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Lefschetz fibrations and Torelli groups

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Abstract

For each \(g \ge 3\) and \(h \ge 2\), we explicitly construct (1) fiber sum indecomposable relatively minimal genus \(g\) Lefschetz fibrations over genus \(h\) surfaces whose monodromies lie in the Torelli group, (2) genus \(g\) Lefschetz fibrations over genus \(h\) surfaces that are not fiber sums of holomorphic ones, and (3) fiber sum indecomposable genus \(g\) surface bundles over genus \(h\) surfaces whose monodromies are in the Torelli group (provided \(g \ge 4\)). The last result amounts to finding explicit irreducible embeddings of surface groups into Torelli groups; in fact we find such embeddings into arbitrary terms of the Johnson filtration.

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Notes

  1. Mustafa Korkmaz pointed out to us another construction of surface subgroups of the Torelli group, obtained by doubling the monodromies induced from including the unit tangent bundle of the surface into the mapping class group as in our proof of Theorem 1.1.

  2. A technical point we will suppress is that the monodromy is really an anti-homomorphism, since elements of \(\pi _1(\Sigma \setminus \{f(p_i)\})\) are written left-to-right and elements of \({{\mathrm{Mod}}}(F)\) are written right-to-left.

  3. Again, we have the situation where \(\mathcal {P}ush\) is an anti-homomorphism.

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Acknowledgments

We would like to thank Benson Farb, Sang-hyun Kim, and an anonymous referee for helpful comments and conversations.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA

    R. İnanç Baykur

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

    Dan Margalit

Authors
  1. R. İnanç Baykur
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  2. Dan Margalit
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Correspondence to R. İnanç Baykur.

Additional information

The first author was partially supported by the NSF grant DMS-0906912. The second author was partially supported by an NSF CAREER grant and a fellowship from the Sloan Foundation.

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Baykur, R.İ., Margalit, D. Lefschetz fibrations and Torelli groups. Geom Dedicata 177, 275–291 (2015). https://doi.org/10.1007/s10711-014-9989-8

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