The self-amalgamation of high distance Heegaard splittings is always efficient

Du, Kun; Qiu, Ruifeng

doi:10.1016/j.topol.2010.01.013

Cited by 17 publications

(6 citation statements)

References 11 publications

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“…First let us review some notions about Heegaard splittings. (2) there is at least one compressing or ∂-compressing disk on each side of F .…”

Section: Preliminariesmentioning

confidence: 99%

“…By the construction of self-amalgamation of Heegaard splitting, we have g(M f ) ≤ g(M, ∂M ) + 1. So a natural and interesting question is: In [2], Du and Qiu proved that when M is sufficiently complicated, the equality holds. In [3], Guo and Zou proved that when M is irreducible and ∂-irreducible, and satisfies some conditions, and if the gluing map is sufficiently complicated, the equality holds.…”

Section: Introductionmentioning

confidence: 99%

“…(see[2,5]) Let M be an irreducible 3-manifold, and let VS W be a Heegaard splitting of M . Suppose that Q is a properly embedded strongly irreducible surface in M and ∂Q = ∅.…”

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confidence: 99%

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A note on Heegaard genus of self-amalgamated 3-manifold

Guo

Qiu

Zou

2014

Chin. Ann. Math. Ser. B

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Let M be a connected orientable compact irreducible 3-manifold. Suppose that ∂M consists of two homeomorphic surfaces F1 and F2, and both F1 and F2 are compressible in M . Suppose furthermore that g(M, F1) = g(M ) + g(F1), where g(M, F1) is the Heegaard genus of M relative to F1. Let M f be the closed orientable 3-manifold obtained by identifying F1 and F2 using a homeomorphism f : F1 → F2. The authors show that if f is sufficiently complicated, then g(M f ) = g(M, ∂M) + 1.

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“…First let us review some notions about Heegaard splittings. (2) there is at least one compressing or ∂-compressing disk on each side of F .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A note on Heegaard genus of self-amalgamated 3-manifold

Guo

Qiu

Zou

2014

Chin. Ann. Math. Ser. B

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“…Lemma 2.1 (see [4,19]) Let M = V ∪ S W be a Heegaard splitting, and F be an incompressible surface in M . Then either F can be isotoped to be disjoint from S or d(S) ≤ 2 − χ(F ).…”

Section: Preliminarymentioning

confidence: 99%

“…There are some differences from [13], because in [13], Q is closed. In fact, the condition d(P ) ≥ 5 can be deleted (see [4]). The argument in [13] contains this result.…”

Section: Lemma 22 Letmentioning

confidence: 99%