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Contact structures on manifolds fibered by circles above a surface. (Structures de contact sur les variétés fibrées en cercles au-dessus d’une surface.) (French) Zbl 0988.57015

The author’s aim is to classify contact structures on certain 3-dimensional manifolds, namely lens spaces, most torus fibrations over the circle, the solid and the ‘thickened’ torus. This classification combines two techniques: surgery which provides numerous examples, and ‘tomography’ which allows us to analyze a given contact structure \(\xi\) and to form a combinatorial image for it. Thus we first manufacture a contact structure and then study it by means of the foliations cut out by the contact plane field \(\xi\) on parallel 2-dimensional slices. The paper brings together in a systematic way earlier results of the author (and others), and must be regarded as a major contribution to contact geometry. After an opening section in which the results and methods are explained, the author turns to the ‘tomography’ mentioned above. This leads to new proofs of the theorems of Bennequin and Eliashberg describing contact structures on \(S^3.\) Taking \(F = T^2\) the method allows us to construct normalized forms for tight contact structures on a thickened torus, and so to complete the classification on the promised class of closed 3-manifolds.
In the light of the Thurston geometrization programme the reviewer is led to ask what can be proved for surface bundles over \(S^1,\) when the fibre \(F\) has genus at least 2. The algebraic input for lens spaces (the continued fraction expansion of the defining rational number \(p/q\)) suggests that the classification of contact structures on fibrations admitting a hyperbolic structure will be particularly beautiful. So far as the completion of Thurston’s project itself goes one obvious question is whether an arbitrary quotient of \(S^3\) by a free topological finite group action admits a (universally?) tight contact structure.
Combined with work of Jonathan Hillman the results in this paper can be used to study symplectic structures on geometric manifolds in dimension 4. Similar considerations apply to the natural Engel structure on the projectivization of a contact distribution in \(TM^3,\) where \(M^3\) is (say) a nilmanifold. Other potential applications could be formulated.
Any reader of this review is urged to turn to the much longer one written by H. Geiges (MR2001i:53147), one of the merits of which is to place the author’s work in the historical development of contact geometry.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
53D10 Contact manifolds (general theory)
57M50 General geometric structures on low-dimensional manifolds
53D35 Global theory of symplectic and contact manifolds
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