The topology of Stein fillable manifolds in high dimensions. II (with an appendix by Bernd C. Kellner). (English) Zbl 1380.32016

Bowden, Jonathan; Crowley, Diarmuid; Stipsicz, András I.

Geom. Topol. 19, No. 5, 2995-3030 (2015).

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
57R17	Symplectic and contact topology in high or arbitrary dimension
57R65	Surgery and handlebodies

Keywords:

Stein fillability; surgery; contact structures; bordism theory

Citations:

Zbl 1314.53134

References:

[1]	H J Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics 628, Springer (1977) · Zbl 0361.55017
[2]	M S Borman, Y Eliashberg, E Murphy, Existence and classification of overtwisted contact structures in all dimensions, · Zbl 1344.53060
[3]	R Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959) 313 · Zbl 0129.15601
[4]	F Bourgeois, Odd-dimensional tori are contact manifolds, Int. Math. Res. Not. 2002 (2002) 1571 · Zbl 1021.53055
[5]	J Bowden, D Crowley, A I Stipsicz, The topology of Stein fillable manifolds in high dimensions, III · Zbl 1314.53134
[6]	J Bowden, D Crowley, A I Stipsicz, Contact structures on \(M\times S^2\), Math. Ann. 358 (2014) 351 · Zbl 1286.57022
[7]	J Bowden, D Crowley, A I Stipsicz, The topology of Stein fillable manifolds in high dimensions, I, Proc. Lond. Math. Soc. 109 (2014) 1363 · Zbl 1314.53134
[8]	L Carlitz, Kummer’s congruence for the Bernoulli numbers, Portugal. Math. 19 (1960) 203 · Zbl 0095.03004
[9]	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) · Zbl 1262.32026
[10]	Y Eliashberg, Filling by holomorphic discs and its applications (editors S K Donaldson, C B Thomas), London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press (1990) 45 · Zbl 0731.53036
[11]	Y Eliashberg, Topological characterization of Stein manifolds of dimension \(> 2\), Internat. J. Math. 1 (1990) 29 · Zbl 0699.58002
[12]	J B Etnyre, K Honda, On symplectic cobordisms, Math. Ann. 323 (2002) 31 · Zbl 1022.53059
[13]	H Geiges, Contact structures on \((n-1)\)-connected \((2n+1)\)-manifolds, Pacific J. Math. 161 (1993) 129 · Zbl 0791.53039
[14]	H Geiges, Applications of contact surgery, Topol. 36 (1997) 1193 · Zbl 0912.57019
[15]	H Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics 109, Cambridge Univ. Press (2008) · Zbl 1153.53002
[16]	H Geiges, K Zehmisch, How to recognize a \(4\)-ball when you see one, Münster J. Math. 6 (2013) 525 · Zbl 1300.53075
[17]	P Ghiggini, K Niederkrüger, C Wendl, Subcritical contact surgeries and the topology of symplectic fillings, · Zbl 1369.57029
[18]	F Hirzebruch, Topological methods in algebraic geometry, Grundl. Math. Wissen. 131, Springer (1966) · Zbl 0138.42001
[19]	D Husemoller, Fibre bundles, Graduate Texts in Mathematics 20, Springer (1994) · Zbl 0202.22903
[20]	K Ireland, M Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics 84, Springer (1990) · Zbl 0712.11001
[21]	M A Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960) 161 · Zbl 0105.35302
[22]	M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504 · Zbl 0115.40505
[23]	W S Massey, Obstructions to the existence of almost complex structures, Bull. Amer. Math. Soc. 67 (1961) 559 · Zbl 0192.29601
[24]	P Massot, K Niederkrüger, C Wendl, Weak and strong fillability of higher dimensional contact manifolds, Invent. Math. 192 (2013) 287 · Zbl 1277.57026
[25]	J Milnor, Lectures on the \(h\)-cobordism theorem, Princeton Univ. Press (1965) · Zbl 0161.20302
[26]	J Milnor, E Spanier, Two remarks on fiber homotopy type, Pacific J. Math. 10 (1960) 585 · Zbl 0109.16301
[27]	J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974) · Zbl 0298.57008
[28]	S Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. 74 (1961) 391 · Zbl 0099.39202
[29]	S Smale, On the structure of \(5\)-manifolds, Ann. of Math. 75 (1962) 38 · Zbl 0101.16103
[30]	R E Stong, Notes on cobordism theory, Princeton Univ. Press (1968) · Zbl 0181.26604
[31]	C T C Wall, Classification of \((n-1)\)-connected \(2n\)-manifolds, Ann. of Math. 75 (1962) 163 · Zbl 0218.57022
[32]	A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241 · Zbl 0737.57012
[33]	H Whitney, The self-intersections of a smooth \(n\)-manifold in \(2n\)-space, Ann. of Math. 45 (1944) 220 · Zbl 0063.08237
[34]	H Yang, Almost complex structures on \((n-1)\)-connected \(2n\)-manifolds, Topology Appl. 159 (2012) 1361 · Zbl 1234.55003