Triangular homotopy equivalences

A map $f:X\to Y$ to a simplicial complex $Y$ is called a $Y$-triangular homotopy equivalence if it has a homotopy inverse $g$ and homotopies $h_1:f\circ g\simeq \mathrm{id}_Y$, $h_2:g\circ f\simeq \mathrm{id}_X$ such that for all simplices $σ\in Y$, $f|_σ:f^{-1}(σ) \to σ$ is a homotopy equivalence with inverse $g|_σ:σ\to f^{-1}(σ)$ and homotopies $h_1|_σ$ and $h_2|_σ$. In this paper we prove that for all pairs $X,Y$ of finite-dimensional locally finite simplicial complexes there is an $ε(X,Y)>0$ such that any $ε$-controlled homotopy equivalence $f:X\to Y$ for $ε<ε(X,Y)$ is homotopic to a $Y$-triangular homotopy equivalence. Conversely, we conjecture that it is possible to `subdivide' a $Y$-triangular homotopy equivalence by finding a homotopic $(Sd\, Y)$-triangular homotopy equivalence, consequently a $Y$-triangular homotopy equivalence would be homotopic to an $ε$-controlled homotopy equivalence for all $ε>0$.

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