The mapping class groups of reducible Heegaard splittings of genus two

Cho, Sangbum; Koda, Yuya

doi:10.1090/tran/7375

The mapping class groups of reducible Heegaard splittings of genus two
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by Sangbum Cho and Yuya Koda PDF

Trans. Amer. Math. Soc. 371 (2019), 2473-2502 Request permission

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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