A twisted first homology group of the handlebody mapping class group (Q2403203)

Let \(H_g\) be a 3-dimensional handlebody of genus \(g\) where the boundary surface \(\partial H_g=\Sigma_g\) has genus \(g \geq 2\). Let \(\mathcal{H}_g\) denote the mapping class group of \(H_g\) and \(\mathcal M_g\) that of \(\Sigma_g\). Given a closed disk \(D\) and a point \(x_0\in \text{Int}(D)\), let us denote by \(\mathcal{H}_g^*\) and \(\mathcal{H}_{g,1}\) the groups of isotopy classes of orientation preserving homeomorphisms of \(H_g\) fixing \(x_0\) and \(D\) pointwise, respectively. In the paper under review the authors compute the twisted first homology groups of \(\mathcal{H}_g\), \(\mathcal{H}_g^*\) and \(\mathcal{H}_{g,1}\) with coefficients in the first integral homology group of the boundary surface \(\Sigma_g\). The main results are: Theorem 1.1. \[ H_1(\mathcal{H}_{g};H_1(\Sigma_g))\cong \begin{cases} \mathbb{Z}/(2g-2)\mathbb{Z} &\text{if }g\geq 4,\\ \mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z} &\text{if }g=3,\\ (\mathbb{Z}/2\mathbb{Z})^2 &\text{if }g=2. \end{cases} \] Furthermore, when \(g \geq 4\), the homomorphism \(H_1(\mathcal{H}_g; H_1(\Sigma_g)) \to H_1(\mathcal M_g; H_1(\Sigma_g))\) induced by the inclusion is an isomorphism. When \(g = 2, 3\), this homomorphism is surjective and the kernel is isomorphic to \(\mathbb{Z}/2\mathbb{Z}\). Theorem 1.2. \[ H_1(\mathcal{H}_{g,1};H_1(\Sigma_g))\cong H_1(\mathcal{H}_g^{*}; H_1(\Sigma_g)) \cong \begin{cases} \mathbb{Z} &\text{if }g\geq 4,\\ \mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z} &\text{if } g=2, 3. \end{cases} \] Furthermore, when \(g \geq 4\), the homomorphism \(H_1(\mathcal{H}_g^*; H_1(\Sigma_g)) \to H_1(\mathcal M_g^*; H_1(\Sigma_g))\) induced by the inclusion is an isomorphism. When g = 2, 3, this homomorphism is surjective and the kernel is isomorphic to \(\mathbb{Z}/2\mathbb{Z}\). They also study relationships between the second homology groups of \(\mathcal{H}_{g}\), \(\mathcal{H}_g^{*}\) and \(\mathcal{H}_{g,1}\) with integral coefficients using the main results. In the introduction the authors provide some quite detailed information of the known results related to the question they solve. Then they review classical facts on homological algebra (mainly cohomology of groups) which are used in the paper, making the exposition more self-contained. The main tools used are the Lyndon-Hochschild-Serre spectral sequence and the Gysin sequence together with relations on the mapping class group of a surface related with Dehn twist homeomorphisms. The exposition is well organized.