Communications in Analysis and Geometry

Volume 27 (2019)

Number 6

$3$-manifolds admitting locally large distance $2$ Heegaard splittings

Pages: 1355 – 1379

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n6.a6

Authors

Ruifeng Qiu (School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China)

Yanqing Zou (School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China)

Abstract

It is known that every closed, orientable $3$-manifold admits a Heegaard splitting. By Thurston’s Geometrization conjecture, proved by Perelman, a $3$-manifold admitting a Heegaard splitting of distance at least $3$ is hyperbolic. So what about $3$-manifolds admitting distance at most $2$ Heegaard splittings?

Inspired by the construction of hyperbolic $3$-manifolds in [Qiu, Zou and Guo, Pacific J. Math. 275 (2015), no. 1, 231-255], we introduce the definition of a locally large geodesic in curve complex and also a locally large distance $2$ Heegaard splitting. Then we prove that if a $3$-manifold admits a locally large distance $2$ Heegaard splitting, then it is either a hyperbolic $3$-manifold or an amalgamation of a hyperbolic $3$-manifold and a small Seifert fiber space along an incompressible torus. After examining those non hyperbolic cases, we give a sufficient and necessary condition to determine a hyperbolic $3$-manifold admitting a locally large distance $2$ Heegaard splitting.

Received 23 August 2016

Accepted 3 June 2017

Published 12 December 2019