Abstract
In this paper we study the spaceT M 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 subsetT M(N) ofT M formed by all triangulations ofM 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 ofT M. As a corollary, we show that for any functional from a very wide class of functionals onT M the number of its “deep” local minima inT M(N) grows at least exponentially withN, whenN→∞.
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Communicated by S.-T. Yau
This work was partially supported by the New York University Research Challenge Fund Grant, by Grant ARO-DAAL-03-92-G-0143 and by NSERC Grant OGP0155879.
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Nabutovsky, A. Geometry of the space of triangulations of a compact manifold. Commun.Math. Phys. 181, 303–330 (1996). https://doi.org/10.1007/BF02101007
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DOI: https://doi.org/10.1007/BF02101007