Finite rigid sets in sphere complexes

TL;DR

This paper proves that for connect sums of at least three copies of S^1×S^2, the sphere complex can be built up from finite strongly rigid sets, unlike the case when n=2.

Contribution

It establishes the existence of finite strongly rigid sets in the sphere complex for all n≥3, extending rigidity results from curve complexes to sphere complexes.

Findings

Finite strongly rigid sets exist in sphere complexes for n≥3.

The sphere complex for n=2 does not admit finite rigid sets.

The result generalizes rigidity phenomena from surface to 3-manifold topology.

Abstract

A subcomplex $X\leq \mathcal{C}$ of a simplicial complex is strongly rigid if every locally injective, simplicial map $X\to\mathcal{C}$ is the restriction of a unique automorphism of $\mathcal{C}$. Aramayona and the second author proved that the curve complex of an orientable surface can be exhausted by finite strongly 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 strongly rigid sets for all $n\ge 3$ and that when $n=2$ the sphere complex does not have finite rigid sets.