The Euler-Poincaré Formula Homology Groups

While the results from the universal coefficient theorem are theoretically beautiful, our motivation in examining them has a computational nature. We have seen that some rings of coefficients, such as R, are unable to capture tor- sion. If a space does not have torsion, then we may be able to craft faster algo- rithms for computing topology by using such rings. The field of real numbers, R , is not an option, because we do not have infinite precision on computers. The field of rational numbers, Q, does not provide any advantage, as we will need to represent each rational exactly with two integers. The simplest prin- cipal ring, Z 2 , however, simplifies computation greatly. Here, the coefficients are either 0 or 1, so there is no need for orienting simplices or maintaining coefficients. A k-chain is simply a list of simplices, those with coefficient 1. Each simplex is its own inverse, reducing the group operation to the symmetric difference , where the sum of two k-chains c , d is c + d = c ∪ d − c ∩ d. Consequently, Z 2 provides us with a best system for computing homology of torsion-free spaces. In fact, nearly all of the spaces in this book are torsion-free. The processes described in Chapter 2 generate subcomplexes of R 3 . R 3 is not compact and creates special cases that need to be handled in algorithms. To avoid these difficulties, we add a point at infinity and compactify R 3 to get S 3 , the three- dimensional sphere. This construction mirrors that of the two-dimensional sphere in Definition 4.3. Algorithmically, the one point compactification of R 3 is easy, as we have a simplicial representation of space. Subcomplexes of a triangulation of S 3 do not have torsion. 5 Morse Theory In the last two chapters, we studied combinatorial methods for describing the topology of a space. One reason for our interest in understanding topology is topological simplification: removing topological “noise,” using a measure that defines what “noise” is. But as we saw in Section 1.2.3, the geometry and topology of a space are intricately related, and modifying one may modify the other. We need to understand this relationship in order to develop intelli- gent methods for topological simplification. Morse theory provides us with a complete analysis of this relationship when the geometry of the space is given by a function. The theory identifies points at which level-sets of the func- tion undergo topological changes and relates these points via a complex. The theory is defined, however, on smooth domains, requiring us to take a radical departure from our combinatorial focus. We need these differential concepts to guide our development of methods for nonsmooth domains. Our exposi- tion of Morse theory, consequently, will not be as thorough and axiomatic as the accounts in the last two chapters. Rather, we rely on the reader’s familiar- ity with elementary calculus to focus on the concepts we need for analyzing 2-manifolds in R 3 . We begin this chapter by extending some ideas from calculus to manifolds in Sections 5.1 and 5.2. These ideas enable us to identify the critical points of a manifold in Section 5.3. The critical points become the vertices of a complex. We define this complex by first decomposing the manifold into regions associ- ated with the critical points in Section 5.4. We then construct the complex in Section 5.5 and look at a couple of examples. Spivak and Well’s notes on Milnor’s lectures provide the basis for Morse theory Milnor, 1963. As an introduction to Riemannian manifolds, Morgan 1998 is beautifully accessible. O’Neill 1997 and Boothby 1986 provide good overviews of differential geometry and differential manifolds, respec- tively. I also use Bruce and Giblin 1992 for inspiration. 83

5.1 Tangent Spaces

In this chapter, we will generally assume that M is a smooth, compact, 2- manifold without boundary, or a surface. We will also assume, for simplicity, that the manifold is embedded in R 3 , that is, M ⊂ R 3 without self-intersections. The embedded manifold derives subspace topology and a metric from R 3 . These assumptions are not necessary, however. The ideas presented in this chapter generalize to higher dimensional abstract manifolds with Riemannian metrics. We begin by attaching tangent spaces to each point of a manifold. As al- ways, we derive our notions about manifolds from the Euclidean spaces. Definition 5.1 T p R 3 A tangent vector v p to R 3 consists of two points of R 3 : its vector part v and its point of application p. The set T p R 3 consists of all tangent vectors to R 3 at p and is called the tangent space of R 3 at p . Note that R 3 has a different tangent space at every point. Each tangent space is a vector space isomorphic to R 3 itself. We may also attach a vector space to each point of a manifold. Definition 5.2 T p M Let p be a point on M in R 3 . A tangent vector v p to R 3 at p is tangent to M at p if v is the velocity of some curve in M. The set of all tangent vectors to M at p is called the tangent plane of M at p and is denoted by T p M. Recall from Chapter 2 that a 2-manifold is covered with a number of charts, which map the neighborhood of a point to an open subset of R 2 . Each map is a homeomorphism, and we may parameterize the manifold using the inverses of these maps, which are often called patches. Theorem 5.1 Let p ∈ M ⊂ R 3 , and let ϕ be a path in M such that ϕu , v = p. A tangent vector v to R 3 at p is tangent to M iff v can be written as a linear combination of ϕ u u , v and ϕ v u , v . In other words, the tangent plane at a point of the manifold is a two-dimensional vector subspace of the tangent space T p R 3 , as shown in Figure 5.1. Based on the properties of derivatives, the tangent plane T p M is the best linear approx- imation of the surface M near p. Given tangent planes, we may select vectors at each point of the manifold to create a vector field. Definition 5.3 vector field A vector field or flow on V is a function that as- signs a vector v p ∈ T p M to each point p ∈ M.