Degeneration of Heegaard genus, a survey

Abstract

We survey known (and unknown) results about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. The constructions we consider are (1) gluing together two 3-manifolds with incompressible boundary, (2) gluing together the boundary components of surface times I, and (3) gluing a handlebody to the boundary of a 3-manifold. We detail those cases in which it is known when the Heegaard genus is less than what is expected after gluing.Comment: This is the version published by Geometry & Topology Monographs on 3 December 200

Stabilization, amalgamation and curves of intersection of Heegaard splittings

“…See eg Bachman and Derby-Talbot [3] or Lackenby [15] for descriptions of the construction in simpler cases.…”

A Lower Bound for the Genus of Self-Amalgamation of Heegaard Splittings

“…There exist examples which show that an amalgamation of two minimal genus Heegaard splittings of M 1 and M 2 is stabilized, see [1], [8] and [19] etc. On the other hand, it has been shown that under some conditions on the manifolds, the gluing maps, or the distances of the factor manifolds, the equality g(M ) = g(M 1 ) + g(M 2 ) − g(F ) holds, see [9], [10], [20], [7] and [21] etc.…”

A sufficient condition for the genus of an amalgamated 3-manifold not to go down

“…There exist examples which show that an amalgamation of two minimal genus Heegaard splittings of M 1 and M 2 could be stabilized (refer to [1,5,13] etc. ), in that case g(M ) < g(M 1 ) + g(M 2 ) − g(F ) does occur.…”