with integral coefficient is zero.
After joining ICTP as a post-doc (in January 2008) I started working on the problem of constructing contact structure on closed manifolds of dimension higher than three. John Etnyre (whom I visited using my travel grant as ICTP post-doc) and I generalized the Lutz's construction to higher dimension. This enabled us to construct new contact structures out of existing ones.
On the other hand, Fran Presas and others generalized a technique due to S. Donaldson (who was interested in invariants of symplectic manifolds) to establish a Lefschetz-pencil type structure on an almost contact manifold. Fran Presas, his student Roger Casals, and I combined these results and the classification of overtwisted contact structures on 3-manifolds due to Eliashberg to establish that on a five-dimensional manifold, there exists a contact structure in every homotopy class of almost contact distributions. This collaboration was also made possible by support from ICTP.
Ramadas Ramakrishnan Trivandrum
