# Geometry of the space of triangulations of a compact manifold

- 61 Downloads
- 7 Citations

## Abstract

In this paper we study the space*T*_{M} of triangulations of an arbitrary compact manifold*M* 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 on*M*.

Our main result can be informally explained as follows: Let*M* 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 constant*C*>1 depending only on the dimension of*M* and for all sufficiently large*N* the subset*T*_{M}(N) of*T*_{M} formed by all triangulations of*M* with ≦*N* simplices can be represented as the union of at least [*C*^{N}] disjoint non-empty subsets such that any two of these subsets are “very far” from each other in the metric of*T*_{M}. As a corollary, we show that for any functional from a very wide class of functionals on*T*_{M} the number of its “deep” local minima in*T*_{M}(N) grows at least exponentially with*N*, when*N*→∞.

### Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics## Preview

Unable to display preview. Download preview PDF.

### References

- [A] Alexander, J.W.: The combinatorial theory of complexes. Ann. Math.
**31**, 292–320 (1930)MathSciNetGoogle Scholar - [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 - [AM] Agishtein, M.E., Migdal, A.A.: Nucl. Phys. B
**385**, 395 (1992)CrossRefGoogle Scholar - [AJK] Ambjorn, F.J., Jurkiewicz, J., Kristjansen, C.F.: Quantum gravity, dynamical triangulations and higher derivative regularization. Nucl. Phys. B
**393**, 601 (1993)CrossRefGoogle Scholar - [B] Barzdin, J.M.: Complexity of programs to determine whether natural numbers not greater than
*n*belong to a recursively enumerable set. Soviet Math. Doklady**9**, 1251–1254 (1968)Google Scholar - [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
- [Ca] Cairns, S.S.: Triangulated manifolds which are not Brower manifolds. Ann. Math.
**41**, 792–795 (1940)Google Scholar - [C] Chaitin, G.: Information, Randomness and Incompleteness. Singapore: World Scientific, 1987Google Scholar
- [CMS] Cheeger, F.J., Muller, W., Schrader, R.: On the curvature of piecewise-flat spaces. Commun. Math. Phys.
**92**, 405–454 (1984)CrossRefGoogle Scholar - [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 - [D] Daley, R.P.: Minimal program complexity of sequences with restricted resources. Inf. and Control
**23**, 301–312 (1973)CrossRefGoogle Scholar - [F] Fomenko, A.T.: Differential Geometry and Topology. N.Y.: Plenum, 1987Google Scholar
- [G] Glaser, L.C.: Geometrical Combinatorial Topology, Vol. 1, Amsterdam: Van Nostrand, 1970Google Scholar
- [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 - [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 Series**182**, 1993Google Scholar - [GV] Gross, M., Varsted, S.: Nucl. Phys. B
**378**, 367 (1992)CrossRefGoogle Scholar - [H] Hopf, H.: Fundamentalgruppe und Zweite Bettische Gruppe. Comm. Math. Helv.
**14**, 257–309 (1941/1942)Google Scholar - [J] Jurkiewicz, J.: Simplicial gravity and random surfaces. Nucl. Phys. B (Proc. Suppl.)
**30**, 108–121 (1993)CrossRefGoogle Scholar - [K] Kervaire, M.: Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc.
**149**, 67–72 (1969)Google Scholar - [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
- [M] Yu I. Manin.: A course in Mathematical Logic. Berlin-Heidelberg-New York: Springer, 1977Google Scholar
- [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
- [Mn] Milnor, J.: Introduction to algebraic
*K*-theory. Annals of Mathematical Studies, Princeton, NJ: Princeton University Press, 1971Google Scholar - [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 - [N0] Nabutovsky, A.: Non-recursive functions in real algebraic geometry. Bull. Amer. Math. Soc.
**20**, 61–65 (1989)Google Scholar - [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 - [N2] Nabutovsky, A.: Disconnectedness of sublevel sets of some Riemannian functionals. To appear in Geom. Funct. Anal.Google Scholar
- [N3] Nabutovsky, A.: Fundamental group and contractible closed geodesics. To appear in Comm. on Pure and Appl. Math.Google Scholar
- [N4] Nabutovsky, A.: Einstein structures: Existence versus uniqueness. Geom. Funct. Anal.
**5**, 76–91 (1995)CrossRefGoogle Scholar - [N5] Nabutovsky, A.: Exponential growth of the number of connected components of sublevel sets of some Riemannian functionals. In preparationGoogle Scholar
- [P1] Pachner, U.: Konstruktionsmethoden und das kombinatorische Homoomorphieproblem fur Triangulationen kompakter semilinearer Mannigfaltigkeiten. Abh. Math. Sem. Univ. Hamburg
**57**, 69–85 (1986)Google Scholar - [P2] Pachner, U.: P.L. Homeomorphic Manifolds are Equivalent by Elementary Shellings. Europ. J. Combinatorics
**12**, 129–145 (1991)Google Scholar - [R] Rotman, J.J.: An introduction to the theory of groups. Boston, MA: Allyn and Bacon, 1984Google Scholar
- [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 - [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