Derivatives and Morse Functions
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 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.Parts
» Spaces Topology for Computing
» New Results Topology for Computing
» Organization Topology for Computing
» Sets and Functions Topological Spaces
» Manifolds Topology for Computing
» Geometric Definition Simplicial Complexes
» Abstract Definition Simplicial Complexes
» Filtrations and Signatures Simplicial Complexes
» Manifold Sweeps Topology for Computing
» Binary Operations Groups Introduction to Groups
» Subgroups and Cosets Introduction to Groups
» Structure-Relating Maps Characterizing Groups
» Cyclic Groups Characterizing Groups
» Finitely Generated Abelian Groups
» Factor Groups Characterizing Groups
» Free Abelian Groups Advanced Structures
» Rings, Fields, Integral Domains, and Principal Ideal Domains
» Modules, Vector Spaces, and Gradings
» Structure Theorem Advanced Structures
» The Fundamental Group Justification
» Chains and Cycles Homology Groups
» Simplicial Homology Homology Groups
» Understanding Homology Homology Groups
» The Euler-Poincaré Formula Homology Groups
» Arbitrary Coefficients Topology for Computing
» Tangent Spaces Topology for Computing
» Derivatives and Morse Functions
» Critical Points Stable and Unstable Manifolds
» Morse-Smale Complex Topology for Computing
» Quasi Morse-Smale Complex Hierarchical Morse-Smale Complexes
» Piece-Wise Linear Artifacts Hierarchical Morse-Smale Complexes
» Filtration Hierarchical Morse-Smale Complexes
» Hierarchy Hierarchical Morse-Smale Complexes
» Knots and Links The Linking Number
» Marking Algorithm Topology for Computing
» Abstract Algorithm Algorithm for Z
» Cycle Search Algorithm for Z
» Canonization Algorithm for Z
» Reduction Algorithm for Fields
» Derivation Algorithm for Fields
» Algorithm Algorithm for Fields
» Algorithm for PIDs Topology for Computing
» Approach and Goals Motivation
» Persistent Betti Number Algorithm
» Lazy Migration Reordering Algorithms
» Topology Maps Topology for Computing
» Complex with Junctions The Quasi Morse-Smale Complex Algorithm
» Extending Paths The Quasi Morse-Smale Complex Algorithm
» Handle Slide Local Transformations
» Steepest Ascent Local Transformations
» Motivation Algorithm Topology for Computing
» Component Tree Enumeration Algorithm
» Methodology Topology for Computing
» Libraries and Packages Organization
» Surfaces Miscellaneous Three-Dimensional Data
» Implementation Algorithm for Fields
» Framework and Data Algorithm for Fields
» Field Coefficients Algorithm for Fields
» Higher Dimensions Algorithm for Fields
» A Case Study Timings and Statistics
» Implementation The Linking Number Algorithm
» Timings and Statistics The Linking Number Algorithm
» Topological Feature Detection Computational Structural Biology
» Knotting Computational Structural Biology
» Structure Determination Computational Structural Biology
» Hierarchical Clustering Topology for Computing
» Terrain Simplification Denoising Density Functions
» Iso-Surface Denoising Denoising Density Functions
» Time-Varying Data Denoising Density Functions
» Medial Axis Simplification Denoising Density Functions
» Surface Reconstruction Topology for Computing
Show more