Peter Saveliev Topology Illustrated With 1000 Illustrations Peter Saveliev Department of Mathematics Marshall University Huntington, WV 25755 USA Topology Illustrated by Peter Saveliev 657 pages, includes index ISBN 978-1-4951-8875-6 Mathematics Subject Classification (2010): 55-01, 57-01, 54-01, 58-01, 39A12 c2016 Peter Saveliev (cid:13) Dedicated to the memory of my parents Preface Afirstcourseintopologyisusuallyasemesterinpoint-settopology. Sometimesachapteronthe fundamental group is included at the end, with very little time left. For the student, algebraic topology often never comes. The main part of the present text grew from the course Topology I and II that I have taught at Marshall University in recent years. This material follows a two-semester first course in topology withemphasisonalgebraictopology. Somecommontopicsaremissing,though: thefundamental group, classification of surfaces, and knots. Point-set topology is presented only to a degree that seems necessary in order to develop algebraic topology; the rest is likely to appear in a, typically required, real analysis course. The focus is on homology. Such tools of algebraic topology as chains and cochains form a foundation of discrete calculus. A through introduction is provided. The presentation is often more detailed than one normally sees in a textbook on this subject, which makes the text useful for self-study or as a companion source. There are over 1000 exercises. They appear just as new material is being developed. Some of them are quite straight-forward; their purpose is to slow down your reading. There are over 1000 pictures. They are used – but only as metaphors – to illustrate topological ideas and constructions. When a picture is used to illustrate a proof, the proof still remains complete without it. Applications are present throughout the book. However, they are neither especially realistic nor (with the exception of a few spreadsheets to be found on the author’s website) computational in nature; they are mere illustrations of the topological ideas. Some of the topics are: the shape of the universe, configuration spaces, digital image analysis, data analysis, social choice, exchange economy, and, of course, anything related to calculus. As the core content is independent of the applications, the instructor can pick and choose what to include. Thewaytheideasaredevelopedmaybecalled“historical”,butnotinthesenseofwhatactually happened – it’s been too messy – but rather what ought to have happened. All of this makes the book a lot longer than a typical book with a comparable coverage. Don’t be discouraged! A rigorous course in linear algebra is an absolute necessity. In the early chapters, one may be able to avoid the need for a modern algebra course but not the maturity it requires. Chapter I: Cycles contains an informal introduction to homology as well as a sample: • homology of graphs. Chapter II: Topologies is the starting point of point-set topology, developed as much as • is needed for the next chapter. Chapter III: Complexes introduces first cubical complexes, cubical chains, unoriented • and then oriented, and cubical homology. Then a thorough introduction to simplicial homology is provided. Chapter IV: Spaces continues to develop the necessary concepts of point-set topology • and uses them to advance the ideas of algebraic topology, such as homotopy and cell complexes. Chapter V: Maps presents homology theory of polyhedra and their maps. • Chapter VI: Forms introduces discrete differential forms as cochains, calculus of forms, • cohomology, and metric tensors on cell complexes. Chapter VII: FlowspresentsapplicationsofcalculusofformstoODEs,PDEs,andsocial • choice. By the end of the first semester, one is expected to reach the section on simplicial homology in Chapter III, but maybe not the section on the homology maps yet. For a single-semester first course, one might try this sequence: Chapter II, Sections III.4 - III.6, Chapter IV (except IV.3), SectionV.1. Foraone-semestercoursethatfollowspoint-settopology(andmodernalgebra),one cantakeanacceleratedroute: ChaptersIII-V(skippingtheapplications). Fordiscretecalculus, follow: Sections III.1 - III.3, Chapters VI and VII. The book is mostly undergraduate; it takes the student to the point where the tough proofs are about to start to become unavoidable. Where the book leaves off, one usually proceeds to such topicsas: theaxiomsofhomology,singularhomology,products,homotopygroups,orhomological algebra. Geometry and Topology by Bredon is a good choice. Peter Saveliev Aboutthecover. OnceuponatimeItookabetterlookattheposterofDrawingHandsbyEscher hanging in my office and realized that what it depicts looks symmetric but isn’t! I decided to fix the problem and the result is called Painting Hands. This juxtaposition illustrates how the antipodal map (aka the central symmetry) reverses the orientation in the odd dimensions and preserves it in the even dimensions. That’s why to be symmetric the original would have to have two right hands! Contents I Cycles 9 1 Topology around us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1 Topology – Algebra – Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The integrity of everyday objects . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The shape of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Patterns in data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Social choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Homology classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1 Topological features of objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 How to define and count 0-dimensional features . . . . . . . . . . . . . . . . . . 22 2.3 How to define and count 1-dimensional features . . . . . . . . . . . . . . . . . . 24 2.4 Homology as an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Homology in calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Topology of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Graphs and their realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Connectedness and components . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Holes vs. cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 The Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Holes of planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 The Euler Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4 Homology groups of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 The algebra of plumbing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Chains of nodes and chains of edges . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 The boundary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4 Holes vs. cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Components vs. boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Quotients in algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7 Homology as a quotient group. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.8 An example of homological analysis . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Maps of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1 What is the discrete counterpart of continuity? . . . . . . . . . . . . . . . . . . 58 5.2 Graph maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3 Chain maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4 Commutative diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.5 Cycles and boundaries under chain maps . . . . . . . . . . . . . . . . . . . . . 68 5.6 Quotient maps in algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.7 Homology maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6 Binary calculus on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.1 The algebra of plumbing, continued . