Tangent Spaces Topology for Computing

Theorem 5.2 Integral lines have the following properties: a Two integral lines are either disjoint or the same. b The integral lines cover all of M. c And the limits org p and dest p are critical points of h. The properties follow from standard differential calculus. Definition 5.14 stable and unstable manifolds The stable manifold S p and the unstable manifold U p of a critical point p are defined as S p = {p} ∪ {y ∈ M | y ∈ im γ, dest γ = p}, 5.4 U p = {p} ∪ {y ∈ M | y ∈ im γ, org γ = p}, 5.5 where im γ is the image of the path γ ∈ M. Both sets of manifolds decompose M into open cells. Definition 5.15 open cell An open d-cell σ is a space homeomorphic to R d . We can predict the dimension of the open cell associated to a critical point p. Theorem 5.3 The stable manifold S p of a critical point p with index i = ip is an open cell of dimension dim S p = i. The unstable manifolds of h are the stable manifolds of −h as ∇−h = −∇h. Therefore, the two types of manifolds have the same structural properties. That is, the unstable manifolds of h are also open cells, but with dimension dimU p = 2 − i, where i is the index of a critical point. The closure of a stable or unstable manifold, however, is not necessarily homeomorphic to a closed ball. We see this in Figure 5.4, where a stable 2-cell is pinched at a minimum. By the properties in Theorem 5.2, the stable manifolds are pairwise disjoint and decompose M into open cells. The cells form a complex, as the bound- ary of every cell S a is a union of lower dimensional cells. We may view a cellular complex as a generalization of a simplicial complex, where we allow for arbitrarily shaped cells and relax restrictions on how they are connected to each other. The unstable manifolds similarly decompose M into a complex dual to the complex of stable manifolds: For a , b ∈ M, dim Sa = 2 − dimUa and Sa is a face of S b iff Ub is a face of Ua. Example 5.1 manifolds Figure 5.4 displays the stable and unstable mani- folds of a sphere and a Morse function h. We show an uncompactified sphere: The boundary of the terrain is a minimum at negative infinity. Note that the stable manifold of a minimum and the unstable manifold of a maximum, are the critical points themselves, respectively. On the other hand, both the unsta- ble manifold of a minimum and the stable manifold of a maximum are 2-cells. A saddle has 1-cells as both stable and unstable manifolds. Also, observe that the stable manifolds of the saddles decompose M into the stable manifolds of the maxima. The unstable manifolds provide such a decomposition for the minima.

5.5 Morse-Smale Complex

We place one more restriction on Morse functions in order to be able to con- struct Morse-Smale complexes. Definition 5.16 Morse-Smale A Morse function is a Morse-Smale function if the stable and unstable manifolds intersect only transversally. In two dimensions, this means that stable and unstable 1-manifolds cross when they intersect. Their crossing point is necessarily a saddle, since crossing at a regular point would contradict property a in Theorem 5.2. Given a Morse- Smale function h, we intersect the stable and unstable manifolds to obtain the Morse-Smale complex. Definition 5.17 Morse-Smale complex Connected components of sets U p ∩ Sq for all critical points p, q ∈ M are Morse-Smale cells. We refer to the cells of dimension 0, 1, and 2 as vertices, arcs, and regions, respec- tively. The collection of Morse-Smale cells form a complex, the Morse-Smale complex . Note that U p ∩ Sp = {p}, and if p = q, then Up ∩ Sq is the set of regular points r ∈ M that lie on integral lines γ with org γ = p and dest γ = q. It is possible that the intersection of stable and unstable manifolds consists of more than one component, as seen in Figure 5.5. Example 5.2 Morse-Smale complex We continue with the manifold and Morse function in Example 5.1. Figure 5.5 shows the Morse-Smale com- plex we get by intersecting the stable and unstable manifolds displayed in Fig- ure 5.4. Each vertex of the Morse-Smale complex is a critical point, each arc is