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Volume 26, Issue 1
A Sufficient Condition for the Genus of an Annulus Sum of Two 3-Manifolds to Be Non-Degenerate

Fengling Li & Fengchun Lei

Commun. Math. Res., 26 (2010), pp. 85-96.

Published online: 2021-05

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  • Abstract

Let $M_i$ be a compact orientable 3-manifold, and $A_i$ a non-separating incompressible 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 the minimal Heegaard splitting of $M$ is unique, which is the natural Heegaard splitting of $M$ induced from $V_1 ∪_{S_1} W_1$ and $V_2 ∪_{S_2} W_2$.

  • Keywords

Heegaard genus, annulus sum, distance.

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COPYRIGHT: © Global Science Press

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@Article{CMR-26-85, author = {Li , Fengling and Lei , Fengchun}, title = {A Sufficient Condition for the Genus of an Annulus Sum of Two 3-Manifolds to Be Non-Degenerate}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {26}, number = {1}, pages = {85--96}, abstract = {

Let $M_i$ be a compact orientable 3-manifold, and $A_i$ a non-separating incompressible 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 the minimal Heegaard splitting of $M$ is unique, which is the natural Heegaard splitting of $M$ induced from $V_1 ∪_{S_1} W_1$ and $V_2 ∪_{S_2} W_2$.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19180.html} }
TY - JOUR T1 - A Sufficient Condition for the Genus of an Annulus Sum of Two 3-Manifolds to Be Non-Degenerate AU - Li , Fengling AU - Lei , Fengchun JO - Communications in Mathematical Research VL - 1 SP - 85 EP - 96 PY - 2021 DA - 2021/05 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19180.html KW - Heegaard genus, annulus sum, distance. AB -

Let $M_i$ be a compact orientable 3-manifold, and $A_i$ a non-separating incompressible 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 the minimal Heegaard splitting of $M$ is unique, which is the natural Heegaard splitting of $M$ induced from $V_1 ∪_{S_1} W_1$ and $V_2 ∪_{S_2} W_2$.

Li , Fengling and Lei , Fengchun. (2021). A Sufficient Condition for the Genus of an Annulus Sum of Two 3-Manifolds to Be Non-Degenerate. Communications in Mathematical Research . 26 (1). 85-96. doi:
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