A controlled local-global theorem for simplicial complexes

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.

13 pages, 4 figures, part of my PhD thesis, minor TeX corrections in abstract

References in corpus (1)

• Triangular homotopy equivalences

Cited by in corpus (1)

• Algebraic subdivision in simplicially controlled categories