Triangulation and general position of PL diagrams

Abstract

We give conditions under which a diagram R ← g P → g Q of polyhedra and PL maps is triangulable. Suppose there is an integer m such that if q 1,…, q r are distinct points of Q with gf -1( q i )∩ gf -1( q i+1 )≠ ∅, i = 1,…, r - 1, then r ⩽ m. Then there are triangulations K, H, and J of P, Q, and R, respectively, with respect to which f and g are simplicial. Moreover, if f and g are surjections, then the relation q ∼ q' if gf -1( q) ∩ gf -1( q') ≠ ∅ defines a quotient complex L of H and simplicial maps ψ : H → L and φ : J → L such that the diagram ▪ is commutative. Suppose R is a PL n-manifold, and for some triangulation T of P, dim σ + max{dim f -1( q) ∩ σ | q ∈ Q}< n for each σ ∈ T. Then g may be approximated by a PL map h : P → R such that the diagram R ← g P → g Q satisfies the hypothesis above with m = 4 n 2, and, hence, is triangulable. These are adaptations and generalizations of results of T. Homma, which we use to give a proof of R. Miller's codimension-three PL approximation theorem: If f : M m → N n , n − m ⩾ 3, is a topological embedding of a PL m-manifold M into a PL n-manifold N, then f can be approximated by PL embeddings.