Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. J. Amorós, F. Bogomolov, L. Katzarkov, and T. Pantev, Symplectic Lefschetz fibrations with arbitrary fundamental groups, J. Diff. Geom. 54 (2000), 489-545. MR 1823313 (2002g:57051)
  • 2. M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135-206. MR 1555438
  • 3. H. Endo, A generalization of Chakiris' fibrations, Groups of Diffeomorphisms, Advanced Studies in Pure Mathematics, 52, Mathematical Society of Japan, Tokyo, 2008, pp. 251-282.
  • 4. H. Endo and Y. Z. Gurtas, Positive Dehn twist expression for a $ \mathbb{Z}_3$ action on $ \Sigma_g$, preprint, arXiv:0808.0752.
  • 5. H. Endo and S. Nagami, Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations, Trans. Amer. Math. Soc. 357 (2005), 3179-3199. MR 2135741 (2006g:57051)
  • 6. R. Fintushel and R. J. Stern, Rational blowdowns of $ 4$-manifolds, J. Diff. Geom. 46 (1997), 181-235. MR 1484044 (98j:57047)
  • 7. S. Gervais, Presentation and central extensions of mapping class groups, Trans. Amer. Math. Soc. 348 (1996), 3097-3132. MR 1327256 (96j:57016)
  • 8. R. E. Gompf and A. I. Stipsicz, $ 4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20, American Mathematical Society, Providence, RI, 1999. MR 1707327 (2000h:57038)
  • 9. D. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), 119-125. MR 529227 (80h:57008)
  • 10. D. Kotschick, J. W. Morgan, and C. H. Taubes, Four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants, Math. Res. Letters 2 (1995), 119-124. MR 1324695 (96i:57024)
  • 11. F. Luo, A presentation of mapping class groups, Math. Res. Lett. 4 (1997), 735-739. MR 1484704 (99b:57031)
  • 12. M. Matsumoto, A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities, Math. Ann. 316 (2000), 401-418. MR 1752777 (2001e:57002)
  • 13. Y. Matsumoto, Lefschetz fibrations of genus two - a topological approach, Proceedings of the 37th Taniguchi Symposium on ``Topology and Teichmüller Spaces'', World Scientific, Singapore, 1996, pp. 123-148. MR 1659687 (2000h:14038)
  • 14. B. Ozbagci, Signatures of Lefschetz fibrations, Pacific J. Math. 202 (2002), 99-118. MR 1883972 (2002k:57066)
  • 15. I. Smith, Lefschetz pencils and divisors in moduli space, Geometry $ \&$ Topology 5 (2001), 579-608. MR 1833754 (2002f:57056)
  • 16. C. H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Letters 1 (1994), 809-822. MR 1306023 (95j:57039)
  • 17. C. H. Taubes, Counting pseudo-holomorphic submanifolds in dimension $ 4$, J. Diff. Geom. 44 (1996), 818-893. MR 1438194 (97k:58029)
  • 18. C. H. Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Letters 2 (1995), 221-238. MR 1324704 (96a:57076)
  • 19. M. Usher, Minimality and symplectic sums, Int. Math. Res. Not. 2006, Art. ID 49857, 17 pp. MR 2250015 (2007h:53139)
  • 20. K. Yasui, Elliptic surfaces without $ 1$-handles, Journal of Topology 1 (2008), 857-878. MR 2461858 (2009i:57065)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57R17, 57N13, 20F38

Retrieve articles in all journals with MSC (2010): 57R17, 57N13, 20F38

Additional Information

Hisaaki Endo
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

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

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

American Mathematical Society