Finite rigid sets in sphere complexes

Edgar A. Bering IV, Christopher J. Leininger

Published 2022-04-05Version 1

A subcomplex $X\leq \mathcal{C}$ of a simplicial complex is rigid if every locally injective, simplicial map $X\to\mathcal{C}$ is the restriction of an automorphism of $\mathcal{C}$. Aramayona and the second author proved that the curve complex of an orientable surface can be exhausted by finite rigid sets. The Hatcher sphere complex is an analog of the curve complex for isotopy classes of essential spheres in a connect sum of $n$ copies of $S^1\times S^2$. We show that there is an exhaustion of the sphere complex by finite rigid sets for all $n\ge 3$ and that when $n=2$ the sphere complex does not have finite rigid sets.