The normal closure of a homological genus 0 bounding pair map

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 $H_0$.