# Circle bundles and surface bundles which admit no strongly irreducible Heegaard splittings

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?

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