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 subdivison of R(n). In the first part, we prove that a PA-map can be defined without referring to a subdivision of R(n) as a continuous mapping which coincides at every point x is an element of R(n) 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 R(n) such that for some number k is an element of 2, ..., n every face of codimension k is contained in a most 2k polyhedra, provided the subdivision contains at least one face of codimension k.