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The topology of Stein fillable manifolds in high dimensions. II (with an appendix by Bernd C. Kellner). (English) Zbl 1380.32016
Summary: We continue our study of contact structures on manifolds of dimension at least five using surgery-theoretic methods. Particular applications include the existence of “maximal” almost contact manifolds with respect to the Stein cobordism relation as well as the existence of weakly fillable contact structures on the product $$M\times S^2$$. We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are not.
Concerning obstructions to Stein fillability, we show for all $$k >1$$ that there are almost contact structures on the $$(8k{-}1)$$-sphere which are not Stein fillable. This implies the same result for all highly connected $$(8k{-}1)$$-manifolds which admit almost contact structures. The proofs rely on a new number-theoretic result about Bernoulli numbers.
For Part I see [the authors, Proc. Lond. Math. Soc. (3) 109, No. 6, 1363–1401 (2014; Zbl 1314.53134)].

##### MSC:
 32E10 Stein spaces, Stein manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension 57R65 Surgery and handlebodies
##### Keywords:
Stein fillability; surgery; contact structures; bordism theory
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