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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lantern relations and rational blowdowns


Authors: Hisaaki Endo and Yusuf Z. Gurtas
Journal: Proc. Amer. Math. Soc. 138 (2010), 1131-1142
MSC (2010): Primary 57R17; Secondary 57N13, 20F38
Published electronically: October 26, 2009
MathSciNet review: 2566578
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Abstract: We discuss a connection between the lantern relation in mapping class groups and the rational blowing down process for $ 4$-manifolds. More precisely, if we change a positive relator in Dehn twist generators of the mapping class group by using a lantern relation, the corresponding Lefschetz fibration changes into its rational blowdown along a copy of the configuration $ C_2$. We exhibit examples of such rational blowdowns of Lefschetz fibrations whose blowup is homeomorphic but not diffeomorphic to the original fibration.


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Additional Information

Hisaaki Endo
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: endo@math.sci.osaka-u.ac.jp

Yusuf Z. Gurtas
Affiliation: Department of Mathematics, DePauw University, 602 S. College Avenue, Greencastle, Indiana 46135
Address at time of publication: Department of Mathematics and Computer Science, Queensborough Community College–CUNY, 222-05 56th Avenue, Room S-245, Bayside, New York 11364
Email: yusufgurtas@depauw.edu, ygurtas@qcc.cuny.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10128-4
PII: S 0002-9939(09)10128-4
Keywords: 4-manifold, mapping class group, symplectic topology, Lefschetz fibration, lantern relation, rational blowdown
Received by editor(s): November 21, 2008
Received by editor(s) in revised form: July 20, 2009
Published electronically: October 26, 2009
Additional Notes: The first author is partially supported by Grant-in-Aid for Scientific Research (No. 21540079), Japan Society for the Promotion of Science.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2009 American Mathematical Society