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Transactions of the American Mathematical Society

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The mapping class groups of reducible Heegaard splittings of genus two


Authors: Sangbum Cho and Yuya Koda
Journal: Trans. Amer. Math. Soc. 371 (2019), 2473-2502
MSC (2010): Primary 57N10, 57M60
DOI: https://doi.org/10.1090/tran/7375
Published electronically: October 23, 2018
MathSciNet review: 3896087
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Abstract: A $ 3$-manifold which admits a genus-$ 2$ reducible Heegaard splitting is one of the $ 3$-sphere, $ \mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-$ 2$ splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces and show that the mapping class group of the genus-$ 2$ Heegaard splitting is finitely presented for any lens space by giving its explicit presentation. As an application, we show that the fundamental groups of the spaces of the genus-$ 2$ Heegaard splittings of lens spaces are all finitely presented.


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Additional Information

Sangbum Cho
Affiliation: Department of Mathematics Education, Hanyang University, Seoul 133-791, Republic of Korea
Email: scho@hanyang.ac.kr

Yuya Koda
Affiliation: Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi- Hiroshima, 739-8526, Japan
Email: ykoda@hiroshima-u.ac.jp

DOI: https://doi.org/10.1090/tran/7375
Received by editor(s): February 21, 2017
Received by editor(s) in revised form: July 12, 2017, and August 21, 2017
Published electronically: October 23, 2018
Additional Notes: The first-named author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (NRF-2015R1A1A1A05001071), and by the Ministry of Education (NRF-201800000001768).
The second author was supported by JSPS KAKENHI Grant Numbers 15H03620, 17K05254, 17H06463, and JST CREST Grant Number JPMJCR17J4.
Article copyright: © Copyright 2018 American Mathematical Society