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Small Triangulations of 4-Manifolds and the 4-Manifold Census

Authors Rhuaidi Antonio Burke , Benjamin A. Burton , Jonathan Spreer

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ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Random search heuristics
  • Mathematics of computing → Mathematical software
Keywords
  • computational low-dimensional topology
  • triangulations
  • census of triangulations
  • 4-manifolds
  • PL standard 4-sphere
  • Pachner graph
  • mathematical software
  • experiments in low-dimensional topology

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Abstract

We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated 4-manifolds with up to 6 pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, CP², and the rational homology sphere QS⁴(2), where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations - which we deem interesting in their own rights.

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Rhuaidi Antonio Burke, Benjamin A. Burton, and Jonathan Spreer. Small Triangulations of 4-Manifolds and the 4-Manifold Census. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.28

Author Details

Rhuaidi Antonio Burke
  • School of Mathematics and Physics, The University of Queensland, Brisbane, Australia
Benjamin A. Burton
  • School of Mathematics and Physics, The University of Queensland, Brisbane, Australia
Jonathan Spreer
  • School of Mathematics and Statistics F07, The University of Sydney, Australia

Funding

  • Burke, Rhuaidi Antonio: This paper was finished whilst the first author was visiting the University of Sydney (supported by the Australian Research Council’s Discovery funding scheme DP190102259), and thanks the University of Sydney for their hospitality.
  • Spreer, Jonathan: Supported by the Australian Research Council’s Discovery funding scheme (project no. DP190102259).

Acknowledgements

The authors thank Ryan Budney for his contributions at an early stage of this project, and the anonymous referees for their insightful comments.

Supplementary Materials

References

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