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An Upper Bound on Pachner Moves Relating Geometric Triangulations

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Abstract

We show that any two geometric triangulations of a closed hyperbolic, spherical, or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes, and bound on the lengths of edges of the triangulation. This leads to an algorithm to check from the combinatorics of the triangulation and bounds on lengths of edges, if two geometrically triangulated closed hyperbolic or low dimensional spherical manifolds are isometric or not.

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Acknowledgements

Tejas Kalelkar was supported by the MATRICS grant of the Science and Engineering Research Board, GoI and Advait Phanse was supported by an award from the National Board of Higher Mathematics, GoI.

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  1. Mathematics Department, Indian Institute of Science Education and Research, Pune, 411008, India

    Tejas Kalelkar & Advait Phanse

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  1. Tejas Kalelkar
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  2. Advait Phanse
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Correspondence to Tejas Kalelkar.

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Kalelkar, T., Phanse, A. An Upper Bound on Pachner Moves Relating Geometric Triangulations. Discrete Comput Geom 66, 809–830 (2021). https://doi.org/10.1007/s00454-021-00283-7

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  • DOI: https://doi.org/10.1007/s00454-021-00283-7

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