The Heegaard genus of amalgamated 3-manifolds

Marc Lackenby

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arXiv · math/0306320

The Heegaard genus of amalgamated 3-manifolds

Marc Lackenby

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Abstract

Let M and M' be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that M and 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 M and 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 of M to some surface S, followed by a sufficiently high power of a pseudo-Anosov on S, 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.

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BibTeX RIS

Marc Lackenby. 2003-06-23. The Heegaard genus of amalgamated 3-manifolds. https://arxiv.org/abs/math/0306320

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