#### Volume 12, issue 4 (2008)

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Connected sums of unstabilized Heegaard splittings are unstabilized

### David Bachman

Geometry & Topology 12 (2008) 2327–2378
##### 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
Primary: 57N10
Secondary: 57M27