Stability of the homology of the mapping class groups of orientable surfaces (Q1067257)

Let \(F^ s_{g,r}\) be the surface of genus g with r boundary components and an (arbitrary) number s of punctures and let \(\Gamma^ s_{g,r}=\pi_ 0(\Lambda)\) where \(\Lambda\) is the topological group of orientation preserving diffeomorphisms of \(F^ s_{g,r}\) onto itself. Inclusion \(\Phi\) : \(F^ s_{g,r}\to F^ s_{g,r+1}\) (r\(\geq 1)\), \(\Psi\) : \(F^ s_{g,r}\to F^ s_{g+1,r-1}\) (r\(\geq 2)\) are obtained by adding a copy of \(F_{0,3}\), a pair of pants, to \(F^ s_{g,r}\) sewn along one boundary component for \(\Phi\), two for \(\Psi\). Moreover define \(\eta\) : \(F^ s_{g,r}\to F^ s_{g+1,r-2}\), \(r\geq 2\), by gluing two boundary components of \(F^ s_{g,r}\) together. The stability theorem is that the following homomorphisms are isomorphisms: \(\Phi_*: H_ k(\Gamma^ s_{g,r})\to H_ k(\Gamma^ s_{g,r+1})\) for \(k>1\), when \(g\geq 3k-2\), \(r\geq 1\), and for \(k=1\), when \(g\geq 2\), \(r\geq 1;\) \(\Psi_*: H_ k(\Gamma^ s_{g,r})\to H_ k(\Gamma^ s_{g+1,r- 1})\) for \(k>1\), when \(g\geq 3k-1\), \(r\geq 2\), and for \(k=1\), when \(g\geq 3\), \(r\geq 2;\) \(\eta_*: H_ k(\Gamma^ s_{g,r};{\mathbb{Q}})\to H_ k(\Gamma^ s_{g+1,r-2};{\mathbb{Q}})\) for \(g\geq 3k-1\), \(r\geq 2\) and \(\eta_*: H_ k(\Gamma^ s_{g,r})\to H_ k(\Gamma^ s_{g+1,r- 2})\) when \(g\geq 3k\), \(r\geq 2\). Thus \(H_ k(\Gamma^ s_{g,r})\) is independent of g and r when \(g\geq 3k+1\). - For the proof a simplicial complex of Thurston's lamination space is constructed.