Many Haken Heegaard splittings

TL;DR

This paper provides a simple criterion for Heegaard splittings to produce Haken manifolds and constructs many such manifolds with specific properties, offering simpler proofs for several existing results.

Contribution

It introduces a straightforward criterion for Haken manifolds from Heegaard splittings and constructs numerous examples with prescribed properties, simplifying previous proofs.

Findings

Constructed many Haken manifolds with specified properties

Provided simpler proofs for existing theorems

Achieved better constants in subsurface projection results

Abstract

We give a simple criterion for a Heegaard splitting to yield a Haken manifold. As a consequence, we construct many Haken manifolds, in particular homology spheres, with prescribed properties, namely Heegaard genus, Heegaard distance and Casson invariant. Along the way we give simpler and shorter proofs of the existence of splittings with specified Heegaard distance, originally proven by Ido-Jang-Kobayashi, of the existence of hyperbolic manifolds with prescribed Casson invariant, originally due to Lubotzky-Maher-Wu, and of a result about subsurface projections of disc sets (for which we even get better constants), originally due to Masur-Schleimer.