Geometry & Topology

The topology of Stein fillable manifolds in high dimensions, II

Jonathan Bowden, Diarmuid Crowley, and András I Stipsicz

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Geom. Topol., Volume 19, Number 5 (2015), 2995-3030.

Received: 28 October 2014
Revised: 23 February 2015
Accepted: 28 March 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32E10: Stein spaces, Stein manifolds
Secondary: 57R17: Symplectic and contact topology 57R65: Surgery and handlebodies

Stein fillability surgery contact structures bordism theory


Bowden, Jonathan; Crowley, Diarmuid; Stipsicz, András I. The topology of Stein fillable manifolds in high dimensions, II. Geom. Topol. 19 (2015), no. 5, 2995--3030. doi:10.2140/gt.2015.19.2995.

Export citation


  • H J Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics 628, Springer, Berlin (1977)
  • M S Borman, Y Eliashberg, E Murphy, Existence and classification of overtwisted contact structures in all dimensions
  • R Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959) 313–337
  • F Bourgeois, Odd-dimensional tori are contact manifolds, Int. Math. Res. Not. 2002 (2002) 1571–1574
  • J Bowden, D Crowley, A I Stipsicz, The topology of Stein fillable manifolds in high dimensions, III in preparation
  • J Bowden, D Crowley, A I Stipsicz, Contact structures on $M\times S\sp 2$, Math. Ann. 358 (2014) 351–359
  • J Bowden, D Crowley, A I Stipsicz, The topology of Stein fillable manifolds in high dimensions, I, Proc. Lond. Math. Soc. 109 (2014) 1363–1401
  • L Carlitz, Kummer's congruence for the Bernoulli numbers, Portugal. Math. 19 (1960) 203–210
  • K Cieliebak, Y Eliashberg, From Stein to Weinstein and back: Symplectic geometry of affine complex manifolds, Amer. Math. Soc. Colloq. Publ. 59, Amer. Math. Soc. (2012)
  • Y Eliashberg, Filling by holomorphic discs and its applications, from: “Geometry of low-dimensional manifolds, 2”, (S K Donaldson, C B Thomas, editors), London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press (1990) 45–67
  • Y Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990) 29–46
  • J B Etnyre, K Honda, On symplectic cobordisms, Math. Ann. 323 (2002) 31–39
  • H Geiges, Contact structures on $(n-1)$–connected $(2n+1)$–manifolds, Pacific J. Math. 161 (1993) 129–137
  • H Geiges, Applications of contact surgery, Topol. 36 (1997) 1193–1220
  • H Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics 109, Cambridge Univ. Press (2008)
  • H Geiges, K Zehmisch, How to recognize a $4$–ball when you see one, Münster J. Math. 6 (2013) 525–554
  • P Ghiggini, K Niederkrüger, C Wendl, Subcritical contact surgeries and the topology of symplectic fillings
  • F Hirzebruch, Topological methods in algebraic geometry, Grundl. Math. Wissen. 131, Springer, New York (1966)
  • D Husemoller, Fibre bundles, 3rd edition, Graduate Texts in Mathematics 20, Springer, New York (1994)
  • K Ireland, M Rosen, A classical introduction to modern number theory, 2nd edition, Graduate Texts in Mathematics 84, Springer, New York (1990)
  • M A Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960) 161–169
  • M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504–537
  • W S Massey, Obstructions to the existence of almost complex structures, Bull. Amer. Math. Soc. 67 (1961) 559–564
  • P Massot, K Niederkrüger, C Wendl, Weak and strong fillability of higher dimensional contact manifolds, Invent. Math. 192 (2013) 287–373
  • J Milnor, Lectures on the $h$–cobordism theorem, Princeton Univ. Press (1965)
  • J Milnor, E Spanier, Two remarks on fiber homotopy type, Pacific J. Math. 10 (1960) 585–590
  • J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974)
  • S Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. 74 (1961) 391–406
  • S Smale, On the structure of $5$–manifolds, Ann. of Math. 75 (1962) 38–46
  • R E Stong, Notes on cobordism theory, Princeton Univ. Press (1968)
  • C T C Wall, Classification of $(n-1)$–connected $2n$–manifolds, Ann. of Math. 75 (1962) 163–189
  • A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241–251
  • H Whitney, The self-intersections of a smooth $n$–manifold in $2n$–space, Ann. of Math. 45 (1944) 220–246
  • H Yang, Almost complex structures on $(n-1)$–connected $2n$–manifolds, Topology Appl. 159 (2012) 1361–1368