Minimal Heegaard genus of amalgamated 3-manifolds

Kun Du; Ruifeng Qiu

doi:10.1142/S0218216517500638

Minimal Heegaard genus of amalgamated 3-manifolds

Kun Du^*
, Ruifeng Qiu

^*Corresponding author for this work

School of Mathematical Sciences

Lanzhou University

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let M_i (i = 1, 2) be a perfect 3-manifold, F_i be a component of ∂M_i, f: F₁ → F₂ be a homeomorphic map, M = M₁ ∪_f M₂ and F = M₁ ∩ M₂(≅F_i). In this paper, we show that if d(M_i) ≥ 3 (i = 1, 2) and d(f) ≥ 2, then g(M) = g(M₁) + g(M₂) - g(F). As a corollary, if F_i^j (i = 1, 2, 1 ≤ j ≤ n) is a component of ∂M_i, f_j: F₁^j → F₂^j is a homeomorphic map, M = M₁ ∪_f1,⋯,fn M₂, U_j=1ⁿ F^j = M₁ ∩ M₂ (≅F_i^j), d(M_i) ≥ 3 (i = 1, 2) and d(f_j) ≥ 2 (1 ≤ j ≤ n), then g(M) = g(M₁) + g(M₂) - g(F¹) - ⋯ - g(Fⁿ) + n - 1.

Original language	English
Article number	1750063
Journal	Journal of Knot Theory and its Ramifications
Volume	26
Issue number	11
DOIs	https://doi.org/10.1142/S0218216517500638
State	Published - 1 Oct 2017

Keywords

Heegaard distance
amalgamation
homeomorphic map
perfect 3-manifold

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title = "Minimal Heegaard genus of amalgamated 3-manifolds",

abstract = "Let Mi (i = 1, 2) be a perfect 3-manifold, Fi be a component of ∂Mi, f: F1 → F2 be a homeomorphic map, M = M1 ∪f M2 and F = M1 ∩ M2(≅Fi). In this paper, we show that if d(Mi) ≥ 3 (i = 1, 2) and d(f) ≥ 2, then g(M) = g(M1) + g(M2) - g(F). As a corollary, if Fij (i = 1, 2, 1 ≤ j ≤ n) is a component of ∂Mi, fj: F1j → F2j is a homeomorphic map, M = M1 ∪f1,⋯,fn M2, Uj=1n Fj = M1 ∩ M2 (≅Fij), d(Mi) ≥ 3 (i = 1, 2) and d(fj) ≥ 2 (1 ≤ j ≤ n), then g(M) = g(M1) + g(M2) - g(F1) - ⋯ - g(Fn) + n - 1.",

keywords = "Heegaard distance, amalgamation, homeomorphic map, perfect 3-manifold",

author = "Kun Du and Ruifeng Qiu",

note = "Publisher Copyright: {\textcopyright} 2017 World Scientific Publishing Company.",

year = "2017",

month = oct,

day = "1",

doi = "10.1142/S0218216517500638",

language = "英语",

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journal = "Journal of Knot Theory and its Ramifications",

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TY - JOUR

T1 - Minimal Heegaard genus of amalgamated 3-manifolds

AU - Du, Kun

AU - Qiu, Ruifeng

PY - 2017/10/1

Y1 - 2017/10/1

N2 - Let Mi (i = 1, 2) be a perfect 3-manifold, Fi be a component of ∂Mi, f: F1 → F2 be a homeomorphic map, M = M1 ∪f M2 and F = M1 ∩ M2(≅Fi). In this paper, we show that if d(Mi) ≥ 3 (i = 1, 2) and d(f) ≥ 2, then g(M) = g(M1) + g(M2) - g(F). As a corollary, if Fij (i = 1, 2, 1 ≤ j ≤ n) is a component of ∂Mi, fj: F1j → F2j is a homeomorphic map, M = M1 ∪f1,⋯,fn M2, Uj=1n Fj = M1 ∩ M2 (≅Fij), d(Mi) ≥ 3 (i = 1, 2) and d(fj) ≥ 2 (1 ≤ j ≤ n), then g(M) = g(M1) + g(M2) - g(F1) - ⋯ - g(Fn) + n - 1.

AB - Let Mi (i = 1, 2) be a perfect 3-manifold, Fi be a component of ∂Mi, f: F1 → F2 be a homeomorphic map, M = M1 ∪f M2 and F = M1 ∩ M2(≅Fi). In this paper, we show that if d(Mi) ≥ 3 (i = 1, 2) and d(f) ≥ 2, then g(M) = g(M1) + g(M2) - g(F). As a corollary, if Fij (i = 1, 2, 1 ≤ j ≤ n) is a component of ∂Mi, fj: F1j → F2j is a homeomorphic map, M = M1 ∪f1,⋯,fn M2, Uj=1n Fj = M1 ∩ M2 (≅Fij), d(Mi) ≥ 3 (i = 1, 2) and d(fj) ≥ 2 (1 ≤ j ≤ n), then g(M) = g(M1) + g(M2) - g(F1) - ⋯ - g(Fn) + n - 1.

KW - Heegaard distance

KW - amalgamation

KW - homeomorphic map

KW - perfect 3-manifold

UR - https://www.scopus.com/pages/publications/85030851256

U2 - 10.1142/S0218216517500638

DO - 10.1142/S0218216517500638

M3 - 文章

AN - SCOPUS:85030851256

SN - 0218-2165

VL - 26

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 11

M1 - 1750063

ER -

Minimal Heegaard genus of amalgamated 3-manifolds

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Keywords

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