A controlled local-global theorem for simplicial complexes

In this paper we prove that a simplicial map of finite-dimensional locally finite simplicial complexes has contractible point inverses if and only if it is an $ε$-controlled homotopy equivalence for all $ε>0$ if and only if $f\times \mathrm{id}_\mathbb{R}$ is a bounded homotopy equivalence measured in the open cone over the target. This confirms for such a space $X$ the slogan that arbitrarily fine control over $X$ corresponds to bounded control over the open cone $O(X_+)$. For the proof a one parameter family of cellulations $\{X_ε^\prime\}_{0<ε<ε(X)}$ is constructed which provides a retracting map for $X$ which can be used to compensate for sufficiently small control.

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