Piecewise Linear Morse Theory

Classical Morse Theory [8] considers the topological changes of the level sets Mh = {x ∈ M | f(x) = h } of a smooth function f defined on a manifold M as the height h varies. At critical points, where the gradient of f vanishes, the topology changes. These changes can be classified locally, and they can be related to global topological properties of M . Between critical values, the level sets vary smoothly. This talk concerns Morse Theory of piecewise linear functions, and in particular, the “uninteresting” part of Morse theory, the level sets between the critical values, where “nothing happens”. Spatial data coming from data acquisition processes (like medical imaging) or numerical simulations (like fluid dynamics) need to be represented for the purpose of storage on a computer, visualization, or further processing. Commonly they are represented as piecewise linear functions. My interest in Morse theory arose out of a fast and simple algorithm [4] for constructing the contour tree (or Reeb graph) of a piecewise linear function, a tree that represents how the connected components of the level sets, the contours, split and merge, are created and destroyed. While writing up this algorithm, I felt that I should say something about the obvious absence of topological changes when passing over “non-critical” vertices, but I could not find any results in the literature that I could readily apply. The results below are a contribution towards the foundations of Morse theory for piecewise linear functions of up to three variables.