Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

Abstract

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving 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). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds. Mathematics Subject Classifications (2000). 53D35, 53D20.A five-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 1 × RP 2 , S 1 × S 2 or S 1 ×S 2 := R × S 2 /∼, where (t, p) ∼ (t + 1, −p). The set of regular orbits contains a canonical submanifold R of dimension 3 (the so-called crosssection), and one has that M (reg) ∼ = SO(3) × S 1 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.There is a similar decomposition of five-dimensional contact SU(2)-manifolds. Here all singular orbits are fixed points, the components of M (sing) are embedded circles, and the set M (reg) can be described by SU(2) × S 1 R. In contrast to SO(3)-manifolds, there is no freedom in gluing M (reg) to M (sing) .

Every Contact Manifolds can be given a Nonfillable Contact Structure

“…(2) Using similar computations as the ones in [KN05] or [Nie06a], it can be seen that (S 2n−1 , α − ) is compatible with the open book (B = f −1 (0), ϑ = f /|f |), which is equivalent to the abstract open book with page (T * S n−1 , dλ can ) and monodromy map consisting of a single left-handed Dehn-twist. In particular, the 3-dimensional case (S 3 , α − ) is overtwisted, and it is not difficult to localize an overtwisted disk: The intersection F between the 3-sphere S 3 and the hyperplane (x 1 + iy 1 , x 2 + iy 2 ) y 1 = x 2 is diffeomorphic to a 2-sphere, which is foliated by α − .…”