#### Volume 10, issue 1 (2006)

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Alternate Heegaard genus bounds distance

### Martin Scharlemann and Maggy Tomova

Geometry & Topology 10 (2006) 593–617
 arXiv: math.GT/0501140
##### 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) $d\left(P\right)\le 2genus\left(Q\right)$.

##### Keywords
Heegaard splitting, Heegaard distance, strongly irreducible, handlebody, weakly incompressible
Primary: 57N10
Secondary: 57M50