Deformation of singular fibers of genus two fibrations and small exotic symplectic 4 -manifolds

Anar Akhmedov, Sümeyra Sakalll

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Abstract

We introduce the 2-nodal spherical deformation of certain singular fibers of genus two fibrations, and use such deformations to construct various examples of simply connected minimal symplectic 4-manifolds with small topology. More specifically, we construct new exotic minimal symplectic 4-manifolds homeomorphic but not diffeomorphic to ℂℙ 2 #6ℂℙ 2 , ℂℙ 2 #7ℂℙ 2 , and 3ℂℙ 2 #kℂℙ 2 for k = 16, 17, 18, 19 using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using g-nodal spherical deformations of certain singular fibers of genus g ≥ 3 fibrations.

Original languageEnglish (US)
Article number1950017
JournalInternational Journal of Mathematics
Volume30
Issue number3
DOIs
StatePublished - Mar 1 2019

Bibliographical note

Funding Information:
We would like to thank Ronald Stern for very useful discussions, dating back to first author's Ph.D. studies at University of California, Irvine. Our work is greatly motivated and inspired by some of these discussions. We also would like to thank Cagri Karakurt, Tian-Jun Li, and Sai-Kee Yeung for their interest in our work, and pointing out some references. The authors are also very grateful to the referee for a careful reading of the paper and many valuable comments and suggestions, which helped us to improve our manuscript. We have also greatly benefited from IPE drawing program (http://ipe.otfried.org) in drawing the various figures in this paper. A. Akhmedov was partially supported by NSF grants DMS-1065955, DMS-1005741, Sloan Research Fellowship, Simons Research Fellowship and Collaboration Grants for Mathematicians by Simons Foundation. S. Sakalli was partially supported by NSF grants DMS-1065955.

Funding Information:
We would like to thank Ronald Stern for very useful discussions, dating back to first author’s Ph.D. studies at University of California, Irvine. Our work is greatly motivated and inspired by some of these discussions. We also would like to thank Cagri Karakurt, Tian-Jun Li, and Sai-Kee Yeung for their interest in our work, and pointing out some references. The authors are also very grateful to the referee for a careful reading of the paper and many valuable comments and suggestions, which helped us to improve our manuscript. We have also greatly benefited from IPE drawing program (http://ipe.otfried.org) in drawing the various figures in this paper. A. Akhmedov was partially supported by NSF grants DMS-1065955, DMS-1005741, Sloan Research Fellowship, Simons Research Fellowship and Collaboration Grants for Mathematicians by Simons Foundation. S. Sakallı was partially supported by NSF grants DMS-1065955.

Publisher Copyright:
© 2019 World Scientific Publishing Company.

Keywords

  • Lefschetz fibration
  • Symplectic 4-manifold
  • genus two fibration
  • mapping class group
  • rational blowdown

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