Filling triangulated surfaces

  • Ryan Kowalick
  • Jean-François LafontEmail author
  • Barry Minemyer
Original Paper


Given a triangulated closed oriented surface \((M, {\mathcal {T}}_M)\), we provide upper bounds on the number of tetrahedra needed to construct a triangulated 3-manifold \((N, {\mathcal {T}}_N)\) which bounds \((M, {\mathcal {T}}_M)\). Along the way, we develop a technique to translate (in all dimensions) between the famous Riemannian systolic inequalities of Gromov and combinatorial analogues of these inequalities.


Systolic geometry Combinatorial systolic inequality Fat triangulation Nash embedding theorem Whitney triangulation Bounding manifold Efficient filling 



The authors would like to thank Dylan Thurston for some helpful comments. We would also like to thank the various anonymous referees for remarks which greatly aided in the exposition of this paper, specifically with substantially shortening the proof of Proposition 6, suggesting the addition of Proposition 8, and pointing us towards references [1, 16]. The work of the second author was partially supported by the NSF, under Grants DMS-1207782, DMS-1510640, and DMS-1812028. The research of the third author was partially supported by an AMS-Simons travel Grant.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.ColumbusUSA
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA
  3. 3.Department of Mathematical and Digital SciencesBloomsburg UniversityBloomsburgUSA

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