Open book decompositions for almost contact manifolds.

Abstract

A contact form on a (2n+1) -dimensional differential manifold M is a 1 -form α such that α∧(dα) n is a volume form; the hyperplane field ξ=kerα is the associated (co-oriented) contact structure. A contact form induces a reduction of the structure group of the tangent bundle TM to the group U(n)×1 . Such a reduction of the structure group is called an almost contact structure. The basic question on the existence of contact structures is whether every almost contact structure is induced from a (co-oriented) contact structure. In dimension three, where an almost contact structure is simply an oriented and co-oriented 2 -plane field, this question is answered in the affirmative by what is known as the Lutz-Martinet theorem. In higher dimensions there are such existence results for some specific manifolds, for instance simply connected 5 -manifolds, as shown by this reviewer [Mathematika 38 (1991), no. 2, 303–311 (1992) The paper under review may well be a first important step towards a general answer to the above question. An h -principle à la Gromov shows the existence of, on any almost contact manifold of dimension 2n+1 , a closed 2 -form ω of maximal rank 2n , where the cohomology class [ω] may be prescribed. In particular, ω may be chosen to be exact. The authors call such an ω an (exact) "quasi-contact'' manifold. Notice that the name "odd-symplectic form'' has also been employed in the literature. The first main theorem is an existence theorem for codimension 2 quasi-contact submanifolds. This is analogous to a theorem of S. K. Donaldson [J. Differential Geom. 44 (1996), no. 4, 666–705 for symplectic manifolds and L. A. Ibort, D. Martínez Torres and F. Presas [J. Differential Geom. 56 (2000), no. 2, 235–283; for contact manifolds. This is then used to prove the existence of an open book decomposition adapted to an exact quasi-contact structure. Here "adapted'' is to be understood in the sense as introduced for contact structures by E. Giroux [in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 405–414, Higher Ed. Press, Beijing, 2002. The open book found by the authors is not quite special enough to carry a contact structure, as one would need to solve the general existence question: the pages have the homotopy type of an (n+1) -dimensional complex, and the monodromy cannot be controlled. For the open book to support a contact structure, one would need homotopically n -dimensional pages and symplectic monodromy. As a bonus, the authors obtain an open book decomposition for almost contact 5 -manifolds. Most of the known topological methods for constructing open books on arbitrary odd-dimensional manifolds (the classical 3-dimensional case apart) only apply to dimension 7 and higher. Contrary to the authors' claim, the existence of open book decompositions is known for arbitrary (closed, smooth) 5-manifolds, by a result of F. Quinn [Topology 18 (1979), no. 1, 55–73.