A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities
A Coxeter matrix is a symmetric \(n\times n\) matrix \(M= (m_{ij})\) with \(m_{ij}\) an integer \(\geq 2\) or \(\infty\) for \(1\leq i\neq j\leq n\) and \(m_{ii}= 1\). Its Artin group has generators \(a_1,a_2,\dots, a_n\) and relations \(a_ia_ja_i\dots= a_ja_ia_j\dots\) where both sides are words of length \(m_{ij}\neq \infty\) and \(1\leq i\neq j\leq n\). The Coxeter matrix corresponds to a graph with vertices \(I= \{1,2,\dots, n\}\) with \(i\) and \(j\) joined by \(m_{ij}-2\) edges. This paper considers only graphs with no edges from a vertex to itself. Any graph, \(\Gamma\), then yields a Coxeter matrix and its Artin group is denoted \(A(\Gamma)\). An induced subgraph \(H\) of \(\Gamma\) has the property that any edge with ends in \(H\) is itself in \(H\). The obvious homomorphism \(A(H)\to A(\Gamma)\) is injective, so \(A(H)\) can be considered to be a subgroup of \(A(\Gamma)\). Let \(T_g\) be a straight path with \(2g\) vertices \(a_1,\dots, a_{2g}\) and a vertex \(a_0\) with an edge from \(a_0\) to \(a_4\). There is a natural surjection from \(A(T_g)\) onto the mapping class group \(M^{\langle 1\rangle}_g\), the group of isotopy classes of orientation preserving self-diffeomorphisms on a surface of genus \(g\) with \(1\) boundary component. The surjection takes each \(a_i\) to the corresponding element in the Dehn-Lickorish-Humphries generators of the mapping class group. A presentation of \(M^{\langle 1\rangle}_g\) is then derived from the presentation given by Wajnryb as a quotient of \(A(T_g)\) by relations among the centers of groups corresponding to \(A(H)\) for specific subgraphs, \(H\), of \(T_g\). A topological interpretation is then given. The second section of the paper studies the geometric monodromy of simple singularities and provides the proofs of theorems leading to the main result.
scientific article; zbMATH DE number 1441716