Explicit Dehn filling and Heegaard splittings

TL;DR

This paper provides an explicit criterion ensuring that Heegaard surfaces in Dehn fillings of hyperbolic 3-manifolds behave predictably, preserving the original splitting structure under certain geometric conditions.

Contribution

It offers a quantitative, effective criterion for the behavior of Heegaard splittings after Dehn filling, extending previous qualitative results.

Findings

Heegaard surfaces in certain Dehn fillings are isotopic to original manifold splittings.

The criterion depends on meridian and longitude lengths exceeding 2pi(2g-1).

The results apply to fillings of multiple boundary tori.

Abstract

We prove an explicit, quantitative criterion that ensures the Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic manifold X, and a Dehn filling whose meridian and longitude curves are longer than 2pi(2g-1), we show that every genus g Heegaard splitting of the filled manifold is isotopic to a splitting of the original manifold X. The analogous statement holds for fillings of multiple boundary tori. This gives an effective version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.