Recognising elliptic manifolds

TL;DR

This paper establishes the computational complexity of recognizing elliptic three-manifolds, showing that the problem is in NP and FNP, and provides a constructive method involving barycentric subdivisions and Heegaard splittings.

Contribution

It proves that recognizing elliptic manifolds is in NP and FNP, and introduces a method to find specific loops in barycentric subdivisions that identify Heegaard structures.

Findings

Recognition of elliptic manifolds is in NP.

Determining homeomorphism type is in FNP.

Existence of specific loops in barycentric subdivisions for lens spaces.

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 $M$.