Automorphism groups of punctured surfaces (Q1313936)

Let \(F\) be an orientable surface of genus \(g\) with \(s > 0\) punctures. Let \(\Gamma = \Gamma^ s_ g\) denote the mapping class group of \(F\). Recall that \(\Gamma\) is the group of isotopy classes of orientation preserving diffeomorphisms of \(F\) preserving the \(s\) punctures individually. Denote by \(\Gamma^*\) the isotopy classes of diffeomorphisms, where reversing orientation and permuting punctures is allowed. In [\textit{J. Harer}, Invent. Math. 84, 157-176 (1986; Zbl 0592.57009)] a simplicial complex \(X\) is constructed to study the mapping class group. In particular, it is shown in [ibid.] that \(X\) is contractible and retracts onto a space \(Y\) of smaller dimension. However it is also shown in [ibid.] that the v.c.d. \(d\) of \(\Gamma\) is less than the dimension of the space \(Y\). Now the group \(\Gamma^*\), which has the same v.c.d. as \(\Gamma\), also acts on the complex \(X\) and the retraction of \(X\) to \(Y\) is \(\Gamma^*\)- equivariant. However as pointed out above \(d\) is less than the dimension of \(Y\). This paper is concerned with an explicit construction of a \(\Gamma^*\)-equivariant retraction of \(Y\) to a complex of dimension \(d\). This is done explicitly for the cases of the 4-punctured sphere, the 5- punctured sphere and the twice punctured torus. From this group theoretic information about the relevant \(\Gamma^*\) is deduced. The methods used follow closely those of Harer, using the combinatorics of arc systems on these surfaces.