Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation

Details Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation

2009-04-02

arxiv.org/abs/0904.0485v1

Mathematics

Geometric Topology

16 pages, 4 figures

Scientific paper

Let M_1 and M_2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of \bdy M_1 to some component of \bdy M_2 by a homeomorphism \phi. We show that when \phi is "sufficiently complicated" then (1) the amalgamation of low genus, unstabilized, boundary-unstabilized Heegaard splittings of M_i is an unstabilized splitting of M, (2) every low genus, unstabilized Heegaard splitting of M can be expressed as an amalgamation of unstabilized, boundary-unstabilized splittings of M_i, and possibly a Type II splitting of F \times I, and (3) if there is no Type II splitting in such an expression then it is unique.

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