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Connected sums of unstabilized Heegaard splittings are
unstabilized
David Bachman
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Geometry & Topology 12 (2008)
2327–2378
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Abstract
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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.
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Keywords
Heegaard splitting, connected sum,
incompressible surface
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Mathematical Subject Classification
Primary: 57N10
Secondary: 57M27
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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
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