Topology Illustrated

I Cycles
1 Topology around us
1.1 Topology { Algebra { Geometry
1.2 The integrity of everyday objects
1.3 The shape of the Universe
1.4 Patterns in data
1.5 Social choice
2 Homology classes
2.1 Topological features of objects
2.2 How to define and count 0-dimensional features
2.3 How to define and count 1-dimensional features
2.4 Homology as an equivalence relation
2.5 Homology in calculus
3 Topology of graphs
3.1 Graphs and their realizations
3.2 Connectedness and components
3.3 Holes vs cycles
3.4 The Euler characteristic
3.5 Holes of planar graphs
3.6 The Euler Formula
4 Homology groups of graphs
4.1 The algebra of plumbing
4.2 Chains of nodes and chains of edges
4.3 The boundary operator
4.4 Holes vs cycles
4.5 Components vs boundaries
4.6 Quotients in algebra
4.7 Homology as a quotient group
4.8 An example of homological analysis
5 Maps of graphs
5.1 What is the discrete counterpart of continuity?
5.2 Graph maps
5.3 Chain maps
5.4 Commutative diagrams
5.5 Cycles and boundaries under chain maps
5.6 Quotient maps in algebra
5.7 Homology maps
6 Binary calculus on graphs
6.1 The algebra of plumbing, continued
6.2 The dual of a group
6.3 Cochains of nodes and cochains of edges
6.4 Maps of cochains
6.5 The coboundary operator
II Topologies
1 A new look at continuity
1.1 From accuracy to continuity
1.2 Continuity in a new light
1.3 Continuity restricted to subsets
1.4 The intrinsic definition of continuity
2 Neighborhoods and topologies
2.1 Bases of neighborhoods
2.2 Open sets
2.3 Path-connectedness
2.4 From bases to topologies
2.5 From topologies to bases
3 Topological spaces
3.1 Open and closed sets
3.2 Proximity of a point to a set
3.3 Interior - frontier - exterior
3.4 Convergence of sequences
3.5 Metric spaces
3.6 Spaces of functions
3.7 The order topology
4 Continuous functions
4.1 Continuity as preservation of proximity
4.2 Continuity and preimages
4.3 Continuous functions everywhere
4.4 Compositions and path-connectedness
4.5 Categories
4.6 Vector fields
4.7 The dynamics of a market economy
4.8 Maps
4.9 Homeomorphisms
4.10 Examples of homeomorphic spaces
4.11 Topological equivalence
5 Subspaces
5.1 How a subset inherits its topology
5.2 The topology of a subspace
5.3 Relative neighborhoods vs relative topology
5.4 New maps
5.5 The extension problem
5.6 Social choice: looking for compromise
5.7 Discrete decompositions of space
5.8 Cells
III Complexes
1 The algebra of cells
1.1 Cells as building blocks
1.2 Cubical cells
1.3 Boundaries of cubical cells
1.4 Binary chains and their boundaries
1.5 The chain groups and the chain complex
1.6 Cycles and boundaries
1.7 Cycles = boundaries?
1.8 When is every cycle a boundary?
2 Cubical complexes
2.1 The definition
2.2 Realizations
2.3 The boundary operator
2.4 The chain complex
2.5 Examples
2.6 Computing boundaries with a spreadsheet
3 The algebra of oriented cells
3.1 Are chains just \combinations" of cells?
3.2 The algebra of chains with coefficients
3.3 The role of oriented chains in calculus
3.4 Orientations and boundaries
3.5 Computing boundaries with a spreadsheet, continued
3.6 A homology algorithm for dimension 2
3.7 The boundary of a cube in the N-dimensional space
3.8 The boundary operator
4 Simplicial complexes
4.1 From graphs to multi-graphs
4.2 Simplices in the Euclidean space
4.3 Realizations
4.4 Refining simplicial complexes
4.5 The simplicial complex of a partially ordered set
4.6 Data as a point cloud
4.7 Social choice: the lottery of life
5 Simplicial homology
5.1 Simplicial complexes
5.2 Boundaries of unoriented chains
5.3 How to orient a simplex
5.4 The algebra of oriented chains
5.5 The boundary operator
5.6 Homology
6 Simplicial maps
6.1 The definition
6.2 Chain maps of simplicial maps
6.3 How chain maps interact with the boundary operators
6.4 Homology maps
6.5 Computing homology maps
6.6 How to classify simplicial maps
6.7 Realizations
6.8 Social choice: no compromise
7 Parametric complexes
7.1 Topology under uncertainty
7.2 Persistence of homology classes
7.3 The homology of a gray scale image
7.4 Homology groups of filtrations
7.5 Maps of filtrations
7.6 The \sharp" homology classes of a gray scale image
7.7 Persistent homology groups of filtrations
7.8 More examples of parametric spaces
7.9 Multiple parameters
1 Compacta
1.1 Open covers and accumulation points
1.2 The definitions
1.3 Compactness of intervals
1.4 Compactness in finite and infinite dimensions
2 Quotients
2.1 Gluing things together
2.2 Quotient sets
2.3 Quotient spaces and identification maps
2.4 Examples
2.5 Topology of pairs of spaces
3 Cell complexes
3.1 Gluing cells together
3.2 Examples and definitions
3.3 The topology
3.4 Transition to algebra
3.5 What we can make from a square
3.6 The nth homology of the n-dimensional balls and spheres
3.7 Quotients of chain complexes
4 Triangulations
4.1 Simplicial vs cell complexes
4.2 How to triangulate topological spaces
4.3 How good are these triangulations?
4.4 Social choice: ranking
4.5 The Simplicial Extension Theorem
5 Manifolds
5.1 What is the topology of the physical Universe?
5.2 The locally Euclidean property
5.3 Two locally Euclidean monstrosities
5.4 The separation axioms
5.5 Manifolds and manifolds with boundary
5.6 The connected sum of surfaces
5.7 Triangulations of manifolds
5.8 Homology of curves and surfaces
5.9 The nth homology of n-manifolds
5.10 Homology relative to the boundary
6 Products
6.1 How products are built
6.2 Products of spaces
6.3 Properties
6.4 The projections
6.5 Products of complexes
6.6 Chains in products
6.7 The Universe: 3-sphere, 3-torus, or something else?
6.8 Configuration spaces
6.9 Homology of products: the Kunneth formula
V Maps
1 Homotopy
1.1 Deforming spaces vs deforming maps
1.2 Properties and examples
1.3 Types of connectedness
1.4 Homotopy theory
1.5 Homotopy equivalence
1.6 Homotopy in calculus
1.7 Is there a constant homotopy between constant maps?
1.8 Homotopy equivalence via cell collapses
1.9 Invariance of homology under cell collapses
2 Cell maps
2.1 The definition
2.2 Examples of cubical maps
2.3 Modules
2.4 The topological step
2.5 The algebraic step
2.6 Examples of homology maps
2.7 Homology theory
2.8 Functors
2.9 New maps from old
3 Maps of polyhedra
3.1 Maps vs cell maps
3.2 Cell approximations of maps
3.3 The Simplicial Approximation Theorem
3.4 Simplicial approximations are homotopic
3.5 Homology maps of homotopic maps
3.6 The up-to-homotopy homology theory
3.7 How to classify maps
3.8 Means
3.9 Social choice: no impartiality
3.10 The Chain Approximation Theorem
4 The Euler and Lefschetz numbers
4.1 The Euler characteristic
4.2 The Euler-Poincare Formula
4.3 Fixed points
4.4 An equilibrium of a market economy
4.5 Chain maps of self-maps
4.6 The Lefschetz number
4.7 The degree of a map
4.8 Zeros of vector fields
4.9 Social choice: gaming the system
5 Set-valued maps
5.1 Coincidences of a pair of maps
5.2 Set-valued maps
5.3 The Vietoris Mapping Theorem
5.4 The Lefschetz number of a pair of maps
5.5 Motion planning in robotics
5.6 Social choice: standstill in hostilities
VI Forms
1 Discrete differential forms and cochains
1.1 Calculus and differential forms
1.2 How do we approximate continuous processes?
1.3 Cochains as building blocks of discrete calculus
1.4 Calculus of data
1.5 The spaces of cochains
1.6 The coboundary operator
1.7 The cochain complex and cohomology
1.8 Calculus I, the discrete version
1.9 Social choice: ratings and comparisons
2 Calculus on cubical complexes
2.1 Visualizing cubical forms
2.2 Forms as integrands
2.3 The algebra of forms
2.4 Derivatives of functions vs derivatives of forms
2.5 The exterior derivative of a 1-form
2.6 Representing cubical forms with a spreadsheet
2.7 A bird’s-eye view of calculus
2.8 Algebraic properties of the exterior derivative
2.9 Tangent spaces
3 Cohomology
3.1 Vectors and covectors
3.2 The dual basis
3.3 The dual operators
3.4 The cochain complex
3.5 Social choice: higher order voting
3.6 Cohomology
3.7 Computing cohomology
3.8 Homology maps vs cohomology maps
3.9 Computing cohomology maps
3.10 Functors, continued
4 Metric tensor
4.1 The need for measuring in calculus
4.2 Inner products
4.3 The metric tensor
4.4 Metric tensors in dimension 1
4.5 Metric complexes in dimension 1
4.6 Hodge duality of forms
4.7 The first derivative
4.8 The second derivative
4.9 Newtonian physics, the discrete version
VII Flows
1 Metric complexes
1.1 Hodge-dual complexes
1.2 Metric complexes
1.3 The Hodge star operator
1.4 The Laplace operator
2 ODEs
2.1 Motion: location from velocity
2.2 Population growth
2.3 Motion: location from acceleration
2.4 Oscillating spring
2.5 ODEs of forms
2.6 Vector fields and systems of ODEs
2.7 Simulating a flow with a spreadsheet
2.8 The derivative of a cell map
2.9 ODEs of cell maps
2.10 ODEs of chain maps
2.11 Cochains are chain maps
2.12 Simulating advection with a spreadsheet
3 PDEs
3.1 The PDE of diffusion
3.2 Simulating diffusion with a spreadsheet
3.3 Diffusion on metric complexes
3.4 How diffusion patterns depend on the sizes of cells
3.5 How diffusion patterns depend on the angles between cells
3.6 The PDE of wave propagation
3.7 Solutions of the wave equation
3.8 Waves in higher dimensions
3.9 Simulating wave propagation with a spreadsheet
4 Social choice
4.1 The paradox of social choice
4.2 Ratings, comparisons, ranking, and preferences
4.3 The algebra of vote aggregation
4.4 Aggregating rating votes
4.5 Aggregating comparison votes
4.6 Google’s PageRank
4.7 Combining ratings with comparisons
4.8 Decycling: how to extract ratings from comparisons
4.9 Is there a fair electoral system?
1 Appendix: Student’s guide to proof writing
2 Appendix: Notation