The normal closure of a homological genus 0 bounding pair map

TL;DR

This paper investigates the structure of the kernel of the Casson--Morita d-map in the genus zero case, showing it is generated by a single bounding pair map and that the Chillingworth subgroup is normally generated by one element.

Contribution

It extends previous results to genus zero, establishing the generation of the kernel and Chillingworth subgroup by specific elements in the mapping class group.

Findings

Kernel of d-map equals the commutator of Chillingworth subgroup and mapping class group

Kernel is generated by a single homological genus 0 bounding pair map

Chillingworth subgroup is normally generated by a specific element H_0

Abstract

Justin Lanier and the authors recently determined the group normally generated by a single bounding pair map of genus $n$. We related this subgroup with the Chillingworth subgroup and the Casson--Morita's $d$ map. In this paper, we extend the results to the case when $n=0$. Let $\mathcal{M}_g^1$ be the mapping class group, $\text{Ch}_g^1$ be the Chillingworth subgroup and $d$ be the Casson--Morita's $d$-map. We show that $\text{Ker}(d)=[\text{Ch}_g^1,\mathcal{M}_g^1]$ and it is generated by a single homological genus 0 bounding pair map. We also construct an element $H_0\in \text{Ch}_g^1$, and show that $\text{Ch}_g^1$ is normally generated by this single element $H_0$.