Geometriae Dedicata

, Volume 109, Issue 1, pp 139–145

The Heegaard Genus of Amalgamated 3-Manifolds

  • Marc Lackenby
Article

DOI: 10.1007/s10711-004-6553-y

Cite this article as:
Lackenby, M. Geom Dedicata (2004) 109: 139. doi:10.1007/s10711-004-6553-y

Abstract

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.

Keywords

3-manifoldHeegaard genusHaken manifoldminimal surfaceshyperbolic geometry

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Marc Lackenby
    • 1
  1. 1.Mathematical InstituteOxford UniversityOxfordUnited Kingdom