The Goeritz group of a Heegaard splitting of genus two of a Seifert manifold whose base orbifold is sphere with three exceptional points of sufficiently complex coefficients

TL;DR

This paper investigates the structure of the Goeritz group associated with a specific genus two Heegaard splitting of a Seifert manifold, providing new examples and insights into the mapping class groups of such 3-manifolds.

Contribution

It introduces new examples of Goeritz groups for a class of Seifert manifolds with complex coefficients, expanding understanding of their mapping class groups.

Findings

Added new examples of Goeritz groups for these manifolds.

Characterized the properties of the Goeritz groups in this setting.

Provided conditions for the complexity of the coefficients in the base orbifold.

Abstract

In this paper, we add examples to Goeritz groups, the mapping class groups of given Heegaard splittings of 3-manifolds. We focus on a Heegaard splitting of genus two of a Seifert manifold whose base orbifold is sphere with three exceptional points of sufficiently complex coefficients, where "sufficiently complex" means that every surgery coefficient p_{l} over q_{l} of each exceptional fiber (in a surgery description) satisfies that q_{l} is congruent to neither +1 or -1 modulo p_{l}.