The Heegaard genus of amalgamated 3-manifolds

Journal article

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.

Authors: Lackenby, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Kluwer Academic Publishers
• Journal: Geometriae Dedicata
• Volume: 109
• Issue: 1
• Pages: 139-145
• Publication date: 2004-12-01
• DOI: 10.1007/s10711-004-6553-y
• EISSN: 1572-9168
• ISSN: 0046-5755
• Keywords: 3-manifold; Haken manifold; minimal surfaces; Heegaard genus; hyperbolic geometry