Triangular homotopy equivalences

Abstract

A map f : X → Y to a simplicial complex Y is called a Y -triangular homotopy equivalence if it has a homotopy inverse g and homotopies h1 : and homotopies h1|σ and h2|σ. In this paper we prove that for all pairs X, Y of finite-dimensional locally finite simplicial complexes there is an ǫ(X, Y ) > 0 such that any ǫ-controlled homotopy equivalence f : X → Y for ǫ < ǫ(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 ǫ-controlled homotopy equivalence for all ǫ > 0.