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

Anar Akhmedov; Sümeyra Sakalll

doi:10.1142/S0129167X19500174

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

Anar Akhmedov, Sümeyra Sakalll

School of Mathematics

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2 Scopus citations

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 language	English (US)
Article number	1950017
Journal	International Journal of Mathematics
Volume	30
Issue number	3
DOIs	https://doi.org/10.1142/S0129167X19500174
State	Published - 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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10.1142/S0129167X19500174

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title = "Deformation of singular fibers of genus two fibrations and small exotic symplectic 4 -manifolds",

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.",

keywords = "Lefschetz fibration, Symplectic 4-manifold, genus two fibration, mapping class group, rational blowdown",

author = "Anar Akhmedov and S{\"u}meyra Sakalll",

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{\textquoteright}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: {\textcopyright} 2019 World Scientific Publishing Company.",

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AU - Akhmedov, Anar

AU - Sakalll, Sümeyra

N1 - 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.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - 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.

AB - 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.

KW - Lefschetz fibration

KW - Symplectic 4-manifold

KW - genus two fibration

KW - mapping class group

KW - rational blowdown

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DO - 10.1142/S0129167X19500174

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SN - 0129-167X

VL - 30

JO - International Journal of Mathematics

JF - International Journal of Mathematics

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M1 - 1950017

ER -

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

Abstract

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