Haken spheres for genus two Heegaard splittings

TL;DR

This paper investigates the structure of Haken sphere complexes for genus-2 Heegaard splittings, revealing that unlike other cases, these complexes for lens spaces are often disconnected and non-contractible.

Contribution

It provides a detailed description of Haken sphere complexes for lens spaces, showing their non-contractibility and disconnectedness, which contrasts with previously studied cases.

Findings

Complexes are often not contractible.

Complexes are often disconnected.

Detailed descriptions for lens space cases.

Abstract

A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \times S^1$ or the connected sum whose summands are lens spaces or $S^2 \times S^1$, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-$2$ Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.