2013/10/10 by Adams-Florou, Spiros
#57Qxx #57R67 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1310.2768
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:f∘ g≃ idY, h2:g∘ f≃ idX such that for all simplices σ∈ Y, f|σ:f-1(σ) → σ is a homotopy equivalence with inverse g|σ:σ→ f-1(σ) 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 ε(X,Y)>0 such that any ε-controlled homotopy equivalence f:X→ Y for ε0.