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Nielsen equivalence and trisections

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Abstract

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every \(k \ge 2\), we exhibit infinitely many manifolds with \(2^{k}-1\) non-diffeomorphic (3k, k)-trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier’s spun trisections. The technique used to distinguish the trisections parallels a common setup for distinguishing Heegaard splittings. In particular, we show that the Nielsen classes of the generators of the fundamental group, obtained from spines of the 4-dimensional 1-handlebodies of the trisection, are isotopy invariants of the trisection. If we additionally consider the action of the automorphism group on the Nielsen classes we obtain diffeomorphism invariants. Once the invariance is established, we analyze the Nielsen classes of the spun trisections in terms of the Nielsen classes of the original Heegaard splitting, and leverage work of Lustig, Moriah, and Rosenberger on Nielsen classes of Fuchsian groups in order to distinguish the trisections.

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References

  1. Baykur, R. I., Saeki, O.: Simplifying indefinite fibrations on 4-manifolds. ArXiv e-prints (2017)

  2. Boileau, M., Collins, D.J., Zieschang, H.: Genus \(2\) Heegaard decompositions of small Seifert manifolds. Ann. Inst. Fourier (Grenoble) 41(4), 1005–1024 (1991)

    Article  MathSciNet  Google Scholar 

  3. Boileau, M., Zieschang, H.: Heegaard genus of closed orientable Seifert \(3\)-manifolds. Invent. Math. 76(3), 455–468 (1984)

    Article  MathSciNet  Google Scholar 

  4. Bonhan, F., Otal, J.: Scindements de Heegaard des espaces lenticulaires. Ann. Sci. École Norm. Sup. (4), 16(3):451–466 (1984), 1983

  5. Cerf, J.: Sur les difféomorphismes de la sphère de dimension trois \((\Gamma _{4}=0)\). Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York (1968)

  6. Engmann, R.: Nicht-homöomorphe Heegaard-Zerlegungen vom Geschlecht \(2\) der zusammenhängenden Summe zweier Linsenräume. Abh. Math. Sem. Univ. Hamburg 35, 33–38 (1970)

    Article  MathSciNet  Google Scholar 

  7. Gay, D., Robion, K.: Trisecting 4-manifolds. Geom. Topol. 20(6), 3097–3132 (2016)

    Article  MathSciNet  Google Scholar 

  8. Gay, D.T.: Trisections of Lefschetz pencils. Algebr. Geom. Topol. 16(6), 3523–3531 (2016)

    Article  MathSciNet  Google Scholar 

  9. Hatcher, Allen E..: A proof of the Smale conjecture, \({\rm Diff}(S^{3})\simeq {\rm O}(4)\). Ann. of Math. (2) 117(3), 553–607 (1983)

  10. Hayano, K.: On diagrams of simplified trisections and mapping class groups. ArXiv e-prints (2017)

  11. Johnson, J.: Horizontal Heegaard splittings of Seifert fibered spaces. Pacific J. Math. 245(1), 151–165 (2010)

    Article  MathSciNet  Google Scholar 

  12. Koenig, D.: Trisections of 3-manifold bundles over \(S^1\). ArXiv e-prints (2017)

  13. Laudenbach, F.: Poénaru, Valentin: a note on \(4\)-dimensional handlebodies. Bull. Soc. Math. France 100, 337–344 (1972)

    Article  MathSciNet  Google Scholar 

  14. Lustig, M., Moriah, Y.: Nielsen equivalence in Fuchsian groups and Seifert fibered spaces. Topology 30(2), 191–204 (1991)

    Article  MathSciNet  Google Scholar 

  15. Lustig, M., Moriah, Y., Rosenberger, G.: Automorphisms of Fuchsian groups and their lifts to free groups. Canad. J. Math. 41(1), 123–131 (1989)

    Article  MathSciNet  Google Scholar 

  16. Meier, J.: Trisections and spun four-manifolds. Math. Res. Lett. 25(5), 1497–1524 (2018)

    Article  MathSciNet  Google Scholar 

  17. Meier, J., Schirmer, T., Zupan, A.: Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc. 144(11), 4983–4997 (2016)

    Article  MathSciNet  Google Scholar 

  18. Moriah, Y., Schultens, J.: Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal. Topology 37(5), 1089–1112 (1998)

    Article  MathSciNet  Google Scholar 

  19. Nielsen, J.: Die Isomorphismengruppe der freien Gruppen. Math. Ann. 91(3–4), 169–209 (1924)

    Article  MathSciNet  Google Scholar 

  20. Reidemeister, Kurt: Zur dreidimensionalen Topologie. Abh. Math. Sem. Univ. Hamburg 9(1), 189–194 (1933)

    Article  MathSciNet  Google Scholar 

  21. Rosenberger, G.: Automorphismen ebener diskontinuierlicher Gruppen. In Riemann surfaces and related topics. In: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 439–455. Princeton Univ. Press, Princeton, N.J (1981)

  22. Rubinstein, J. Hyam., Tillmann, Stephan: Multisections of piecewise linear manifolds. Indiana Univ. Math. J. 69(6), 2209–2239 (2020)

  23. Scharlemann, Martin: Thompson, Abigail: Heegaard splittings of \(({\rm surface})\times I\) are standard. Math. Ann. 295(3), 549–564 (1993)

    Article  MathSciNet  Google Scholar 

  24. Singer, James: Three-dimensional manifolds and their Heegaard diagrams. Trans. Amer. Math. Soc. 35(1), 88–111 (1933)

    Article  MathSciNet  Google Scholar 

  25. Waldhausen, Friedhelm: Heegaard-Zerlegungen der \(3\)-Sphäre. Topology 7, 195–203 (1968)

    Article  MathSciNet  Google Scholar 

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  1. University of Waterloo, Waterloo, ON, Canada

    Gabriel Islambouli

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  1. Gabriel Islambouli
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Correspondence to Gabriel Islambouli.

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Islambouli, G. Nielsen equivalence and trisections. Geom Dedicata 214, 303–317 (2021). https://doi.org/10.1007/s10711-021-00617-y

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