Let be a compact, orientable, –irreducible –manifold and be a connected closed essential surface in with which cuts into and . In the present paper, we show the following theorem: Suppose that there is no essential surface with boundary in satisfying , . Then . As a consequence, we further show that if has a Heegaard splitting with distance , , then .
The main results follow from a new technique which is a stronger version of Schultens’ Lemma.
"Amalgamations of Heegaard splittings in $3$–manifolds without some essential surfaces." Algebr. Geom. Topol. 9 (4) 2041 - 2054, 2009. https://doi.org/10.2140/agt.2009.9.2041