Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Practical Software for Triangulating and Simplifying 4-Manifolds (2402.15087v1)

Published 23 Feb 2024 in math.GT and cs.CG

Abstract: Dimension 4 is the first dimension in which exotic smooth manifold pairs appear -- manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of 4-manifolds from Kirby diagrams, as well as a new heuristic for simplifying 4-manifold triangulations. Using these tools, we present new triangulations of several bounded exotic pairs, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the fundamental K3 surface. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (46)
  1. Colin C. Adams. The knot book. W. H. Freeman and Company, New York, 1994. An elementary introduction to the mathematical theory of knots.
  2. S. I. Adyan. Algorithmic unsolvability of problems of recognition of certain properties of groups. Dokl. Akad. Nauk SSSR (N.S.), 103:533–535, 1955.
  3. Selman Akbulut. An exotic 4444-manifold. J. Differential Geom., 33(2):357–361, 1991. URL: http://projecteuclid.org/euclid.jdg/1214446321.
  4. Selman Akbulut. A fake compact contractible 4444-manifold. J. Differential Geom., 33(2):335–356, 1991. URL: http://projecteuclid.org/euclid.jdg/1214446320.
  5. Selman Akbulut. The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture. Comment. Math. Helv., 87(1):187–241, 2012. doi:10.4171/CMH/252.
  6. Selman Akbulut. 4-manifolds, volume 25 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2016. doi:10.1093/acprof:oso/9780198784869.001.0001.
  7. Corks, plugs and exotic structures. J. Gökova Geom. Topol. GGT, 2:40–82, 2008.
  8. Random discrete Morse theory and a new library of triangulations. Exp. Math., 23(1):66–94, 2014. doi:10.1080/10586458.2013.865281.
  9. Rhuaidi Antonio Burke. Combinatorial structures of small 4-manifold triangulations (in preparation). 2024+.
  10. Benjamin A. Burton. Computational topology with Regina: algorithms, heuristics and implementations. In Geometry and topology down under, volume 597 of Contemp. Math., pages 195–224. Amer. Math. Soc., Providence, RI, 2013. doi:10.1090/conm/597/11877.
  11. Regina: Software for low-dimensional topology. http://regina-normal.github.io/, 1999–2023.
  12. Computationally proving triangulated 4-manifolds to be diffeomorphic, 2014. arXiv:1403.2780.
  13. Stewart S. Cairns. Triangulation of the manifold of class one. Bull. Amer. Math. Soc., 41(8):549–552, 1935. doi:10.1090/S0002-9904-1935-06140-3.
  14. Stewart S. Cairns. A simple triangulation method for smooth manifolds. Bull. Amer. Math. Soc., 67:389–390, 1961. doi:10.1090/S0002-9904-1961-10631-9.
  15. Maria Rita Casali. From framed links to crystallizations of bounded 4-manifolds. J. Knot Theory Ramifications, 9(4):443–458, 2000. doi:10.1142/S0218216500000220.
  16. Kirby diagrams and 5-colored graphs representing compact 4-manifolds. Rev. Mat. Complut., 36(3):899–931, 2023. doi:10.1007/s13163-022-00438-x.
  17. G-degree for singular manifolds. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112(3):693–704, 2018. doi:10.1007/s13398-017-0456-x.
  18. A triangulated K⁢3𝐾3K3italic_K 3 surface with the minimum number of vertices. Topology, 40(4):753–772, 2001. doi:10.1016/S0040-9383(99)00082-8.
  19. SnapPy, a computer program for studying the geometry and topology of 3333-manifolds. Available at http://snappy.computop.org.
  20. A decomposition theorem for hℎhitalic_h-cobordant smooth simply-connected compact 4444-manifolds. Invent. Math., 123(2):343–348, 1996. doi:10.1007/s002220050031.
  21. S. K. Donaldson. Irrationality and the hℎhitalic_h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987. URL: http://projecteuclid.org/euclid.jdg/1214441179.
  22. A graph-theoretical representation of PL-manifolds—a survey on crystallizations. Aequationes Math., 31(2-3):121–141, 1986. doi:10.1007/BF02188181.
