Critical Points Stable and Unstable Manifolds

a Stable b Unstable minimum saddle maximum Fig. 5.4. The stable a and unstable b 1-manifolds, with dotted iso-lines h −1 c, for constants c. In the diagrams, all the saddle points have height between all minima and maxima. Regions of the 2-cells of maxima and minima are shown, including the critical point, and bounded by the dotted iso-lines. The underlying manifold is S 2 , and the outer 2-cell in b corresponds to the minimum at negative infinity. minimum saddle maximum Fig. 5.5. The Morse-Smale complex of Figure 5.4. a A single cell on a gray-scale image b Graph of the cell Fig. 5.6. The Morse-Smale complex of the graph of sin x+siny is a tiling into copies of the cell shown in a, along with its reflections and rotations. Each cell has simple geometry b. half of a stable or unstable 1-manifold of a saddle, and each region is a compo- nent of the intersection of a stable 2-manifold of a maximum and an unstable 2-manifold of a minimum. Example 5.3 sin x + siny Figure 5.6 shows a single cell of the Morse- Smale complex for the graph of h x, y = sinx + siny. The cell is super- imposed on a gray-scale image, mapping h x, y to an intensity value for pixel x, y. The figure shows that each cell has simple geometry: The gradient flows from the maximum to the minimum, after being attracted by the sad- dles on each side. We saw the Morse-Smale complex for this function on a triangulated domain in Figure 1.9. 6 New Results This chapter concludes the first part of this book by introducing the nonalgo- rithmic 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 numer- ous areas, as we shall see in Chapter 13. Primarily, persistence enables us to simplify spaces topologically. The meaning of this simplification, how- ever, 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. Finally, in Section 6.3, we extend the linking number, a topological invariant detecting entanglings, to simplicial complexes. Naturally, we care about the computational aspects of these ideas and their applications. We dedicate Parts Two and Three of this book to examining these concerns. 94

6.1 Persistence

In this section, we introduce a new concept called persistence Edelsbrun- ner et al., 2002; Zomorodian and Carlsson, 2004. This notion may be placed within the framework of spectral sequences, the by-product of a divide-and- conquer method for computing homology McCleary, 2000. We will show how persistence arises out of our need for feature discernment in Section 6.1.1. This discussion motivates the formulation of persistence in terms of homology groups in Section 6.1.2. In order to better comprehend the meaning of persis- tence, we visualize the theoretical definition in Section 6.1.3. We next briefly discuss persistence in relation to spaces we are most interested in: subspaces of R 3 . In the last section, we take a more algebraic view of persistent homol- ogy using the advanced structures we discussed in Section 3.3. This view is necessary for understanding the persistence algorithm for spaces of arbitrary dimensions and arbitrary coefficient rings, as developed in Chapter 7. The reader may skip this section safely, however, without any loss of understand- ing of the algorithms for subspaces of R 3 .

6.1.1 Motivation

In Chapter 2, we examined an approach for exploring the topology of a space. This approach used a geometrically grown filtration as the representation of the space. In Chapter 4, we studied a combinatorial method for computing topology using homology groups. Applying homology to filtrations, we get some signature functions for a space. Definition 6.1 homology of filtration Let K l be a filtration of a space X. Let Z l k = Z k K l and B l k = B k K l be the kth cycle and boundary group of K l , respectively. The kth homology group of K l is H l k = Z l k B l k . The kth Betti number β l k of K l is the rank of H l k . The kth Betti numbers describe the topology of a growing simplicial complex by a sequence of integers. Our hope is that these numbers contain topological information about the original space. Unfortunately, as Figure 6.1 illustrates, our representation scheme generates a lot of additional topological attributes, all of which are captured by homology. We cannot distinguish between the features of the original space and the noise spawned by the representation. The primary topological feature of the space in the figure is a single tunnel. The graph of β l k in Figure 6.1, however, gives up to 43 tunnels for complexes in the filtration of this space. The evidence of the feature is buried in a heap of topological noise. To be able to derive any meaningful information about a