This chapter concludes the first part of this book by introducing the nonalgorithmic aspects of some of the recent results in computational topology. In Chapter 1, we established the primary goal of this book: the computational exploration of topological spaces. Having laid the mathematical foundation required for this study in the previous four chapters, we now take steps toward this goal through

• persistence;

• hierarchical Morse-Smale complexes;

• and the linking number for simplicial complexes.

The three sections of this chapter elaborate on these topics. In Section 6.1, we introduce a new measure of importance for topological attributes called persistence. Persistence is simple, immediate, and natural. Perhaps precisely because of its naturalness, this concept is powerful and applicable in numerous areas, as we shall see in Chapter 13. Primarily, persistence enables us to simplify spaces topologically. The meaning of this simplification, however, changes according to context. For example, topological simplification of Morse-Smale complexes corresponds to geometric smoothing of the associated function. To apply persistence to sampled density functions, we extend Morse-Smale complexes to piece-wise linear (PL) manifolds in Section 6.2. This extension will allow us to construct hierarchical PL Morse-Smale complexes, providing us with an intelligent method for noise reduction in sampled data.