Alternate Heegaard genus bounds distance

Scharlemann, Martin; Tomova, Maggy

doi:10.2140/gt.2006.10.593

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Alternate Heegaard genus bounds distance

Martin Scharlemann and Maggy Tomova

Geometry & Topology 10 (2006) 593–617

DOI: 10.2140/gt.2006.10.593

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 $d (P) \leq 2 genus (Q)$ .

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) $d (P) \leq 2 genus (Q)$ .

Keywords

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

Mathematical Subject Classification 2000

Primary: 57N10

Secondary: 57M50

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Publication

Received: 1 June 2005

Accepted: 27 March 2006

Published: 4 May 2006

Proposed: David Gabai

Seconded: Joan Birman, Cameron Gordon

Authors

Martin Scharlemann
Mathematics Department University of California Santa Barbara, CA 93106 USA

Maggy Tomova
Mathematics Department University of Iowa Iowa City, IA 52242 USA

Volume 10, issue 1 (2006)