The genus two Goeritz group of $\mathbb{S}^2 \times \mathbb{S}^1$

The genus-$g$ Goeritz group is the group of isotopy classes of orientation-preserving homeomorphisms of a closed orientable $3$-manifold that preserve a given genus-$g$ Heegaard splitting of the manifold. In this work, we show that the genus-$2$ Goeritz group of $\mathbb{S}^2 \times \mathbb{S}^1$ is finitely presented, and give its explicit presentation.

2010 Mathematics Subject Classification

57M60, 57N10

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Accepted 5 January 2014

Published 13 October 2014