Lantern relations and rational blowdowns
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- by Hisaaki Endo and Yusuf Z. Gurtas PDF
- Proc. Amer. Math. Soc. 138 (2010), 1131-1142 Request permission
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
- J. Amorós, F. Bogomolov, L. Katzarkov, and T. Pantev, Symplectic Lefschetz fibrations with arbitrary fundamental groups, J. Differential Geom. 54 (2000), no. 3, 489–545. With an appendix by Ivan Smith. MR 1823313, DOI 10.4310/jdg/1214339791
- M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), no. 1, 135–206 (German). Das arithmetische Feld auf Flächen. MR 1555438, DOI 10.1007/BF02547712
- 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.
- H. Endo and Y. Z. Gurtas, Positive Dehn twist expression for a $\mathbb {Z}_3$ action on $\Sigma _g$, preprint, arXiv:0808.0752.
- Hisaaki Endo and Seiji Nagami, Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3179–3199. MR 2135741, DOI 10.1090/S0002-9947-04-03643-8
- Ronald Fintushel and Ronald J. Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. MR 1484044
- Sylvain Gervais, Presentation and central extensions of mapping class groups, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3097–3132. MR 1327256, DOI 10.1090/S0002-9947-96-01509-7
- Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
- Dennis L. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), no. 1, 119–125. MR 529227, DOI 10.1090/S0002-9939-1979-0529227-4
- D. Kotschick, J. W. Morgan, and C. H. Taubes, Four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants, Math. Res. Lett. 2 (1995), no. 2, 119–124. MR 1324695, DOI 10.4310/MRL.1995.v2.n2.a1
- Feng Luo, A presentation of the mapping class groups, Math. Res. Lett. 4 (1997), no. 5, 735–739. MR 1484704, DOI 10.4310/MRL.1997.v4.n5.a11
- Makoto Matsumoto, A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities, Math. Ann. 316 (2000), no. 3, 401–418. MR 1752777, DOI 10.1007/s002080050336
- Yukio Matsumoto, Lefschetz fibrations of genus two—a topological approach, Topology and Teichmüller spaces (Katinkulta, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 123–148. MR 1659687
- Burak Ozbagci, Signatures of Lefschetz fibrations, Pacific J. Math. 202 (2002), no. 1, 99–118. MR 1883972, DOI 10.2140/pjm.2002.202.99
- Ivan Smith, Lefschetz pencils and divisors in moduli space, Geom. Topol. 5 (2001), 579–608. MR 1833754, DOI 10.2140/gt.2001.5.579
- Clifford Henry Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809–822. MR 1306023, DOI 10.4310/MRL.1994.v1.n6.a15
- Clifford Henry Taubes, Counting pseudo-holomorphic submanifolds in dimension $4$, J. Differential Geom. 44 (1996), no. 4, 818–893. MR 1438194
- Clifford Henry Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Lett. 2 (1995), no. 2, 221–238. MR 1324704, DOI 10.4310/MRL.1995.v2.n2.a10
- Michael Usher, Minimality and symplectic sums, Int. Math. Res. Not. , posted on (2006), Art. ID 49857, 17. MR 2250015, DOI 10.1155/IMRN/2006/49857
- Kouichi Yasui, Elliptic surfaces without 1-handles, J. Topol. 1 (2008), no. 4, 857–878. MR 2461858, DOI 10.1112/jtopol/jtn026
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
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1131-1142
- MSC (2010): Primary 57R17; Secondary 57N13, 20F38
- DOI: https://doi.org/10.1090/S0002-9939-09-10128-4
- MathSciNet review: 2566578