Connected sums of unstabilized Heegaard splittings are unstabilized
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 either H1 or H2 comes from stabilizing a lower genus splitting. If Hi and Gi are non-isotopic Heegaard surfaces in Mi, and Hi is unstabilized, then we show H1 # H2 is not isotopic to G1 # G2 in M1 # M2. The former result answers a question of C. Gordon (Problem 3.91 from Kirby's problem list).
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