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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations


Authors: Hisaaki Endo and Seiji Nagami
Journal: Trans. Amer. Math. Soc. 357 (2005), 3179-3199
MSC (2000): Primary 57R20; Secondary 57N13, 20F05, 14D06
Published electronically: September 2, 2004
MathSciNet review: 2135741
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Abstract: We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus $3, 4$ and $5$ which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.


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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.wani.osaka-u.ac.jp

Seiji Nagami
Affiliation: 3-6-3-10 Sakuranchou, Toyonaka, Osaka 560-0054, Japan
Email: nagami-s@est.hi-ho.ne.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03643-8
PII: S 0002-9947(04)03643-8
Keywords: Signature, mapping class group, Lefschetz fibration, relation, signature cocycle, slope
Received by editor(s): November 16, 2003
Published electronically: September 2, 2004
Dedicated: Dedicated to Professor Yukio Matsumoto for his 60th birthday
Article copyright: © Copyright 2004 American Mathematical Society