Recognising elliptic manifolds

Journal article

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 RP³ nor a prism manifold. Suppose that T is a triangulation of M. Then there is a loop, in the one-skeleton of the 86th iterated barycentric subdivision of T, whose simplicial neighbourhood is a Heegaard solid torus for M.

Authors: Lackenby, M; Schleimer, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: EMS Press
• Journal: Commentarii Mathematici Helvetici
• Volume: 101
• Issue: 1
• Pages: 1–45
• Publication date: 2025-11-19
• Acceptance date: 2024-12-22
• DOI: 10.4171/CMH/597
• EISSN: 1420-8946
• ISSN: 0010-2571
• Language: English
• Keywords: elliptic; three-manifold; NP; triangulation; Heegaard splitting; core curves