Connected sums of unstabilized Heegaard splittings are unstabilized

Bachman, David

doi:10.2140/gt.2008.12.2327

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Connected sums of unstabilized Heegaard splittings are unstabilized

David Bachman

Geometry & Topology 12 (2008) 2327–2378

DOI: 10.2140/gt.2008.12.2327

Abstract

Let $M_{1}$ and $M_{2}$ be closed, orientable 3–manifolds. Let $H_{i}$ denote a Heegaard surface in $M_{i}$ . We prove that if $H_{1} # H_{2}$ comes from stabilizing a lower genus splitting of $M_{1} # M_{2}$ then one of $H_{1}$ or $H_{2}$ comes from stabilizing a lower genus splitting. This answers a question of C Gordon (Problem 3.91 from Kirby’s problem list). We also show that every unstabilized Heegaard splitting has a unique expression as the connected sum of Heegaard splittings of prime 3–manifolds.

Keywords

Heegaard splitting, connected sum, incompressible surface

Mathematical Subject Classification 2000

Primary: 57N10

Secondary: 57M27

References

Publication

Received: 23 November 2004

Revised: 3 May 2005

Accepted: 25 August 2008

Published: 15 September 2008

Proposed: David Gabai

Seconded: Colin Rourke, Joan Birman

Authors

David Bachman
Pitzer College 1050 N Mills Ave Claremont CA 91711 USA

Volume 12, issue 4 (2008)