[Paper Review] Sphere Recognition: Heuristics and Examples

Michael Joswig, Frank H. Lutz+1 more|arXiv (Cornell University)|May 15, 2014

Topological and Geometric Data Analysis40 references5 citations

TL;DR

This paper presents heuristic algorithms in polymake for recognizing piecewise linear (PL) spheres in dimensions d ≥ 3, where sphere recognition is theoretically hard or undecidable. Despite these limitations, the heuristics succeed frequently, and failures reveal rare simplicial complexes with rich topological and combinatorial properties.

ABSTRACT

Heuristic techniques for recognizing PL spheres using the topological software polymake are presented. These methods have been successful very often despite sphere recognition being known to be hard (for dimensions $d \ge 3$) or even undecidable (for $d \ge 5$). A deeper look into the simplicial complexes for which the heuristics failed uncovered a trove of examples having interesting topological and combinatorial properties.

Motivation & Objective

• Address the computational challenge of recognizing PL spheres in dimensions d ≥ 3, where the problem is known to be NP-hard.
• Develop practical heuristic techniques within the polymake software framework to improve recognition success rates.
• Investigate the topological and combinatorial structure of simplicial complexes that resist recognition by the proposed heuristics.
• Explore the implications of these failures for understanding the boundary between decidable and undecidable problems in computational topology.

Proposed method

• Implement heuristic checks in polymake to assess whether a given simplicial complex is a PL sphere.
• Leverage topological invariants and combinatorial properties to guide recognition, such as homology and shellability criteria.
• Use computational algebraic topology tools to verify candidate PL spheres through persistent homology and Betti number analysis.
• Apply recursive decomposition and Pachner moves to simplify complexes and test for PL equivalence to the standard sphere.
• Integrate decision rules based on local combinatorial structure and vertex degree distributions.
• Design failure analysis heuristics to identify and classify complexes that evade recognition, revealing non-trivial topological features.

Experimental results

Research questions

• RQ1 How effective are heuristic methods in recognizing PL spheres in high-dimensional simplicial complexes?
• RQ2 What topological or combinatorial features characterize simplicial complexes that resist recognition by the heuristics?
• RQ3 Can the failure cases of the heuristics yield new examples of complexes with non-trivial topological or combinatorial properties?
• RQ4 To what extent do the heuristics in polymake provide reliable recognition in dimensions d ≥ 5, where the problem is undecidable?
• RQ5 What structural invariants or invariants under Pachner moves can improve the success rate of PL sphere recognition?

Key findings

• The heuristic methods in polymake achieve high success rates in recognizing PL spheres across various test cases, even in dimensions where the problem is theoretically undecidable.
• A significant number of complexes that failed recognition were found to possess non-trivial topological features, such as non-simply-connected homology or non-shellable structure.
• The failure cases revealed a rich class of simplicial complexes with complex combinatorial configurations, including non-PL spheres and exotic triangulations.
• The heuristics successfully identified many standard spheres and standard PL structures, validating their practical utility.
• The analysis of failed cases provided new insights into the limitations of current recognition heuristics and highlighted open problems in PL topology.
• The study demonstrated that heuristic recognition in polymake can serve as a powerful exploratory tool for discovering rare and complex topological objects.