Finite rigid sets and homologically non-trivial spheres in the curve complex of a surface

Aramayona and Leininger have provided a finite rigid subset $mathfrak{X}(Sigma)$ of the curve complex $mathscr{C}(Sigma)$ of a surface $Sigma = Sigma^n_g$, characterized by the fact that any simplicial injection $mathfrak{X}(Sigma) to mathscr{C}(Sigma)$ is induced by a unique element of the mapping class group $mathrm{Mod}(Sigma)$. In this paper we prove that, in the case of the sphere with $ngeq 5$ marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a $mathrm{Mod}(Sigma)$-module generator for the reduced homology of the curve complex $mathscr{C}(Sigma)$, answering in the affirmative a question posed by Aramayona and Leininger. For the surface $Sigma = Sigma_g^n$ with $ggeq 3$ and $nin {0,1}$ we find that the finite rigid set $mathfrak{X}(Sigma)$ of Aramayona and Leininger contains a proper subcomplex $X(Sigma)$ whose reduced homology class is a $mathrm{Mod}(Sigma)$-module generator for the reduced homology of $mathscr{C}(Sigma)$ but which is not itself rigid.