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Topology Illustrated

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Topology Illustrated by Peter Saveliev

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Foreword

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 9
  2. Homology classes 16
  3. Topology of graphs 12
  4. Homology groups of graphs 16
  5. Maps of graphs 10

Chapter 2. Topologies

  1. Point-set topology 8
  2. Neighborhoods and topologies 11
  3. Topological spaces 14
  4. Continuous functions 14
  5. Topological equivalence 8
  6. Relative topology 12

Chapter 3. Complexes

  1. Euclidean space with discrete structure 10
  2. Algebra of cells 18
  3. Cubical complexes 14
  4. Oriented chains 13
  5. Data with Euclidean topology 15
  6. Simplicial complexes 17
  7. Simplicial maps 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 19
  2. Cell maps 22
  3. Maps of polyhedra 26
  4. Euler and Lefschetz numbers 21
  5. Homology of parametric spaces 18


Estimated:

  • 464 pages
  • 628 illustrations
  • 508 exercises


Appendix: Student's guide to proof writing


Notation


References

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