The compatibility of the filtration of mapping class groups of two surfaces pasted along the boundaries
Let \(\Sigma^n_g\) be an orientable surface of genus \(g\geq 0\) with \(n\geq 0\) punctures, \(\Sigma^n_{g,1}\) the same with one boundary component, and \(\Gamma^n_g\), \(\Gamma^n_{g,1}\), their respective pure mapping class groups. The weight filtration of Oda and Kaneko of \(\Gamma^n_g\) induces a filtration \(\Gamma^n_g= \Gamma^n_g[0]\supset \Gamma^n_g[1]\supset\cdots\)\ . Let \(\Sigma_1\), \(\Sigma_2\), and \(\Sigma\) have genus \(g_1\), \(g_2\), and \(g_1+ g_2\), respectively with \(n_1- 1\), \(n_2-1\), and \(n_1+ n_2-2\) punctures, respectively where \(\Sigma_1\) and \(\Sigma_2\) each have one boundary component. Let \(\Gamma_1\), \(\Gamma_2\), and \(\Gamma\) be their pure mapping class groups. Pasting \(\Sigma_1\) and \(\Sigma_2\) to form \(\Sigma\) yields a homomorphism \(\Gamma_1\times \Gamma_2\to\Gamma\). The exact sequence \(1\to \mathbb{Z}\to \Gamma_i\to \Gamma^{n_i}_{g_i}\to 1\) induces a homomorphism \(p:\Gamma^{n_1}_{g_1}[3]\times \Gamma^{n_2}_{g_2}[3]\to \Gamma^n_g\) which preserves the filtration. Nakamura showed that \(p^{-1}(\Gamma^n_g[m])= \Gamma^{n_1}_{g_1}[m]\times \Gamma^{n_2}_{g_2}[m]\) for \(n>0\) and \(m\geq 3\). The first result in this paper extends that to the case of \(n= 0\). Let \(\Gamma= \Gamma^0_g\). The graded quotient module \(\text{gr}^m(\Gamma)= \Gamma[m]/\Gamma[m+ 1]\) is a free \(\mathbb{Z}\)-module of finite rank. The technique of pasting two surfaces gives a method of obtaining a lower bound for the rank of \(\text{gr}^m(\Gamma)\).
scientific article; zbMATH DE number 1545862