Moving between 3-manifold triangulations is NP-hard

2026-06-12

AI Summary

This study addresses the decision problem of whether two triangulations of a three-dimensional manifold can be transformed into one another via a finite sequence of bistellar moves and sparse degree-two edge collapses. By integrating techniques from computational complexity theory and combinatorial topology, the authors establish for the first time that this problem is NP-hard, even when restricted to the 3-sphere or any arbitrary 3-manifold. This result resolves a long-standing gap in the theoretical understanding of triangulation transformation operations in three dimensions and provides a foundational contribution to computational topology by rigorously characterizing the inherent computational difficulty of such equivalence problems.

Abstract

We show that \textsc{number of bistellar moves and sparse degree-two edge collapses for 3-sphere} is NP-hard. It follows that a similar problem for an arbitrary 3-manifold is NP-hard as well. This is the first NP-hardness result concerning moves between two triangulations of a 3-manifold.

Problem

• 3-manifold
• triangulation
• bistellar moves
• NP-hard
• edge collapses

Innovation

• bistellar moves
• edge collapses
• 3-manifold triangulation
• NP-hardness
• computational topology