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On Subcritically Stein Fillable 5-manifolds

Published online by Cambridge University Press:  20 November 2018

Fan Ding ,
Hansjörg Geiges  and
Guangjian Zhang
Fan Ding
Affiliation:
School of Mathematical Sciences and LMAM, Peking University, Beijing 100871, P. R. China, e-mail: dingfan@math.pku.edu.cn
Hansjörg Geiges
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 K?ln, Germany, e-mail: geiges@math.uni-koeln.de
Guangjian Zhang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China, e-mail: zhangguangjian8888@163.com
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Abstract

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We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

Keywords

53D3532Q2857M2057Q1057R17subcritically Stein fillable5-manifoldalmost contact structurethickening
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 1 , 01 March 2018 , pp. 85 - 96
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Barden, D., Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. http://dx.doi.Org/10.2307/1970702.Google Scholar
[2] Barth, K., Geiges, H. and Zehmisch, K., The diffeomorphism type of symplectic fillings, arXiv:1607.03310.Google Scholar
[3] Bowden, J., Crowley, D. and Stipsicz, A. I., The topology of Stein fillable manifolds in high dimensions I, Proc. London Math. Soc. (3) 109 (2014), 1363–1401. http://dx.doi.org/10.1112/plms/pdu028.Google Scholar
[4] Cieliebak, K. and Eliashberg, Ya., From Stein to Weinstein and Back - Symplectic geometry of affine complex manifolds, Amer. Math. Soc. Colloq. Publ. 59 (American Mathematical Society, Providence, RI, 2012).Google Scholar
[5] Ding, F., Geiges, H. and Van Koert, O., Diagrams for contact 5-manifolds, J. London Math. Soc. (2) 86 (2012), 657–682. http://dx.doi.Org/10.1112/jlms/jdsO2OGoogle Scholar
[6] Geiges, H., An Introduction to Contact Topology, Cambridge Stud. Adv. Math. 109 (Cambridge University Press, Cambridge, 2008).Google Scholar
[7] Hambleton, I. and Kreck, M., Cancellation of lattices and finite two-complexes, J. Reine Angew. Math. 442 (1993), 91–109.Google Scholar
[8] Hamilton, M., On symplectic 4-manifolds and contact 5-manifolds, Ph.D. thesis, LMU München (2008); available at https ://edoc.üb.uni-muenchen.de/8779/.Google Scholar
[9] Kervaire, M. A. and Milnor, J. W., Groups of homotopy spheres I, Ann. of Math. (2) 77 (1963), 504–537. http://dx.doi.org/10.2307/197012 8.Google Scholar
[10] Kosinski, A. A., Differential Manifolds, Pure Appl. Math. 138 (Academic Press, Boston, MA, 1993).Google Scholar
[11] Mazur, B., Differential topology from the point of view of simple homotopy theory, Inst. Hautes Etudes Sei. Publ. Math. 15 (1963), 5–93.Google Scholar
[12] Milnor, J., Lectures on the h-Cobordism Theorem (Princeton University Press, 1965).Google Scholar
[13] Milnor, J., Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. http://dx.doi.org/10.1090/S0002-9904-1966-11484-2.Google Scholar
[14] Oancea, A. and Viterbo, C., On the topology of fillings of contact manifolds and applications, Comment. Math. Helv. 87 (2012), 41–69. http://dx.doi.Org/10.41 71/CMH/248.Google Scholar
[15] Steenrod, N., The Topology of Fibre Bundles, Princeton Math. Ser. 14 (Princeton University Press, 1951).Google Scholar
[16] Wall, C. T. C., Classification problems in differential topology - IV. Thickenings, Topology 5 (1966), 73–94. http://dx.doi.Org/10.1016/0040-9383(66)90005-XGoogle Scholar
[17] Wall, C. T. C., Classification problems in differential topology - V. On certain 6-manifolds, Invent. Math. 1 (1966), 355–374. http://dx.doi.Org/10.1007/BF01425407.Google Scholar
[18] Whitehead, G. W., Elements of Homotopy Theory, Grad. Texts in Math. 61 (Springer-Verlag, Berlin, 1978).Google Scholar