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 The dual of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.3 Cochains of nodes and cochains of edges . . . . . . . . . . . . . . . . . . . . . . 76 6.4 Maps of cochains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1 2 CONTENTS 6.5 The coboundary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 II Topologies 83 1 A new look at continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1.1 From accuracy to continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1.2 Continuity in a new light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 1.3 Continuity restricted to subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1.4 The intrinsic definition of continuity . . . . . . . . . . . . . . . . . . . . . . . . 89 2 Neighborhoods and topologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.1 Bases of neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.2 Open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3 Path-connectedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.4 From bases to topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.5 From topologies to bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.1 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2 Proximity of a point to a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.3 Interior - frontier - exterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.4 Convergence of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.5 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.6 Spaces of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.7 The order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.1 Continuity as preservation of proximity . . . . . . . . . . . . . . . . . . . . . . 112 4.2 Continuity and preimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.3 Continuous functions everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.4 Compositions and path-connectedness . . . . . . . . . . . . . . . . . . . . . . . 118 4.5 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.6 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.7 The dynamics of a market economy . . . . . . . . . . . . . . . . . . . . . . . . 124 4.8 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.9 Homeomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.10 Examples of homeomorphic spaces . . . . . . . . . . . . . . . . . . . . . . . . 129 4.11 Topological equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1 How a subset inherits its topology . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2 The topology of a subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Relative neighborhoods vs. relative topology . . . . . . . . . . . . . . . . . . . 138 5.4 New maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.5 The extension problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.6 Social choice: looking for compromise . . . . . . . . . . . . . . . . . . . . . . . 144 5.7 Discrete decompositions of space . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.8 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 III Complexes 151 1 The algebra of cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1.1 Cells as building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1.2 Cubical cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 1.3 Boundaries of cubical cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 1.4 Binary chains and their boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 158 1.5 The chain groups and the chain complex . . . . . . . . . . . . . . . . . . . . . . 160 1.6 Cycles and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 1.7 Cycles = boundaries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 CONTENTS 3 1.8 When is every cycle a boundary? . . . . . . . . . . . . . . . . . . . . . . . . . . 165 2 Cubical complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 2.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 2.2 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 2.3 The boundary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 2.4 The chain complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 2.6 Computing boundaries with a spreadsheet . . . . . . . . . . . . . . . . . . . . . 182 3 The algebra of oriented cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.1 Are chains just “combinations” of cells? . . . . . . . . . . . . . . . . . . . . . . 183 3.2 The algebra of chains with coefficients . . . . . . . . . . . . . . . . . . . . . . . 185 3.3 The role of oriented chains in calculus . . . . . . . . . . . . . . . . . . . . . . . 187 3.4 Orientations and boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.5 Computing boundaries with a spreadsheet, continued. . . . . . . . . . . . . . . 191 3.6 A homology algorithm for dimension 2 . . . . . . . . . . . . . . . . . . . . . . . 193 3.7 The boundary of a cube in the N-dimensional space . . . . . . . . . . . . . . . 195 3.8 The boundary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 4 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.1 From graphs to multi-graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.2 Simplices in the Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.3 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.4 Refining simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.5 The simplicial complex of a partially ordered set . . . . . . . . . . . . . . . . . 210 4.6 Data as a point cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4.7 Social choice: the lottery of life . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5 Simplicial homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.1 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.2 Boundaries of unoriented chains. . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5.3 How to orient a simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.4 The algebra of oriented chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.5 The boundary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.6 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6 Simplicial maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 6.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 6.2 Chain maps of simplicial maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 6.3 How chain maps interact with the boundary operators . . . . . . . . . . . . . . 236 6.4 Homology maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.5 Computing homology maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 6.6 How to classify simplicial maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.7 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.8 Social choice: no compromise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7 Parametric complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.1 Topology under uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.2 Persistence of homology classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.3 The homology of a gray scale image . . . . . . . . . . . . . . . . . . . . . . . . 254 7.4 Homology groups of filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 7.5 Maps of filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 7.6 The “sharp” homology classes of a gray scale image . . . . . . . . . . . . . . . 259 7.7 Persistent homology groups of filtrations . . . . . . . . . . . . . . . . . . . . . . 262 7.8 More examples of parametric spaces . . . . . . . . . . . . . . . . . . . . . . . . 265 7.9 Multiple parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266