  23. Rational blowdowns of smooth 4444-manifolds. J. Differential Geom., 46(2):181–235, 1997. URL: http://projecteuclid.org/euclid.jdg/1214459932.
  24. Constructions of smooth 4444-manifolds. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), pages 443–452, 1998.
  25. Robert E. Gompf. Nuclei of elliptic surfaces. Topology, 30(3):479–511, 1991. doi:10.1016/0040-9383(91)90027-2.
  26. 4444-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.
  27. Smoothings of piecewise linear manifolds, volume No. 80 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974.
  28. 00-efficient triangulations of 3-manifolds. J. Differential Geom., 65(1):61–168, 2003. URL: http://projecteuclid.org/euclid.jdg/1090503053.
  29. Robion Kirby. A calculus for framed links in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Invent. Math., 45(1):35–56, 1978. doi:10.1007/BF01406222.
  30. M. Kreck. Some closed 4444-manifolds with exotic differentiable structure. In Algebraic topology, Aarhus 1982 (Aarhus, 1982), volume 1051 of Lecture Notes in Math., pages 246–262. Springer, Berlin, 1984. doi:10.1007/BFb0075570.
  31. Greg Kuperberg. Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pacific J. Math., 301(1):189–241, 2019. doi:10.2140/pjm.2019.301.189.
  32. Marc Lackenby. Algorithms in 3-manifold theory. In Surveys in differential geometry 2020. Surveys in 3-manifold topology and geometry, volume 25 of Surv. Differ. Geom., pages 163–213. Int. Press, Boston, MA, [2022] ©2022.
  33. A note on 4444-dimensional handlebodies. Bull. Soc. Math. France, 100:337–344, 1972. URL: http://www.numdam.org/item?id=BSMF_1972__100__337_0.
  34. A. Markov. The insolubility of the problem of homeomorphy. Dokl. Akad. Nauk SSSR, 121:218–220, 1958.
  35. R. Matveyev. A decomposition of smooth simply-connected hℎhitalic_h-cobordant 4444-manifolds. J. Differential Geom., 44(3):571–582, 1996. URL: http://projecteuclid.org/euclid.jdg/1214459222.
  36. James Munkres. Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. of Math. (2), 72:521–554, 1960. doi:10.2307/1970228.
  37. Hironobu Naoe. Corks with large shadow-complexity and exotic four-manifolds. Exp. Math., 30(2):157–171, 2021. doi:10.1080/10586458.2018.1514332.
  38. U. Pachner. Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten. Abh. Math. Sem. Univ. Hamburg, 57:69–86, 1987. doi:10.1007/BF02941601.
  39. Jongil Park. Non-complex symplectic 4-manifolds with b2+=1superscriptsubscript𝑏21b_{2}^{+}=1italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT = 1, 2001. arXiv:math/0108220.
  40. Jongil Park. Simply connected symplectic 4-manifolds with b2+=1subscriptsuperscript𝑏21b^{+}_{2}=1italic_b start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 and c12=2subscriptsuperscript𝑐212c^{2}_{1}=2italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2. Invent. Math., 159(3):657–667, 2005. doi:10.1007/s00222-004-0404-1.
  41. Michael O. Rabin. Recursive unsolvability of group theoretic problems. Ann. of Math. (2), 67:172–194, 1958. doi:10.2307/1969933.
  42. Alexandru Scorpan. The wild world of 4-manifolds. American Mathematical Society, Providence, RI, 2005.
  43. Combinatorial properties of the K⁢3𝐾3K3italic_K 3 surface: simplicial blowups and slicings. Exp. Math., 20(2):201–216, 2011. doi:10.1080/10586458.2011.564546.
  44. Face numbers of triangulations of manifolds, 2024. arXiv:2401.11152.
  45. J. H. C. Whitehead. On C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-complexes. Ann. of Math. (2), 41:809–824, 1940. doi:10.2307/1968861.
  46. Kouichi Yasui. Corks, exotic 4-manifolds and knot concordance, 2017. arXiv:1505.02551.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now