Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)

Details Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report) Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)

2008-06-28

arxiv.org/abs/0806.4689v1

Mathematics

Geometric Topology

10 pages, 2 figures

Scientific paper

We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a "sufficiently complicated" map, the resulting splitting is unstabilized. As a corollary, we produce a manifold that has distance one Heegaard splittings of arbitrarily high genus. Finally, we show that in a 3-manifold formed by a sufficiently complicated gluing, a low genus, unstabilized Heegaard splitting can be expressed in a unique way as an amalgamation over the gluing surface.

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