239
Views
20
CrossRef citations to date
0
Altmetric
Original Articles

Random Discrete Morse Theory and a New Library of Triangulations

&

References

  • K. [Adiprasito et al. 14] Adiprasito, B. Benedetti, and F. H. Lutz. “Random Discrete Morse Theory II and a Collapsible 5-Manifold Different from the 5-Ball.” In preparation, 2014.
  • S. [Akbulut and Kirby 85] Akbulut and R. Kirby. “A Potential Smooth Counterexample In Dimension 4 to the Poincar´e Conjecture, the Schoenflies Conjecture, and the Andrews–Curtis Conjecture.” Topology 24 (1985), 375–390.
  • A. [Akhmedov and Park 10] Akhmedov and B. D. Park. “Exotic Smooth Structures on Small 4-Manifolds with Odd Signatures.” Invent. Math. 181, (2010), 577–603.
  • A. [Altshuler et al. 96] Altshuler, J. Bokowski, and P. Schuchert. “Neighborly 2-Manifolds with 12 Vertices.” J. Comb. Theory, Ser. A 75 (1996), 148–162.
  • L. [Aronshtam et al. 13] Aronshtam, N. Linial, T. Łuczak, and R. Meshulam. “Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes.” Discrete Comput. Geom. 49 (2013), 317–334.
  • E. [Babson et al. 10] Babson, C. Hoffman, and M. Kahle. “The Fundamental Group of Random 2-Complexes.” J. Am. Math. Soc. 24 (2010), 1–28.
  • B. [Bagchi and Datta 05] Bagchi and B. Datta. “Combinatorial Triangulations of Homology Spheres.” Discrete Math. 305, 1–17 (2005).
  • D. [Barnette 73] Barnette. “The Triangulations of the 3-Sphere with up to 8 Vertices.” J. Comb. Theory, Ser. A 14 (1973), 37–52.
  • B. [Benedetti 12] Benedetti. “Discrete Morse Theory for Manifolds with Boundary.” Trans. Am. Math. Soc. 364 (2012), 6631–6670.
  • B. [Benedetti 13] Benedetti. “Smoothing Discrete Morse Theory.” arXiv:1212.0885v2, 2013.
  • B. [Benedetti and Lutz 13a] Benedetti and F. H. Lutz. “Knots in Collapsible and Non-Collapsible Balls.” Electron. J. Comb. 20 No. 3, Research Paper P31, 2013.
  • B. [Benedetti and Lutz 13b] Benedetti and F. H. Lutz. “The Dunce Hat and a Minimal Non-extendably Collapsible 3-Ball.” Electronic Geometry Model No. 2013. 10.001. Available online (http://www.eg-models.de/2013.10.001), 2013.
  • R. H. [Bing 64] Bing. “Some Aspects of the Topology of 3-Manifolds Related to the Poincar´e Conjecture.” In Lectures on Modern Mathematics II, edited by T. L. Saaty, pp. 93–128. Wiley, New York, 1964.
  • A. [Bj¨orner and Lutz 00] Bj¨orner and F. H. Lutz. “Simplicial Manifolds, Bistellar Flips and a 16-Vertex Triangulation of the Poincar´e Homology 3-Sphere.” Exp. Math. 9 (2000), 275–289.
  • A. [Bj¨orner and Lutz 03] Bj¨orner and F. H. Lutz. “A 16-Vertex Triangulation of the Poincar´e Homology 3-Sphere and Non-PL Spheres with Few Vertices.” Electronic Geometry Model No. 2003.04.001. Available online (http://www.eg-models.de/2003. 04.001), 2003.
  • M. [Boileau and Zieschang 84] Boileau and H. Zieschang. “Heegaard Genus of Closed Orientable Seifert 3-Manifolds.” Invent. Math. 76 (1984), 455–468.
  • U. [Brehm and K¨uhnel 92] Brehm and W. K¨uhnel. “15-Vertex Triangulations of an 8-Manifold.” Math. Ann. 294 (1992), 167–193.
  • M. [Casella and K¨uhnel 01] Casella and W. K¨uhnel. “A Triangulated K3 Surface with the Minimum Number of Vertices.” Topology 40 (2001), 753–772.
  • M. K. [Chari 00] Chari. “On Discrete Morse Functions and Combinatorial Decompositions.” Discrete Math. 217 (2000), 101–113.
  • D. [Cohen et al. 12] Cohen, A. Costa, M. Farber, and T. Kappeler. “Topology of Random 2-Complexes.” Discrete Comput. Geom. 47 (2012), 117–149.
  • K. [Crowley et al. 03] Crowley, A. Ebin, H. Kahn, P. Reyfman, J. White, and M. Xue. “Collapsing a Simplex to aNoncollapsible Simplicial Complex.” Preprint, 2003.
