Let and be closed, orientable 3–manifolds. Let denote a Heegaard surface in . We prove that if comes from stabilizing a lower genus splitting of then one of or 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.
"Connected sums of unstabilized Heegaard splittings are unstabilized." Geom. Topol. 12 (4) 2327 - 2378, 2008. https://doi.org/10.2140/gt.2008.12.2327