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Topology Illustrated

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Topology Illustrated by Peter Saveliev

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The text follows the content of a fairly typical, two-semester, first course in topology. Some typical topics are missing though: the fundamental group, classification of surfaces, and knots. Point-set topology is presented only as much as seems necessary in order to develop algebraic topology. The focus is on homology.

The presentations are often more detailed than one normally sees in the textbooks on this subject which makes the text appropriate for self-study. Exercises are numerous. Pictures are in the hundreds. Applications serve as another way to illustrate topological ideas. Some of the topics are: the shape of the universe, configuration spaces, digital image analysis, data analysis, social choice, and, of course, calculus.

The way the ideas are developed may be called "historical" but not in the sense of what actually happened -- it's 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.

A rigorous course in linear algebra is an absolute necessity. In the first half of the book, the need for a modern algebra course may be avoided but not the maturity it requires.

Where the book leaves off, one usually proceeds to cohomology or singular homology. If it has to be a single semester course, one might try this: chapter 2, 3.5-3.7, chapter 4 (except 4.3), 5.1.

Work in progress... If you have a question or a comment, drop me a line.

Note: Ignore the occasional links as you read the sections below. Instead, click Back when you are finished and continue to the next section.

Note: Internet Explorer 8 is very slow to render the math (a MathJax issue); Firefox is much better.

Chapter 1. Cycles and boundaries

  1. Topology around us 10
  2. Homology classes 13
  3. Topology of graphs 12
  4. Homology groups of graphs 13
  5. Maps of graphs and their homology 17

Chapter 2. Topologies

  1. The intrinsic definition of continuity 10
  2. Neighborhoods and topologies 11
  3. Topological spaces 14
  4. Continuity and topological equivalence 22
  5. Relative topology 12

Chapter 3. Complexes

  1. Euclidean space with discrete structure 10
  2. The binary algebra of cells 18
  3. Cubical complexes 14
  4. The algebra of oriented cells 13
  5. Datasets with Euclidean topology 15
  6. Simplicial complexes 17
  7. Simplicial maps and their homology 19

Chapter 4. Spaces

  1. Compact spaces 11
  2. Quotients 19
  3. Cell complexes 22
  4. Triangulations 12
  5. Manifolds 27
  6. Products 28

Chapter 5. Maps

  1. Homotopy 25
  2. Cell maps and their homology 23
  3. Maps of polyhedra 25
  4. Euler and Lefschetz numbers 20
  5. Homology of parametric spaces 21

Appendix: Student's guide to proof writing 3

Appendix: Notation 4


  • 488 pages
  • 650 illustrations
  • 547 exercises


These are the books I've used for teaching in the past: