An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step, we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Némethi. We also show that the construction cannot always work in the reverse direction: in fact, the U-filtration depth of contact Ozsváth–Szabó invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsváth–Szabó invariants of those contact structures fillable by an AR plumbing graph, generalizing an earlier work of the first author.
We show that Brieskorn manifolds with their standard contact structures are
contact branched coverings of spheres. This covering maps a contact open book
decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur
A real 3-manifold is a smooth 3-manifold together with an orientation
preserving smooth involution, called a real structure. In this article we study
open book decompositions on smooth real 3-manifolds that are compatible with
the real structure. We call them real open book decompositions. We show that
each real open book carries a real contact structure and two real contact
structures supported by the same real open book decomposition are equivariantly
isotopic. We also show that every real contact structure on a closed
3-dimensional real manifold is supported by a real open book. Finally, we
conjecture that two real open books on a real contact manifold supporting the
same real contact structure are related by positive real stabilizations and
equivariant isotopy and that the Giroux correspondence applies to real
manifolds as well namely that there is a one to one correspondence between the
real contact structures on a real 3-manifold up to equivariant contact isotopy
and the real open books up to positive real stabilization. Meanwhile, we study
some examples of real open books and real Heegaard decompositions in lens
spaces.Comment: 26 pages, 12 figure
We give a partial classification for the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on [Formula: see text] and real lens spaces [Formula: see text]. We prove that there is a unique real tight [Formula: see text] and [Formula: see text]. We show that there is at most one real tight [Formula: see text] with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an [Formula: see text] which cannot be made convex equivariantly.
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