Computational Topology Lecture notes Francis Lazarus and Arnaud de Mesmay 2016-2018 Contents 1 Panorama4 1.0.1 Topology..........................................4 1.0.2 Computational Topology..............................6 2 Planar Graphs8 2.1 Topology..............................................9 2.1.1 The Jordan Curve Theorem............................ 10 2.1.2 Euler’s Formula.................................... 12 2.2 Kuratowski’s Theorem..................................... 15 2.2.1 The Subdivision Version.............................. 15 2.2.2 The Minor Version.................................. 19 2.3 Other Planarity Characterizations............................ 19 2.4 Planarity Test........................................... 22 2.5 Drawing with Straight Lines................................ 27 3 Surfaces and Embedded Graphs 32 3.1 Surfaces............................................... 32 3.1.1 Surfaces and cellularly embedded graphs................. 32 3.1.2 Polygonal schemata................................. 35 3.1.3 Classification of surfaces............................. 36 3.2 Maps................................................. 39 3.3 The Genus of a Map...................................... 43 3.4 Homotopy............................................. 45 3.4.1 Groups, generators and relations....................... 46 3.4.2 Fundamental groups, the combinatorial way............... 46 3.4.3 Fundamental groups, the topological way................. 50 3.4.4 Covering spaces.................................... 52 4 The Homotopy Test 55 4.1 Dehn’s Algorithm........................................ 56 4.2 van Kampen Diagrams.................................... 58 4.2.1 Disk Diagrams..................................... 58 4.2.2 Annular Diagrams.................................. 59 4.3 Gauss-Bonnet Formula.................................... 59 4.4 Quad Systems........................................... 61 4.5 Canonical Representatives................................. 62 4.5.1 The Four Bracket Lemma............................. 62 1 CONTENTS 2 4.5.2 Bracket Flattening.................................. 64 4.5.3 Canonical Representatives............................ 65 4.6 The Homotopy Test....................................... 67 5 Minimum Weight Bases 68 5.1 Minimum Basis of the Fundamental Group of a Graph............. 69 5.2 Minimum Basis of the Cycle Space of a Graph................... 70 5.2.1 The Greedy Algorithm............................... 70 5.3 Uniqueness of Shortest Paths............................... 73 5.4 First Homology Group of Surfaces............................ 74 5.4.1 Back to Graphs..................................... 74 5.4.2 Homology of Surfaces................................ 75 5.5 Minimum Basis of the Fundamental Group of a Surface............ 77 5.5.1 Dual Maps and Cutting............................... 77 5.5.2 Homotopy Basis Associated with a Tree-Cotree Decomposition. 77 5.5.3 The Greedy Homotopy Basis........................... 78 5.6 Minimum Basis of the First Homology Group of a Surface........... 80 5.6.1 Homology Basis Associated with a Tree-Cotree Decomposition. 80 5.6.2 The Greedy Homology Basis........................... 80 6 Homology 83 6.1 Complexes............................................. 83 6.2 Homology............................................. 85 6.2.1 Chain complexes................................... 85 6.2.2 Simplicial homology................................. 86 6.2.3 Examples and the question of the coefficient ring........... 87 6.2.4 Betti numbers and Euler-Poincaré formula................ 88 6.2.5 Homology as a functor............................... 88 6.3 Homology computations................................... 89 6.3.1 Over a field....................................... 89 6.3.2 Computation of the Betti numbers: the Delfinado-Edelsbrunner algorithm......................................... 89 6.3.3 Over the integers: the Smith-Poincaré reduction algorithm.... 90 7 Persistent Homology 93 7.1 Persistence Modules...................................... 94 7.1.1 Classification of Persistence Modules.................... 95 7.1.2 Restrictions of Persistence Modules..................... 97 7.2 Application to Topological Inference.......................... 98 7.3 Computing the Barcode................................... 99 7.3.1 Compatible Boundary Basis........................... 101 7.3.2 Algorithm........................................ 102 7.4 Persistence Diagrams..................................... 103 7.4.1 Stability of Persistence Diagrams....................... 104 CONTENTS 3 8 Knots and 3-Dimensional Computational Topology 106 8.1 Knots................................................. 107 8.2 Knot diagrams.......................................... 108 8.3 The knot complement..................................... 111 8.3.1 Homotopy........................................ 112 8.3.2 Homology........................................ 114 8.3.3 Triangulations..................................... 114 8.4 An algorithm for unknot recognition.......................... 115 8.4.1 Normal surface theory............................... 116 8.4.2 Trivial knot and spanning disks......................... 118 8.4.3 Normalization of spanning disks........................ 120 8.4.4 Haken sum, fundamental and vertex normal surfaces........ 123 8.5 Knotless graphs......................................... 126 9 Undecidability in Topology 128 9.1 The Halting Problem...................................... 129 9.1.1 Turing Machines................................... 129 9.1.2 Undecidability of the Halting Problem................... 130 9.2 Decision Problems in Group Theory.......................... 131 9.3 Decision Problems in Topology.............................. 134 9.3.1 The Contractibility and Transformation Problems........... 134 9.3.2 The Homeomorphism Problem........................ 136 9.4 Proof of the Undecidability of the Group Problems................ 139 2 9.4.1 Z -Machines...................................... 139 9.4.2 Useful Constructs in Combinatorial Group Theory.......... 140 9.4.3 Undecidability of the Generalized Word Problem............ 141 1 Panorama Contents 1.0.1 Topology........................................4 1.0.2 Computational Topology............................6 1.0.1 Topology. Topology deals with the study of spaces. One of its goals is to answer the following broad class of questions: “Are these two spaces the same?” This naturally leads to the following subquestions: What is a space? General topology typically defines topological spaces via open • and closed sets. In order to avoid pathological examples, and with an eye towards applications, we will take a more concrete approach1: in this course, topological spaces will be obtained in the form of complexes, that is, by gluing together fundamental blocks. For example, gluing segments yields a graph, while by gluing together triangles one can obtain a surface (or something more compli- cated). The usual notions of distance on these fundamental blocks naturally induce a notion of proximity on such a complex, and therefore a topology whose properties are convenient to understand geometrically. 1This is by no means original: see introductory textbooks on algebraic topology, for example Hatcher [Hat02] or Stillwell [Sti93]. 4 1. PANORAMA 5 What is “the same” ? It very much depends on the context. The most common • equivalence relation is homeomorphism, which is a continuous map with a continuous inverse function. But in some contexts, when a space is embedded in another space, one will be interested in distinguishing between different embeddings. There, a convenient notion is isotopy : two embedded spaces will be considered the same if one can deform continuously one into the other one. Let us look at examples. Example 1: By gluing triangles or quadrilaterals, one can easily obtain a sphere (left figure), or a torus (right figure). Are these two spaces homeomorphic? Obviously not: the torus has a hole. But what is a hole? Two naive answers will guide us to the two fundamental constructs of algebraic topology: Homotopy: On the sphere, every closed curve can be deformed into a single • point. While on the torus, a curve going around the hole can not. Such a curve is not homotopic to a point. Homology: On the sphere, every closed curve separates the sphere into two • regions. While on the torus, a curve going around the hole is not separating. Such a curve is not trivial in homology. These intuitions can be formalized into algebraic objects which will constitute invariants (actually, functors) that one can use to distinguish topological spaces. 3 Example 2: By gluing segments in R , one can obtain the following knots. 1 Are they homeomorphic? Certainly: they are both homeomorphic to the circle S . But are they isotopic: can one be deformed into the other without crossing itself? The answer is negative, but this is not that easy to prove. One way to see it is that the knot on the left bounds a disk, while the one on the right does not. Studying which surfaces one can find in a 3-dimensional space is the goal of normal surface theory. 1. PANORAMA 6 1.0.2 Computational Topology. Computational topology deals with effective computations on topological spaces. The main question now becomes: “How to compute whether these two spaces are the same?” Note that since we study spaces described by gluings of fundamental blocks, in most instances this can be made into a well-defined algorithmic problem, with a finite input. One can then