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Connected sums of unstabilized Heegaard splittings are unstabilized

David Bachman

Geometry & Topology 12 (2008) 2327–2378
Abstract

Let M1 and M2 be closed, orientable 3–manifolds. Let Hi denote a Heegaard surface in Mi. We prove that if H1#H2 comes from stabilizing a lower genus splitting of M1#M2 then one of H1 or H2 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