The Heegaard Genus of Amalgamated 3-Manifolds

By Marc Lackenby. 2003

73 citations recorded by the source corpus.

Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/09a59703d038a4e314524e4318b577f743e4a8b5

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In Geometry and Topology

1. Tao Li
2. Ruifeng Qiu
3. Tsuyoshi Kobayashi

In Mathematical Physics

1. Tao Li
2. Ruifeng Qiu
3. Tsuyoshi Kobayashi

Citations in the corpus

• Heegaard surfaces and the distance of amalgamation (2008, background): “Intersection of F with sweepout surfaces 32 References 37”
• The amalgamation of high distance Heegaard splittings is always efficient (2008, background): “There exists a 3-manifold M containing separating closed incompressible surfaces Sn of genus gn arbitrarily large, such that the amalgamation of minimal genus Heegaard splittings of two manifolds M − Sn along Sn can be destabilized gn − 3 times to a minimal genus Heegaard splitting of M. (5) The fol”
• Sweepouts of amalgamated 3-manifolds (2005, unknown): “57N10, 57M99; 57M27”