A controlled local-global theorem for simplicial complexes

TL;DR

This paper establishes a equivalence between contractible point inverses of simplicial maps and controlled homotopy equivalences, linking local contractibility to bounded control in the open cone over the space.

Contribution

It proves a new controlled local-global theorem for simplicial complexes, connecting local contractibility with bounded control in the open cone.

Findings

Contractible point inverses correspond to $orall \, ext{epsilon}>0$, epsilon-controlled homotopy equivalences.

The construction of a family of cellulations provides a retracting map for controlled homotopy.

The theorem confirms the equivalence between fine local control and bounded control in the open cone.

Abstract

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 $\epsilon$-controlled homotopy equivalence for all $\epsilon>0$ 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 $\{X_\epsilon^\prime\}_{0<\epsilon<\epsilon(X)}$ is constructed which provides a retracting map for $X$ which can be used to compensate for sufficiently small control.