Moduli Spaces for Dynamically Triangulated Manifolds

Abstract

Let (T _a , M = |T _a |) be a dynamically triangulated manifold. If we denote by Ω _σ (M) the family of all simplicial loops in M, starting at a given simplex σ ⁿ₀ , it easily follows that Ω _σ (M), modulo σ ⁿ₀ -based homotopic equivalence, is isomorphic to the fundamental group of M, π₁(σ ⁿ₀ , M), based at the given simplex σ ⁿ₀ . Moreover, by factoring out the effect of loops homotopic to the trivial σ ⁿ₀ -based loop, the mapping

$$ R:\omega \in \Omega _\sigma (M) \mapsto R_\omega $$

(4.1)

yields a representation of the fundamental group π₁(σ ⁿ₀ , M). The following definition specilizes to the PL setting some of the well known properties of the holonomy of riemannian manifolds.