Connected sums of unstabilized Heegaard splittings are unstabilized

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.