[Paper Review] Computationally proving triangulated 4-manifolds to be diffeomorphic

Benjamin A. Burton, Jonathan Spreer|arXiv (Cornell University)|Mar 12, 2014

Topological and Geometric Data Analysis10 references3 citations

TL;DR

This paper presents novel computational methods to prove diffeomorphism between triangulated 4-manifolds, focusing on the conjecture that the 16-vertex and 17-vertex triangulations of the K3 surface are diffeomorphic. Using advanced algorithms in Regina’s 4-manifold toolkit—including dual-source breadth-first search on the Pachner graph, isomorphism signatures, and composite bistellar moves—it identifies over 1.7 million distinct minimal one-vertex triangulations of the K3 surface, providing strong computational evidence toward resolving the conjecture.

ABSTRACT

We present new computational methods for proving diffeomorphy of triangulated 4-manifolds, including algorithms and topological software that can for the first time effectively handle the complexities that arise in dimension four and be used for large scale experiments.

Motivation & Objective

• To computationally resolve Conjecture 1.1 that the 16-vertex and 17-vertex triangulations of the K3 surface are diffeomorphic.
• To develop effective algorithms for handling the high complexity of bistellar moves in 4-dimensional triangulations.
• To explore the space of minimal triangulations of the K3 surface and detect whether the two candidate triangulations are connected via a sequence of flips.
• To establish a computational framework for studying exotic 4-manifolds and diffeomorphism types in dimension four.

Proposed method

• Employing Pachner’s theorem, the paper uses bistellar flips as local moves to transform one triangulation into another if they are diffeomorphic.
• Implementing a dual-source breadth-first search through the Pachner graph, with both minimal one-vertex triangulations as starting points, to detect a connecting path.
• Using isomorphism signatures—polynomial-time computable hashes—to efficiently test combinatorial isomorphism and avoid redundant exploration.
• Combining greedy reduction, composite moves, simulated annealing, and breadth-first search to overcome local minima and simplify triangulations.
• Leveraging the 4-manifold toolkit in Regina software to perform high-performance computations on large-scale triangulations.
• Applying the Dehn-Sommerville equations and Euler characteristic constraints to derive a tight lower bound on the number of 4-simplices in one-vertex K3 triangulations.

Experimental results

Research questions

• RQ1 Are the 16-vertex and 17-vertex triangulations of the K3 surface diffeomorphic?
• RQ2 What is the total number of distinct minimal one-vertex triangulations of the K3 surface?
• RQ3 Can a sequence of bistellar flips connect the two candidate triangulations, and if so, what is the structure of such a path?
• RQ4 How do the computational challenges of 4-manifold triangulations differ from those in dimension three?
• RQ5 What is the minimal number of 4-simplices required for a one-vertex triangulation of the K3 surface?

Key findings

• The paper establishes a tight lower bound of 140 four-simplices for any one-vertex triangulation of the K3 surface, which is achieved by both (K3)₁₆ and (K3)₁₇.
• Over 1,738,260 distinct minimal one-vertex triangulations of the K3 surface have been detected so far in the search, indicating a vastly larger space than in 3D.
• The dual-source breadth-first search has not yet connected (K3)₁₆ and (K3)₁₇, but the search continues and has not exhausted the space of minimal triangulations.
• The interaction of multiple heuristics—greedy reduction, composite moves, simulated annealing, and breadth-first search—was essential, as no single technique succeeded alone.
• The search runs in O(T log T · n² log n) time, where T is the number of triangulations and n is the number of simplices, and scales well in parallel computing environments.
• The work provides the first effective computational framework for studying exotic 4-manifolds and diffeomorphism types via triangulations.