Geometry & Topology

Heegaard surfaces and the distance of amalgamation

Tao Li

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Let M1 and M2 be orientable irreducible 3–manifolds with connected boundary and suppose M1M2. Let M be a closed 3–manifold obtained by gluing M1 to M2 along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then M is not homeomorphic to S3 and all small-genus Heegaard splittings of M are standard in a certain sense. In particular, g(M)=g(M1)+g(M2)g(Mi), where g(M) denotes the Heegaard genus of M. This theorem is also true for certain manifolds with multiple boundary components.

Article information

Geom. Topol., Volume 14, Number 4 (2010), 1871-1919.

Received: 31 July 2008
Revised: 9 March 2010
Accepted: 7 June 2010
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Heegaard splitting amalgamation curve complex


Li, Tao. Heegaard surfaces and the distance of amalgamation. Geom. Topol. 14 (2010), no. 4, 1871--1919. doi:10.2140/gt.2010.14.1871.

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