Simplicial approximation and complexity growth

TL;DR

This paper introduces new criteria and methods for approximating smooth manifolds using simplicial complexes, analyzing complexity growth in polyhedral sequences, with applications involving group actions and various dimensions.

Contribution

It develops novel criteria for manifold approximation and links complexity growth to geometric and topological observables, extending to infinite and finite dimensions.

Findings

New criteria for manifold approximation via triangulations

Control of complexity growth through geometric and topologic measures

Illustrative examples in finite and infinite dimensions

Abstract

This work is motivated by two problems: 1) The approach of manifolds and spaces by triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problems as related, new criteria and methods for approximating smooth manifolds are deduced. When the sequences of polyhedra are obtained by the action of a discrete group or semigroup, further control is given by geometric, topologic and complexity observables. We give a set of relevant examples to illustrate the results, both in infinite and finite dimensions.