Homology

Overview

The Map of the Journey

The golden line is the reading order. Point at a station (or tap it) to light up everything it builds on, and everything that builds on it. Every chapter in reading order, with the chapters each one builds on.

Your journey 0 of 32 chapters visited

About 27 hours of reading in all, at an unhurried pace.

Begin with 0.1The Shape of a Question
PreludePart I · FoundationsPart II · ShapesPart III · HomologyPart IV · CohomologyPart V · The Bigger Picture0.1The Shape ofa Question0.2How to ReadMathematics1.1Sets andFunctions1.2Equivalenceand Quotients1.3Groups1.4Abelian Groupsand Formal Sums1.5Linear Algebra2.1Spaces andContinuity2.2Gluing SpacesTogether2.3Homotopy2.4Manifoldsand Surfaces2.5SimplicialComplexes2.6The EulerCharacteristic3.1Cycles andBoundaries3.2Chains andthe BoundaryOperator3.3HomologyGroups3.4ComputingHomology3.5Maps,Invariance,and FirstTriumphs3.6ExactSequences andMayer–Vietoris3.7Knotsand Links3.8PersistentHomology4.1Cochains:MeasuringShapes4.2CohomologyGroups4.3DifferentialForms4.4de RhamCohomology4.5The CupProduct4.6PoincaréDuality4.7Sheavesand ČechCohomology4.8Curvature andCharacteristicClasses5.1Categoriesand Functors5.2HomologicalAlgebra5.3Horizons
  1. 0 Prelude
    1. 0.1 The Shape of a Question
      about 35 min
    2. 0.2 How to Read Mathematics
      about 45 min
  2. I Foundations
    1. 1.1 Sets and Functions
      builds on 0.2 · about 45 min
    2. 1.2 Equivalence and Quotients
      builds on 1.1 · about 45 min
    3. 1.3 Groups
      builds on 1.2 · about 55 min
    4. 1.4 Abelian Groups and Formal Sums
      builds on 1.3 · about 55 min
    5. 1.5 Linear Algebra
      builds on 1.4 · about 80 min
  3. II Shapes
    1. 2.1 Spaces and Continuity
      builds on 1.1 · about 55 min
    2. 2.2 Gluing Spaces Together
      builds on 2.1, 1.2 · about 45 min
    3. 2.3 Homotopy
      builds on 2.1 · about 50 min
    4. 2.4 Manifolds and Surfaces
      builds on 2.2 · about 60 min
    5. 2.5 Simplicial Complexes
      builds on 2.2 · about 60 min
    6. 2.6 The Euler Characteristic
      builds on 2.5 · about 40 min
  4. III Homology
    1. 3.1 Cycles and Boundaries
      builds on 2.6 · about 50 min
    2. 3.2 Chains and the Boundary Operator
      builds on 3.1, 1.4 · about 45 min
    3. 3.3 Homology Groups
      builds on 3.2 · about 50 min
    4. 3.4 Computing Homology
      builds on 3.3, 1.5 · about 50 min
    5. 3.5 Maps, Invariance, and First Triumphs
      builds on 3.3, 2.3 · about 60 min
    6. 3.6 Exact Sequences and Mayer–Vietoris
      builds on 3.5 · about 45 min
    7. 3.7 Knots and Links
      builds on 3.6, 2.3, 2.4 · about 70 min
    8. 3.8 Persistent Homology
      builds on 3.4 · about 65 min
  5. IV Cohomology
    1. 4.1 Cochains: Measuring Shapes
      builds on 3.3 · about 55 min
    2. 4.2 Cohomology Groups
      builds on 4.1, 1.5 · about 40 min
    3. 4.3 Differential Forms
      builds on 2.4, 1.5 · about 50 min
    4. 4.4 de Rham Cohomology
      builds on 4.3, 4.2 · about 40 min
    5. 4.5 The Cup Product
      builds on 4.4 · about 50 min
    6. 4.6 Poincaré Duality
      builds on 4.5 · about 40 min
    7. 4.7 Sheaves and Čech Cohomology
      builds on 4.2 · about 60 min
    8. 4.8 Curvature and Characteristic Classes
      builds on 4.4, 4.6 · about 50 min
  6. V The Bigger Picture
    1. 5.1 Categories and Functors
      builds on 3.5 · about 55 min
    2. 5.2 Homological Algebra
      builds on 5.1, 4.2 · about 45 min
    3. 5.3 Horizons
      builds on 5.2 · about 35 min