A note on the connectivity of certain complexes associated to surfaces
Let \(\Sigma_g\) be a genus \(g\) surface and Mod(\(\Sigma_g\)) be the mapping class group of \(\Sigma_g\). The complex of curves C(\(\Sigma_g\)) is the simplicial complex whose simplices are sets of non-trivial isotopy classes of simple closed curves on \(\Sigma_g\) which can be realized disjointly. The complex of separating curves C\(_{sep}(\Sigma_g\)) and the complex of non-separating curves C\(_{nosep}(\Sigma_g\)) are the full subcomplexes of C(\(\Sigma_g\)) spanned by separating and nonseparating curves, respectively. The author exploits the basic combinatorial group-theoretic properties of Mod(\(\Sigma_g\)) and its subgroups, deducing that complexes are connected from the structure of generating sets and deducing that they are simply connected from relations. A simple and unified proof of the following well known theorem is given: Theorem 1.2 C(\(\Sigma_g\)) and C\(_{nosep}(\Sigma_g\)) are connected for \(g\geq 2\), while C\(_{sep}(\Sigma_g\)) is connected for \(g\geq 3\). New is for instance the following: Theorem 1.11 C\(_{sep}(\Sigma_g\)) is simply connected for \(g\geq 4\). Analogous results are achieved for complexes of pants and cut systems. The author also proves that two new types of complexes are connected: one involves curves which split a genus \(2g\) surface into two genus \(g\) pieces, and the other involves curves which are homologous to a fixed curve. The connectivity of the latter complex can be interpreted as saying the ``homology'' relation on the surface is (for \(g\geq 3\)) generated by ``embedded/disjoint homologies''.
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