  • P. [Csorba and Lutz 06] Csorba and F. H. Lutz. “GraphColoringManifolds.” In Algebraic and Geometric Combinatorics, Euroconf. Math., Algebraic and Geometric Combinatorics, Anogia, Crete,Greece, 2005, edited by C. A. Athanasiadis, V. V. Batyrev, D. I. Dais, M. Henk, and F. Santos, Contemporary Mathematics 423, pp. 51–69. American Mathematical Society, 2006.
  • N. M. [Dunfield and Thurston 06] Dunfield and W. P. Thurston. “Finite Covers of Random 3-Manifolds.” Invent. Math. 166 (2006), 457–521.
  • A. [Engstr¨om 09] Engstr¨om. “Discrete Morse Functions from Fourier Transforms.” Exp. Math. 18 (2009), 45–53.
  • P. [Erd˝os and R´enyi 60] Erd˝os and A. R´enyi. “On the Evolution of Random Graphs.” Publ. Math. Inst. Hung. Acad. Sci., Ser. A 5 (1960) 17–61.
  • R. [Forman 98] Forman. “Morse Theory for Cell Complexes.” Adv. Math. 134 (1998), 90–145.
  • R. [Forman 02] Forman. “A User's Guide to Discrete Morse Theory.” S´emin. Lothar. Comb. 48 (2002), B48c (electronic).
  • M. H. [Freedman 82] Freedman. “The Topology of Four-Dimensional Manifolds.” J. Differ. Geom. 17 (1982), 357–453.
  • R. E. [Goodrick 68] Goodrick. “Non-simplicially Collapsible Triangulations of I n.” Proc. Camb. Phil. Soc. 64 (1968), 31–36.
  • M. [Hachimori 01] Hachimori. “Simplicial Complex Library.” Available online (http://infoshako.sk.tsukuba.ac.jp/∼hachi/math/library/index eng.html), 2001.
  • P. [Hersh 05] Hersh. “On Optimizing Discrete Morse Functions.” Adv. Appl. Math. 35 (2005), 294–322.
  • M. [Joswig 04] Joswig. “Computing Invariants of Simplicial Manifolds.” arXiv:math.AT/0401176, 2004.
  • M. [Joswig and Pfetsch 06] Joswig and M. E. Pfetsch. “Computing Optimal Morse Matchings.” SIAM J. Discrete Math. 20 (2006), 11–25.
  • J. [Kahn et al. 84] Kahn, M. Saks, and D. Sturtevant. “A Topological Approach to Evasiveness.” Combinatorica 4 (1984), 297–306.
  • V. [Kaibel and Pfetsch 02] Kaibel and M. E. Pfetsch. “Computing the Face Lattice of a Polytope from Its Vertex–Facet Incidences.” Comput. Geom. 23 (2002), 281–290.
  • V. [Kaibel and Pfetsch 03] Kaibel and M. E. Pfetsch. “Some Algorithmic Problems In Polytope Theory.” In Algebra, Geometry,and Software Systems, edited by M. Joswig and N. Takayama, pp. 23–47. Springer, 2003.
  • H. [King et al. 05] King, K. Knudson, and N. Mramor. “Generating Discrete Morse Functions from Point Data.” Exp. Math. 14 (2005), 435–444.
  • M. [Kreck 84] Kreck. “Some Closed 4-Manifolds with Exotic Differentiable Structure.” In Algebraic Topology, Aarhus 1982, Proc. Conference Held in Aarhus, Denmark, 1982, edited by I. Madsen and B. Oliver, Lecture Notes in Mathematics 1051, pp. 246–262. Springer, 1984.
  • W. [K¨uhnel 87] K¨uhnel. “Minimal Triangulations of Kummer Varieties.” Abh. Math. Sem. Univ. Hamburg 57 (1987), 7–20.
  • W. [K¨uhnel and Banchoff 83] K¨uhnel and T. F. Banchoff. “The 9-Vertex Complex Projective Plane.” Math. Intell. 5 (1983), 11–22.
  • T. [Lewiner 05] Lewiner. “Geometric Discrete Morse Complexes.” PhD thesis, Pontif´ıcia Universidade Cat´olica do Rio de Janeiro, 2005.
  • T. [Lewiner et al. 03a] Lewiner, H. Lopes, and G. Tavares. “Optimal Discrete Morse Functions for 2-Manifolds.” Comput. Geom. 26 (2003), 221–233.
  • T. [Lewiner et al. 03b] Lewiner, H. Lopes, and G. Tavares. “Toward Optimality in Discrete Morse Theory.” Exp. Math. 12 (2003), 271–285.
  • W. B. R. [Lickorish 91] Lickorish. “Unshellable Triangulations of Spheres.” Eur. J. Comb. 12 (1991), 527–530.
