Homeomorphism Conditions for Coherently Oriented Piecewise Affine Mappings

This article is mainly concerned with the homeomorphism problem for piecewise affine mappings (PA-maps), i.e., mappings which coincide with an affine mapping on each polyhedron of some finite polyhedral subdivision of ℝⁿ. In the first part, we prove that a PA-map can be defined without referring to a subdivision of ℝⁿ as a continuous mapping which coincides at every point x ∈ ℝⁿ with at least one function from a finite collection of affine functions. The second part studies the recession function of a PA-map. It is shown that the recession function is piecewise linear and that a coherently oriented PA-map is a homeomorphism if and only if its recession function is a homeomorphism. In the last part we prove that a coherently oriented PA-map is a homeomorphism if it admits a corresponding polyhedral subdivision of ℝⁿ such that for some number k ∈ {2, …, n} every face of codimension k is contained in at most 2k polyhedra, provided the subdivision contains at least one face of codimension k.