Communications in Mathematical Physics

, Volume 181, Issue 2, pp 303–330 | Cite as

Geometry of the space of triangulations of a compact manifold

  • A. Nabutovsky
Article

Abstract

In this paper we study the spaceTM of triangulations of an arbitrary compact manifoldM of dimension greater than or equal to four. This space can be endowed with the metric defined as the minimal number of bistellar operations required to transform one of two considered triangulations into the other. Recently, this space became and object of study in Quantum Gravity because it can be regarded as a “toy” discrete model of the space of Riemannian structures onM.

Our main result can be informally explained as follows: LetM be either any compact manifold of dimension greater than four or any compact four-dimensional manifold from a certain class described in the paper. We prove that for a certain constantC>1 depending only on the dimension ofM and for all sufficiently largeN the subsetTM(N) ofTM formed by all triangulations ofM with ≦N simplices can be represented as the union of at least [CN] disjoint non-empty subsets such that any two of these subsets are “very far” from each other in the metric ofTM. As a corollary, we show that for any functional from a very wide class of functionals onTM the number of its “deep” local minima inTM(N) grows at least exponentially withN, whenN→∞.

Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A] Alexander, J.W.: The combinatorial theory of complexes. Ann. Math.31, 292–320 (1930)MathSciNetGoogle Scholar
  2. [ABB] Acquistapace, F., Benedetti, R., Broglia, F.: Effectiveness-non-effectiveness in semi-algebraic and PL geometry. Inv. Math.102, (1), 141–156 (1990)CrossRefGoogle Scholar
  3. [AM] Agishtein, M.E., Migdal, A.A.: Nucl. Phys. B385, 395 (1992)CrossRefGoogle Scholar
  4. [AJK] Ambjorn, F.J., Jurkiewicz, J., Kristjansen, C.F.: Quantum gravity, dynamical triangulations and higher derivative regularization. Nucl. Phys. B393, 601 (1993)CrossRefGoogle Scholar
  5. [B] Barzdin, J.M.: Complexity of programs to determine whether natural numbers not greater thann belong to a recursively enumerable set. Soviet Math. Doklady9, 1251–1254 (1968)Google Scholar
  6. [BHP] Boone, W.W., Haken, W., Poénaru, V.: On recursively unsolvable problems in topology and their classification. In: “Contributions to Mathematical Logic,” Arnold Schmidt, H., Schutte, K., Thiele, H.-J. (eds.), Amsterdam; North Holland, 1968Google Scholar
  7. [Ca] Cairns, S.S.: Triangulated manifolds which are not Brower manifolds. Ann. Math.41, 792–795 (1940)Google Scholar
  8. [C] Chaitin, G.: Information, Randomness and Incompleteness. Singapore: World Scientific, 1987Google Scholar
  9. [CMS] Cheeger, F.J., Muller, W., Schrader, R.: On the curvature of piecewise-flat spaces. Commun. Math. Phys.92, 405–454 (1984)CrossRefGoogle Scholar
  10. [CH] Connelly, R., Henderson, D.: A convex 3-complex not simplicially isomorphic to a strictly convex complex. Math. Proc. Cambr. Phil. Soc.88, 299–306 (1980)Google Scholar
  11. [D] Daley, R.P.: Minimal program complexity of sequences with restricted resources. Inf. and Control23, 301–312 (1973)CrossRefGoogle Scholar
  12. [F] Fomenko, A.T.: Differential Geometry and Topology. N.Y.: Plenum, 1987Google Scholar
  13. [G] Glaser, L.C.: Geometrical Combinatorial Topology, Vol. 1, Amsterdam: Van Nostrand, 1970Google Scholar
  14. [Gr1] Gromov, M.: Hyperbolic manifolds, groups and actions. Riemannian surfaces and related topics, Kra, I, Maskit, B. (ed.), Ann. of Math. Studies,97, 183–215 (1981)Google Scholar
  15. [Gr2] Gromov, M.: Asymptotic invariants of infinite groups. Geometric Group Theory, Niblo, G.A., Roller, M.A. (ed.), Vol.2, London Math. Soc. Lecture Notes Series182, 1993Google Scholar
  16. [GV] Gross, M., Varsted, S.: Nucl. Phys. B378, 367 (1992)CrossRefGoogle Scholar
  17. [H] Hopf, H.: Fundamentalgruppe und Zweite Bettische Gruppe. Comm. Math. Helv.14, 257–309 (1941/1942)Google Scholar
  18. [J] Jurkiewicz, J.: Simplicial gravity and random surfaces. Nucl. Phys. B (Proc. Suppl.)30, 108–121 (1993)CrossRefGoogle Scholar
  19. [K] Kervaire, M.: Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc.149, 67–72 (1969)Google Scholar
  20. [LV] Li, M., Vitanyi, P.M.B.: Kolmogorov Complexity and its applications. Handbook of Theoretical Computer Science, Jan van Leeuwen, (ed.), Elsevier, 1990, pp. 187–254Google Scholar
  21. [M] Yu I. Manin.: A course in Mathematical Logic. Berlin-Heidelberg-New York: Springer, 1977Google Scholar
  22. [Mi] Miller, C.F.: Decision Problems for groups—Survey and reflections. Algorithms and Classification in Combinatorial group theory, Baumslag, G., Miller, C.F. (eds.) Berlin-Heidelberg-New York: Springer, 1989Google Scholar
  23. [Mn] Milnor, J.: Introduction to algebraicK-theory. Annals of Mathematical Studies, Princeton, NJ: Princeton University Press, 1971Google Scholar
  24. [NBA] Nabutovsky, A., Ben-Av, R.: Non-computability arising in dynamical triangulation model of Four-Dimensional Quantum Gravity. Commun. in Math. Phys.157, 93–98 (1993)CrossRefGoogle Scholar
  25. [N0] Nabutovsky, A.: Non-recursive functions in real algebraic geometry. Bull. Amer. Math. Soc.20, 61–65 (1989)Google Scholar
  26. [N1] Nabutovsky, A.: Non-recursive functions, knots “with thick ropes” and self-clenching “thick” hyperspheres. Comm. in Pure and Appl. Math.48, 381–428 (1995)Google Scholar
  27. [N2] Nabutovsky, A.: Disconnectedness of sublevel sets of some Riemannian functionals. To appear in Geom. Funct. Anal.Google Scholar
  28. [N3] Nabutovsky, A.: Fundamental group and contractible closed geodesics. To appear in Comm. on Pure and Appl. Math.Google Scholar
  29. [N4] Nabutovsky, A.: Einstein structures: Existence versus uniqueness. Geom. Funct. Anal.5, 76–91 (1995)CrossRefGoogle Scholar
  30. [N5] Nabutovsky, A.: Exponential growth of the number of connected components of sublevel sets of some Riemannian functionals. In preparationGoogle Scholar
  31. [P1] Pachner, U.: Konstruktionsmethoden und das kombinatorische Homoomorphieproblem fur Triangulationen kompakter semilinearer Mannigfaltigkeiten. Abh. Math. Sem. Univ. Hamburg57, 69–85 (1986)Google Scholar
  32. [P2] Pachner, U.: P.L. Homeomorphic Manifolds are Equivalent by Elementary Shellings. Europ. J. Combinatorics12, 129–145 (1991)Google Scholar
  33. [R] Rotman, J.J.: An introduction to the theory of groups. Boston, MA: Allyn and Bacon, 1984Google Scholar
  34. [S] Stanley, R.: Subdivisions and localh-vectors. J. of A.M.S.5, 805–852 (1992)Google Scholar
  35. [VKF] Volodin, I.A., Kuznetsov, V.E., Fomenko, A.T.: The problem of discriminating algorithmically the standard three-dimensional sphere. Russ. Math Surv.29, no. 5, 71–172 (1974)Google Scholar
  36. [ZL] Zvonkin, A.K., Levin, L.A.: the complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russ. Math. Surv.25, no. 6, 83–129 (1970)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • A. Nabutovsky
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations