Alternate Heegaard genus bounds distance

Scharlemann, Martin¹ ; Tomova, Maggy²

¹ Mathematics Department, University of California, Santa Barbara, CA 93106, USA
² Mathematics Department, University of Iowa, Iowa City, IA 52242, USA

Geometry & topology, Tome 10 (2006) no. 1, pp. 593-617.

Résumé

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

Détail

DOI : 10.2140/gt.2006.10.593

Keywords: Heegaard splitting, Heegaard distance, strongly irreducible, handlebody, weakly incompressible

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