# Geometry of the space of triangulations of a compact manifold

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## 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

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