Advertisement

Filling triangulated surfaces

  • Ryan Kowalick
  • Jean-François LafontEmail author
  • Barry Minemyer
Original Paper
First Online:
  • 14 Downloads

Abstract

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.

Keywords

Systolic geometry Combinatorial systolic inequality Fat triangulation Nash embedding theorem Whitney triangulation Bounding manifold Efficient filling 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

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.

References

  1. 1.
    Babenko, I.K.: Asymptotic invariants of smooth manifolds. Izv. Ross. Akad. Nauk Ser. Mat. 56(4), 707–751 (1992)zbMATHGoogle Scholar
  2. 2.
    Boissonnat, J.-D., Dyer, R., Ghosh, A.: Delaunay triangulation of manifolds. Found. Comput. Math. 18(2), 399–431 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boissonnat, J.-D., Dyer, R., Ghosh, A., Martynchuk, N.: An obstruction to Delaunay triangulations in Riemannian manifolds. Discrete Comput. Geom. 59(1), 226–237 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Breslin, W.: Thick triangulations of hyperbolic \(n\)-manifolds. Pacific J. Math. 241(2), 215–225 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cairns, S.: On the triangulation of regular loci. Ann. Math. (2) 35(3), 579–587 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cairns, S.: Polyhedral approximations to regular loci. Ann. Math. (2) 37(2), 409–415 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cairns, S.: A simple triangulation method for smooth manifolds. Bull. Am. Math. Soc. 67, 389–390 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Costantino, F., Thurston, D.: 3-manifolds efficiently bound 4-manifolds. J. Topol. 1, 703–745 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Verdière, E.C., Hubard, A., de Mesmay, A.: Discrete systolic inequalities and decompositions of triangulated surfaces. Discrete Comput. Geom. 53(3), 587–620 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gromov, M.: Systoles and intersystolic inequalities. Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, Paris, pp. 291–362 (1996) (English, with English and French summaries)Google Scholar
  12. 12.
    Hamenstädt, U., Hensel, S.: The geometry of handlebody groups I: distortion. J. Topol. Anal. 4, 71–97 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hass, J., Lagarias, J.C.: The minimal number of triangles needed to span a polygon embedded in \(\mathbb{R}^d\). In: Discrete and Computational Geometry, 509526. Algorithms Combin. 25. Springer, Berlin (2003)Google Scholar
  14. 14.
    Hass, J., Lagarias, J.C., Thurston, W.P.: Area inequalities for embedded disks spanning unknotted curves. J. Differ. Geom. 68, 1–29 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hass, J., Snoeyink, J., Thurston, W.P.: The size of spanning disks for polygonal curves. Discrete Comput. Geom. 29, 1–17 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hutchinson, J.P.: On short noncontractible cycles in embedded graphs. SIAM J. Discrete Math. 1, 185–192 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kowalick, R.: Discrete systolic inequalities. Ph.D. Thesis, The Ohio State University (2013)Google Scholar
  18. 18.
    Kowalick, R., Lafont, J.F., Minemyer, B.: Combinatorial systolic inequalities. Preprint. arXiv:1506.07121
  19. 19.
    Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. (2) 63, 20–63 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Peltonen, K. : On the existence of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 85 (1992)Google Scholar
  21. 21.
    Rouxel-Labbé, M., Wintraecken, M., Boissonnat, J.-D.: Discretized Riemannian Delaunay triangulations. IMR25 Proc. Eng. 163, 97–109 (2016)CrossRefGoogle Scholar
  22. 22.
    Saucan, E.: Note on a theorem of Munkres. Mediterr. J. Math. 2, 215–229 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Saucan, E.: The existence of quasimeromorphic mappings in dimension 3. Conform. Geom. Dyn. 10, 21–40 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Saucan, E.: The existence of quasimeromorphic mappings. Ann. Acad. Sci. Fenn. Math. 31, 131–142 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)CrossRefzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations

Cite article

Buy options