Finite rigid sets and homologically nontrivial spheres in the curve complex of a surface

Aramayona and Leininger have provided a "finite rigid subset" (∑) of the curve complex C(Σ) of a surface Σ = Σⁿ_g, characterized by the fact that any simplicial injection X(Σ) → C(Σ) is induced by a unique element of the mapping class group Mod(∑). In this paper we prove that, in the case of the sphere with n ≥ 5 marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a Mod(∑)-module generator for the reduced homology of the curve complex C(Σ), answering in the affirmative a question posed in [1]. For the surface Σ = Σgⁿ with g ≥ 3 and n {0, 1} we find that the finite rigid set (∑) of Aramayona and Leininger contains a proper subcomplex X(∑) whose reduced homology class is a Mod(∑)-module generator for the reduced homology of C}(Σ)$ but which is not itself rigid.