Mapping class groups of Heegaard splittings of surface bundles

TL;DR

This paper classifies the mapping class groups of canonical Heegaard splittings in surface bundles with high-complexity monodromy, advancing understanding of their symmetries and structure.

Contribution

It provides a classification of the mapping class groups for a broad class of surface bundle Heegaard splittings with complicated monodromies.

Findings

Mapping class groups are explicitly classified for complex monodromies.

The classification reveals structural properties of these groups.

Results apply to surface bundles with genus g fibers and genus 2g+1 splittings.

Abstract

Every surface bundle with genus $g$ fiber has a canonical Heegaard splitting of genus $2g+1$. We classify the mapping class groups of such Heegaard splittings in the case when the surface bundle has a sufficiently complicated monodromy map.