Günter Rote:

Piecewise linear Morse theory

In: Oberwolfach Reports, 3, European Mathematical Society - Publishing House, 2006, pp. 696–698. doi:0.4171/OWR/2006/12

Abstract

Classical Morse Theory considers the topological changes of the level sets Mh={x in 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. We prove that the same statement is true for piecewise linear functions in up to three variables: between critical values, all level sets are isotopic.

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Last update: June 12, 2007.