A Sufficient Condition for the Genus of an Annulus Sum of Two 3-Manifolds to Be Non-Degenerate (Communications in Mathematical Research, 2021)
Let $M_i$ be a compact orientable 3-manifold, and $A_i$ a non-separatingincompressible annulus on a component of $∂M_i$, say $F_i, i = 1, 2$. Let $h : A_1 → A_2$ be a homeomorphism, and $M = M_1 ∪_h M_2$, the annulus sum of $M_1$ and $M_2$ along $A_1$ and $A_2$. Suppose that $M_i$ has a Heegaard splitting $V_i ∪_{S_i} W_i$ with distance $d(S_i) ≥ 2g(M_i) + 2g(F_{3−i}) + 1, i = 1, 2$. Then $g(M) = g(M_1) + g(M_2)$, and theminimal Heegaard splitting of $M$ is unique, which is the natural Heegaard splittingof $M$ induced from $V_1 ∪_{S_1} W_1$ and $V_2 ∪_{S_2} W_2$.
Keywords: Heegaard genus; annulus sum; distance.