On the contact boundaries of normal surface singularities

Caubel, Clemént; Popescu-Pampu, Patrick

doi:10.1016/j.crma.2004.04.023

Cited by 12 publications

(9 citation statements)

References 7 publications

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“…[21] and [26]). (c) In [2], the main result 1.2 was established only for Milnor fillable 3-manifolds which are rational homology spheres because the authors were not conscious of 4.6. Instead, they used the fact, easily deducible from results of Stallings and Waldhausen, that in such a 3-manifold an open book is determined up to isotopy by its binding alone.…”

Section: B Vertical Links and Horizontal Open Books In Plumbed Manifmentioning

confidence: 99%

Milnor open books and Milnor fillable contact 3-manifolds

2006

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We say that a contact manifold (M, ξ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X , x). In this article we prove that any 3-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate with any holomorphic function f : (X , x) → (C, 0), with isolated singularity at x (and any euclidian rug function ρ), an open book decomposition of M , and we verify that all these open books carry the contact structure ξ of (M, ξ) -generalizing results of Milnor and Giroux.

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Section: B Vertical Links and Horizontal Open Books In Plumbed Manifmentioning

confidence: 99%

Milnor open books and Milnor fillable contact 3-manifolds

2006

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“…Our work may be considered as a continuation of the efforts to find all possible Stein or, more generally, symplectic fillings of the contact links of normal surface singularities. As a continuation of [8], in [9] we showed with Caubel that such contact structures are determined up to contactomorphism by the topology of the link, that is, they depend only on the topological type of the singularity, and not on its analytical type. Therefore, singularity theory gives the following Stein fillings up to diffeomorphism: the minimal resolution of good representatives (which may be made Stein by deformation of the complex structure, see [6]) and the Milnor fibers of the smoothings of all the analytical realizations of the given topological type.…”

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confidence: 91%

On the Milnor fibres of cyclic quotient singularities

Némethi

Popescu-Pampu

2010

Proceedings of the London Mathematical Society

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The oriented link of the cyclic quotient singularity 𝒳p, q is orientation‐preserving diffeomorphic to the lens space L(p, q) and carries the standard contact structure ξst. Lisca classified the Stein fillings of (L(p, q), ξst) up to diffeomorphisms and conjectured that they correspond bijectively through an explicit map to the Milnor fibres associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of 𝒳p, q. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibres given by de Jong and van Straten, we also canonically identify these fibres with Lisca's fillings. Using these and a newly introduced additional structure (the order) associated with lens spaces, we prove that the above Milnor fibres are pairwise non‐diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.

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“…By the discussion above, there is at least one such Milnor open book. Combining this with the following result of Caubel and Popescu-Pampu [10], it follows that if ( , ) is a surface singularity with a rational homology sphere link, there is a unique Milnor open book up to isotopy corresponding to each element ∈  + , which we will denote by  ( ).…”

Section: Canonical Contact Structures and Milnor Open Booksmentioning

confidence: 82%

Open books decompositions of links of minimally elliptic singularities

Bhupal

2021

Mathematische Nachrichten

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We present an explicit Milnor open book decomposition supporting the canonical contact structure on the link of each minimally elliptic singularity whose fundamental cycle satisfies −3 ≤ ⋅ ≤ −1. For the Milnor open books whose pages have genus less than three, we give a factorization of the monodromy which does not involve any left-handed Dehn twists around interior curves. Necessary results regarding the roots of reducible, and irreducible elements in mapping class groups are proved, and some new relations in the mapping class groups are presented.

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On the contact boundaries of normal surface singularities

Cited by 12 publications

References 7 publications

Milnor open books and Milnor fillable contact 3-manifolds

Milnor open books and Milnor fillable contact 3-manifolds

On the Milnor fibres of cyclic quotient singularities

Open books decompositions of links of minimally elliptic singularities

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