Automorphisms of curve complexes on nonorientable surfaces
Let \(N\) be a compact connected surface of genus \(g\) with \(n\) boundary components. The \textit{curve complex} \(C(N)\) of \(N\) is the abstract simplicial complex whose \(n\)-dimensional simplices are sets consisting of \((n+1)\) isotopy classes of ``non-trivial'' simple closed curves on \(N\) having pairwise disjoint representatives. The mapping class group \(\text{Mod}(N)\) of \(N\) acts in the obvious way on the simplicial complex \(C(N)\). Ivanov proved that, if \(N\) is orientable and if the pair \((g,n)\) does not belong to a short list of ``exceptional'' cases, then the resulting homomorphism from \(\text{Mod}(N)\) to the group \( \text{Aut}(C(N))\) of simplicial automorphisms of \(C(N)\) is bijective [\textit{N. V. Ivanov}, Int. Math. Res. Not. 1997, No. 14, 651--666 (1997; Zbl 0890.57018)]. The paper under review proves the analogous result for a non-orientable surface \(N\): specifically, the authors show that the canonical map \(\text{Mod}(N) \to \text{Aut}(C(N))\) is an isomorphism if \(g+n \geq 5\). The proof goes by induction on \(g\) and it needs Ivanov's result for orientable surfaces. The exceptional cases where \(g+n \leq 3\) are examined in the last section of the paper, leaving the case \(g+n=4\) unsolved. Besides, the authors show that any compact connected surfaces \(N_1\) and \(N_2\) are homeomorphic if and only if the simplicial complexes \(C(N_1)\) and \(C(N_2)\) are isomorphic. Here it is assumed that the curve complexes are not empty and that \(N_1, N_2\) do not constitute one of the following exceptional pairs of orientable surfaces: a sphere with four holes and a torus with at most one hole; a sphere with five holes and a torus with two holes; a sphere with six holes and a closed orientable surface of genus two.
scientific article; zbMATH DE number 6305382
scientific article