For any genus-three reducible Heegaard splitting of a connected sum of two lens spaces, the Goeritz group is finitely generated and the reducing sphere complex is connected.
Strong Haken via Sphere Complexes
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abstract
We give a short proof of Scharlemann's Strong Haken Theorem for closed $3$-manifolds (and manifolds with spherical boundary). As an application, we also show that given a decomposing sphere $R$ for a $3$-manifold $M$ that splits off an $S^2 \times S^1$ summand, any Heegaard splitting of $M$ restricts to the standard Heegaard splitting on the summand.
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Genus three Goeritz groups of connected sums of two lens spaces
For any genus-three reducible Heegaard splitting of a connected sum of two lens spaces, the Goeritz group is finitely generated and the reducing sphere complex is connected.