Extend Saeki’s sphere–torus Morse classification to higher-genus regular fibers

Determine whether Saeki’s Theorem 6.5 (restated as Theorem 3 in this paper), which characterizes 3-dimensional closed, connected, orientable manifolds admitting a Morse function whose regular level sets are disjoint unions of spheres S^2 and tori S^1 × S^1, extends to the case where the regular level sets are disjoint unions of closed connected surfaces of genus greater than 1; specifically, characterize all such 3-manifolds that admit a Morse function whose every regular fiber is a union of closed connected surfaces of genus at most g for some g > 1.

Background

Theorem 3 (citing Saeki 2006, Theorem 6.5) gives a complete classification of 3-dimensional closed, connected, orientable manifolds that admit a Morse function whose regular fibers are unions of spheres and tori; namely, such manifolds are connected sums of copies of S^1 × S^2 and lens spaces (Heegaard genus one manifolds), and conversely these manifolds admit such Morse functions.

Theorem 5 (citing Saeki’s Lemma 6.6) ensures that if a 3-manifold admits a Morse function whose regular fibers have genus at most g > 0, then there exists a simple such function. For g = 0 or 1 this yields SSTF (simple sphere–torus–fibered) Morse functions, which underpin the results proved here.

The authors explicitly note that it is unknown whether the classification of Theorem 3 persists when regular fibers are allowed to have higher genus, prompting the question of extending Saeki’s result beyond the sphere–torus case.