Speaker: Pankaj Kapari
Date: December 04, 2025
Affiliation: TIFR, Mumbai

Abstract: Let $S_g$ be a closed oriented surface of genus $g\geq 2$ and $S_{1,2}$ be a torus with two marked points. For an oriented surface $S$, let $\text{Mod}(S)$ be the \textit{mapping class group} of $S$ consisting of isotopy classes of orientation-preserving homeomorphisms of $S$ that preserve the set of marked points. In this talk, we show that the kernel of the homology representation $\Psi:\text{Mod}(S_{1,2}) \to \mathrm{GL}_3(\mathbb{Z})$ is normally generated by a Dehn twist about a separating simple closed curve, and it is free on a countable basis. Furthermore, we will provide an explicit countable basis consisting of Dehn twists about separating simple closed curves. As an application, we will provide a finite generating set for the \textit{liftable mapping class group} $\text{LMod}_{p_k}(S_{1,2})$ consisting of mapping classes that lift under the cyclic branched cover $p_k:S_k\to S_{1,2}$.