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Let $S$ be a closed connected orientable surface with $g(S)>0$. Jennifer Schultens, in her paper ``The Classification of Heegaard Splittings for (Compact Orientable Surface)$\times S^1$'', proves that $S\times S^1$ does not admit any strongly irreducible Heegaard splitting. My questions are:

  1. Are there other (i.e. non-trivial) closed circle bundles which admit no strongly irreducible Heegaard splittings? Furthermore, is there a classification of closed circle bundles which admit no strongly irreducible Heegaard splittings?
  2. Are there other surface bundles which admit no strongly irreducible Heegaard splittings?
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  1. The only circle bundles with strongly irreducible Heegaard splittings are those with Euler class $\pm 1$. This follows from Corollary 0.5 and Theorem 2.6 of

Moriah, Yoav; Schultens, Jennifer, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal,(http://dx.doi.org/10.1016/S0040-9383(97)00072-4), Topology 37, No. 5, 1089-1112 (1998). ZBL0926.57016.

  1. Cooper and Scharlemann classified Heegaard splittings of solv manifolds. In particular, Proposition 3.1 of the paper implies that any irreducible Heegaard splitting of genus 3 of a solv 3-manifold is weakly reducible. Together with Theorem 4.2 of the paper, which classifies the solv manifolds that have genus 2 splittings (which are strongly irreducible), this shows that "most" solv manifolds have no strongly irreducible Heegaard splittings.

Cooper, Daryl; Scharlemann, Martin, The structure of a solvmanifold’s Heegaard splittings, Turk. J. Math. 23, No. 1, 1-18 (1999). ZBL0948.57015.

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