Suppose $M_1, M_2$ are two closed three manifolds, $g(M_i)$ denote their Heegaard genus, is it true that $g(M_1\#M_2)=g(M_1)+g(M_2)?$ It is true that $g(M_1\#M_2)\leq g(M_1)+g(M_2),$ but I can't figure out how a sphere in $M_1\#M_2$ intersects the handlebodies of a splitting of $M_1\#M_2$, so I don't know if the inequality can be transformed into an equality. Appreciation for any comment!
2026-03-28 14:44:33.1774709073
Additivity of Heegaard genus under connected sum.
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