  • N. [Linial and Meshulam 06] Linial and R. Meshulam. “Homological Connectivity of Random 2-Complexes.” Combinatorica 26 (2006), 475–487.
  • N. [Linial et al. 10] Linial, R. Meshulam, and M. Rosenthal. “Sum Complexes—A New Family of Hypertrees.” Discrete Comput. Geom. 44 (2010), 622–636.
  • F. H. [Lutz 99] Lutz. Triangulated Manifolds with Few Vertices and Vertex-Transitive Group Actions. Shaker Verlag, Aachen, 1999.
  • F. H. [Lutz 02] Lutz. “Examples of Z-Acyclic and Contractible Vertex-Homogeneous Simplicial Complexes.” Discrete Comput. Geom. 27 (2002), 137–154.
  • F. H. [Lutz 03] Lutz. “Triangulated Manifolds with Few Vertices: Geometric 3-Manifolds.” arXiv:math.GT/0311116, 2003.
  • F.H. [Lutz 04a] Lutz. “AVertex-Minimal Non-shellable Simplicial 3-Ball with 9 Vertices and 18 Facets.” Electronic Geometry Model No. 2003.05.004. Available online (http://www.eg-models.de/ 2003.05.004/), 2004.
  • F. H. [Lutz 04b] Lutz. “Small Examples of Nonconstructible Simplicial Balls and Spheres.” SIAM J. Discrete Math. 18 (2004), 103–109.
  • F. H. [Lutz 05] Lutz. “Triangulated Manifolds with Few Vertices: Combinatorial Manifolds.” arXiv:math.CO/0506372, 2005.
  • F. H. [Lutz 08] Lutz. “Combinatorial 3-Manifolds with 10 Vertices.” Beitr. Algebra Geom. 49 (2008), 97–106.
  • F. H. [Lutz and Møller 14] Lutz and J. M. Møller. “A Non-(4, 2)-Colorable Triangulation of the 3-Sphere.” In preparation, 2014.
  • F. H. [Lutz et al. 09] Lutz, T. Sulanke, and E. Swartz. “ f -Vectors of 3-Manifolds.” Electron. J. Comb. 16 (2009), research paper R13.
  • [Milnor 11] J. Milnor. “Differential Topology Forty-Six Years Later.” Notices Am. Math. Soc. 58 (2011), 804–809.
  • L. I. [Nicolaescu 12] Nicolaescu. “Combinatorial Morse Flows Are Hard to Find.” arXiv:1202.2724, 2012.
  • Y. [Ollivier 05] Ollivier. A January 2005 Invitation to Random Groups, Ensaios Matem´aticos 10. Sociedade Brasileira de Matem´atica, Rio de Janeiro, 2005.
  • U. [Pachner 87] Pachner. “Konstruktionsmethoden und das kombinatorische Hom¨oomorphieproblem f¨ur Triangulationen kompakter semilinearer Mannigfaltigkeiten.” Abh. Math. Sem. Univ. Hamburg 57 (1987), 69–86.
  • M. E. [Rudin 58] Rudin. “An Unshellable Triangulation of a Tetrahedron.” Bull. Am. Math. Soc. 64 (1958), 90–91.
  • H. [Seifert and Threlfall 34] Seifert and W. Threlfall. Lehrbuch der Topologie. B. G. Teubner, 1934.
  • J. [Spreer and K¨uhnel 11] Spreer and W. K¨uhnel. “Combinatorial Properties of the K3 Surface: Simplicial Blowups and Slicings.” Exp. Math. 20, 201–216 (2011).
  • T. [Sulanke and Lutz 09] Sulanke and F. H. Lutz. “Isomorphism Free Lexicographic Enumeration of Triangulated Surfaces and 3-Manifolds.” Eur. J. Comb. 30 (2009), 1965–1979.
  • M. [Tancer 12] Tancer. “Recognition of Collapsible Complexes Is NP-Complete.” arXiv:1211.6254, 2012.
  • M. [Tsuruga and Lutz 13] Tsuruga and F. H. Lutz. “Constructing Complicated Spheres.” arXiv:1302.6856, 2013.
  • C. [Weber and Seifert 33] Weber and H. Seifert. “Die beiden Dodekaederr ¨aume.” Math. Z. 37 (1933), 237–253.
  • V. [Welker 99] Welker. “Constructions Preserving Evasiveness and Collapsibility.” Discrete Math. 207 (1999), 243–255.
  • J. H. C. [Whitehead 39] Whitehead. “Simplicial Spaces, Nuclei and m-Groups.” Proc. Lond. Math. Soc., II. Ser. 45 (1939), 243–327.
  • R. F. [Wotzlaw 05] Wotzlaw. “Rudin's Non-shellable Ball.” Electronic Geometry Model No. 2004.08.001. Available online (http://www.eg-models.de/2004.08.001), 2005.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.