Homology of the curve complex and the Steinberg module of the mapping class group
Let \(\Sigma\) denote either the closed genus \(g\) surface \(\Sigma_g\) or the closed genus \(g\) surface with one marked point \(\Sigma_g^1\). The mapping class group \(\mathrm{Mod}(\Sigma)\) is the group of isotopy classes of orientation-preserving self-diffeomorphisms of \(\Sigma\), resp. of diffeomorphisms which fix the marked point when \(\Sigma=\Sigma_g^1\). The curve complex \(\mathcal{C}(\Sigma)\) is the simplicial complex with \(n\)-simplices corresponding to the isotopy class of \(n+1\) disjoint essential simple closed curves in \(\Sigma\). \textit{J. L. Harer} [Invent. Math. 84, 157--176, (1986; Zbl 0592.57009)] showed that \(\mathcal{C}(\Sigma)\) has the homotopy type of a wedge sum of spheres of dimension \(2g-2\). The Steinberg module for \(\mathrm{Mod}(\Sigma)\) is the \(\mathrm{Mod}(\Sigma)\)-module \(\mathrm{St}(\Sigma) = \tilde{H}_{2g-2} (\mathcal{C}(\Sigma) ; \mathbb{Z})\) . In the paper under review, it is shown that \(\mathrm{St}(\Sigma)\) is a cyclic \(\mathrm{Mod}(\Sigma)\)-module (Theorem 4.2).NEWLINENEWLINEAn arc system \(\alpha = \{ \alpha_0, \ldots, \alpha_n \}\) \(k\)-fills \(\Sigma_g^1\) if \(\Sigma_g^1 \setminus \cup \alpha\) is \(k+1\) disks. Let \(\mathcal{F}_k\) be the module of oriented \(k\)-filling arc systems. In Proposition 3.3, it is shown that, as a \(\mathrm{Mod}(\Sigma_g^1)\)-module, \(\mathrm{St}(\Sigma_g^1) \cong \mathcal{F}_0 / \partial \mathcal{F}_1\). The homeomorphism types of filling arc systems are described by chord diagrams. In Theorem 4.2, the generator of \(\mathrm{St}(\Sigma)\) is explicitly described by a chord diagram.
scientific article; zbMATH DE number 6063225
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