5-dimensional Contact So(3)-manifolds and Dehn Twists

In this paper the 5-dimensional contact SO(3)-manifolds are classified up to equivariant contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k, 2, 2, 2). A 5-dimensional contact SO(3)-manifold M can be decomposed into the set of singular orbits M(sing) and the set of regular orbits M(reg). Both parts can be described relatively easily: The singular orbits are the disjoint union of copies of S ×RP2, S × S or S1×̃S2 := R× S/ ∼, where (t, p) ∼ (t + 1,−p). The set of regular orbits contains a canonical submanifold R of dimension 3 (the so-called cross-section), and one has that M(reg) ∼= SO(3)×S1 R. For gluing the singular orbits onto the regular ones, there is an integer invariant that classifies all possibilities. This integer corresponds to the number of Dehn twists. Acknowledgments. This article contains most of the results of my Ph.D. thesis. I would like to thank the Universiteit Leiden and the Universität zu Köln for funding my work. I am indebted to my supervisor Hansjörg Geiges for his support and patience. Furthermore I’m grateful to the following people for many useful discussions: Peter Heinzner, Federica Pasquotto, Juan Souto, and Kai Zehmisch, but most of all to Otto van Koert.