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The topology of Stein fillable manifolds in high dimensions, II

Jonathan Bowden, Diarmuid Crowley and András I Stipsicz

Appendix: Bernd C Kellner

Geometry & Topology 19 (2015) 2995–3030

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 × S2. 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 (8k1)–sphere which are not Stein fillable. This implies the same result for all highly connected (8k1)–manifolds which admit almost contact structures. The proofs rely on a new number-theoretic result about Bernoulli numbers.

Stein fillability, surgery, contact structures, bordism theory
Mathematical Subject Classification 2010
Primary: 32E10
Secondary: 57R17, 57R65
Received: 28 October 2014
Revised: 23 February 2015
Accepted: 28 March 2015
Published: 20 October 2015
Proposed: Yasha Eliashberg
Seconded: Peter Teichner, Simon Donaldson
Jonathan Bowden
Ludwig-Maximillians Universität
Mathemathisches Institut
Theresienstr. 39
D-80333 Munich
Diarmuid Crowley
Institute of Mathematics
University of Aberdeen
Aberdeen AB24 3UE
András I Stipsicz
Rényi Institute of Mathematics
Hungarian Academy of Sciences
Réaltanoda utca 13-15
Budapest, H-1053
Bernd C Kellner
Universität Göttingen
Mathematisches Institut
D-37073 Göttingen