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" $\mathfrak{X}(\Sigma )$ of the curve complex $\mathcal{C}(\Sigma )$ of a surface $\Sigma ={\Sigma }_g^n$, characterized by the fact that any simplicial injection $\mathfrak{X}(\Sigma )\to \mathcal{C}(\Sigma )$ 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 $n\ge 5$ 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 $\mathcal{C}(\Sigma )$, answering in the affirmative a question posed by Aramayona and Leininger. For the surface $\Sigma ={\Sigma }_g^n$ with $g\ge 3$ and $n\in \{0,1\}$ 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 $\mathcal{C}(\Sigma )$ but which is not itself rigid.

21 pages, 7 figures; Section 4 revised along with minor corrections throughout

References in corpus (1)

• Homology of the curve complex and the Steinberg module of the mapping class group