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Given a triangulated closed oriented surface $$(M, {\mathcal {T}}_M)$$ ( M , T M ) , we provide upper bounds on the number of tetrahedra needed to construct a triangulated 3-manifold $$(N, {\mathcal {T}}_N)$$ ( N , T N ) which bounds $$(M, {\mathcal {T}}_M)$$ ( M , T M ) . Along the way, we develop a technique to translate (in all dimensions) between the famous Riemannian...
In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are...
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