A necessary and sufficient condition for a Heegaard splitting to be keen weakly reducible

Abstract

In this work, we first present a necessary and sufficient condition to closed orientable 3-manifold admitting keen weakly reducible Heegaard splitting. By an example, we show that condition is not suitable for compact orientable manifold with non-sphere boundary component. At last, we give a necessary and sufficient condition for compact orientable 3-manifold to have a keen weakly reducible Heegaard splitting.