Triangular homotopy equivalences

Abstract: 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 $\sigma\in Y$, $f|\sigma:f^{-1}(\sigma) \to \sigma$ is a homotopy equivalence with inverse $g|\sigma:\sigma \to f^{-1}(\sigma)$ and homotopies $h_1|\sigma$ and $h_2|_\sigma$. In this paper we prove that for all pairs $X,Y$ of finite-dimensional locally finite simplicial complexes there is an $\epsilon(X,Y)>0$ such that any $\epsilon$-controlled homotopy equivalence $f:X\to Y$ for $\epsilon<\epsilon(X,Y)$ 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 $\epsilon$-controlled homotopy equivalence for all $\epsilon>0$.