Recognising elliptic manifolds
Marc Lackenby, Saul Schleimer
TL;DR
The paper proves that recognizing elliptic 3-manifolds and naming their Seifert data lie in NP and FNP, respectively, by reducing to lens-space recognition via a finite cover of degree at most $60$; core curves and their simplicial realizations in iterated barycentric subdivisions underpin the certificates. It develops a robust framework linking normal/almost-normal surfaces, affine handle structures, and parallelity bundles to produce explicit subdivision bounds (e.g., core curves appear in ${\mathcal{T}}^{(51)}$ or ${\mathcal{T}}^{(86)}$), enabling polynomial-time verifiability of certificates. The work classifies elliptic manifolds into lens spaces, prism manifolds, and platonic manifolds, and provides detailed certificate schemes for each family, including supporting subproblems in NP and $\mathsf{P}$. Its certification approach yields a practical pathway for verifying elliptic geometry in 3-manifolds and sharpens the complexity landscape of the homeomorphism problem in this important geometric setting. Overall, the results deliver the first comprehensive NP/FNP–level framework for elliptic-manifold recognition and explicit, verifiable certificates for the constituent geometric types.
Abstract
We show that the problem of deciding whether a closed three-manifold admits an elliptic structure lies in NP. Furthermore, determining the homeomorphism type of an elliptic manifold lies in the complexity class FNP. These are both consequences of the following result. Suppose that $M$ is a lens space which is neither $\mathbb{RP}^3$ nor a prism manifold. Suppose that $\mathcal{T}$ is a triangulation of $M$. Then there is a loop, in the one-skeleton of the 86th iterated barycentric subdivision of $\mathcal{T}$, whose simplicial neighbourhood is a Heegaard solid torus for $M$.