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

Abstract

Aramayona and Leininger have provided a "finite rigid subset" 𝔛(Σ) of the curve complex [Formula: see text] of a surface [Formula: see text], characterized by the fact that any simplicial injection [Formula: see text] is induced by a unique element of the mapping class group Mod(Σ). 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(Σ)-module generator for the reduced homology of the curve complex [Formula: see text], answering in the affirmative a question posed in [1]. For the surface [Formula: see text] with g ≥ 3 and n ∈ {0, 1} we find that the finite rigid set 𝔛(Σ) of Aramayona and Leininger contains a proper subcomplex X(Σ) whose reduced homology class is a Mod(Σ)-module generator for the reduced homology of [Formula: see text] but which is not itself rigid.