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

Abstract

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