A Topological Description of the Topographic Map

Abstract

The aim ofMorse theory is to describe the topological changes of the (iso)level sets of a real valued function in terms of its critical points. Our purpose in this chapter is to describe several different notions of critical values (one of them will be called critical value, the other singular value) and prove that they are equivalent. One of those notions, the one of singular value, is intuitively related to the classical notion of critical value for a smooth function (points where the gradient vanishes), the other notion called critical value is based on a computational approach. When N = 2, this algorithm computes the Morse structure of the image from its upper and lower level sets. When N ≥ 3, our notion of critical value is of topological nature and not equivalent to the notion of critical value used in classical Morse theory. Indeed, a function which embeds in its level sets the transformation of a sphere into a torus by pinching a hole into it may have no critical values (in our sense), besides a maximum. Due to that, we shall refer to it as a weak version of Morse theory of the topographic map. The results of this Chapter will be used in Chapter 5 in order to justify, in any dimension N ≥ 2, the construction of the tree of shapes by merging of the trees of connected components of upper and lower level sets. Let us mention that when N = 3 the topological type of level surfaces could be later computed on the tree. Since this is not essential for our purposes, we shall not pursue it in the present text.