Families of contact 3-manifolds with arbitrarily large Stein fillings. (English) Zbl 1348.57036

Given a closed contact 3-manifold \((Y, \xi)\) a Stein filling is a Stein surface \( (X, J)\) such that \(\partial X=Y\) and such that \(J|Y\) induces the contact structure \(\xi\). There are many different notions of a filling of a contact manifold, but Stein manifolds “exhibit [the] strongest filling properties for a contact manifold”.
It has been conjectured by several authors that given a closed contact 3-manifold the set of signatures and Euler characteristics of all possible Stein fillings is finite. In this paper the authors disprove this conjecture. More precisely, they show that there are infinite families of contact 3-manifolds, where each contact 3-manifold admits a Stein filling whose Euler characteristic is larger and whose signature is smaller than any two given numbers.


57R17 Symplectic and contact topology in high or arbitrary dimension
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