Piecewise linear Morse theory
Classical Morse theory studies smooth manifolds by means of certain smooth real-valued maps defined on them, namely so called Morse functions, whose critical points are non-degenerate. For this study, topological changes of level sets are examined that occur when the level defined by a real function value varies. The results of the theory allow to infer global topological properties of the manifold from local changes at critical points. In this thesis, an analogous theory for combinatorial manifolds and piecewise linear maps defined on them is presented. The focus of the thesis lies on three topics:
Our first aim is a careful step by step transfer of basic results and their proofs based on the study of level sets from classical Morse theory to the piecewise linear setting. A valuable tool is a thorough investigation of how a polyhedral complex with a map linear on its cells induces in a natural way for each level set defined as preimage of a closed interval a polyhedral complex whose domain is that level set.
As another main topic of the thesis, we compare different characterisations of regular and non-degenerate critical points. Several definitions for such points turn out to be equivalent, but two characterisations suggested in the literature impose gradually weaker requirements on such points. In this context, we also present a method to convert a discrete Morse function on a combinatorial manifold into a piecewise linear Morse function whose critical points correspond to the critical cells of the discrete Morse function.
The third topic addresses isotopies between level sets as considered in classical Morse theory as well. At least for sufficiently generic piecewise linear maps on combinatorial manifolds we prove the existence of isotopies across all level sets belonging to an interval provided that the interval contains no critical values.
The thesis concludes with considerations concerning selected computational aspects. First, we discuss the decision problem whether a given point is regular or not. Second, the algorithmic construction of the isotopy between level sets is analysed in order to obtain an upper bound for the number of cells in the combinatorially equivalent complexes that represent the isotopy.