Optimal topological simplification of discrete functions on surfaces
In this note we consider piecewise linear mappings of combinatorial manifolds into the real line. We define—in a purely combinatorial way—a certain class of mappings called nondegenerate and we prove that every continuous mapping may be approximated by a nondegenerate one. Nondegenerate mappings on a combinatorial manifold behave like differentiable nondegenerate mappings on a differentiate manifold. In particular, the index of a singularity can be defined and one can prove the Morse inequalities and an analogue of the Reeb theorem about a function with only two nondegenerate singularities. The approximation theorem can also be extended to height functions on combinatorial submanifolds of Euclidean space. Detailed proofs will be published later. We give here descriptions of the singularities, a statement of the main theorems and a very brief sketch of the approximation theorem.