Alternate Heegaard genus bounds distance

Abstract

Suppose $M$ is a compact orientable irreducible $3$–manifold with Heegaard splitting surfaces $P$ and $Q$. Then either $Q$ is isotopic to a possibly stabilized or boundary-stabilized copy of $P$ or the distance $d\left(P\right)\le 2genus\left(Q\right)$.

More generally, if $P$ and $Q$ are bicompressible but weakly incompressible connected closed separating surfaces in $M$ then either

(i) $P$ and $Q$ can be well-separated or

(ii) $P$ and $Q$ are isotopic or

(iii) $d\left(P\right)\le 2genus\left(Q\right)$.