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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 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.

Work in progress...


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


Chapter 1. Homology classes

  1. Topology around us
  2. Introduction to homology
  3. Topology of graphs
  4. Homology groups of graphs
  5. Maps of graphs

Chapter 2. Topologies

  1. Introduction to point-set topology
  2. Neighborhoods and topologies
  3. Topological spaces
  4. Continuous functions
  5. Topological equivalence
  6. Relative topology

Chapter 3. Complexes

  1. Euclidean space with discrete structure
  2. Chains of cubes
  3. The chain complex
  4. Cubical complexes
  5. Oriented chains
  6. Data with Euclidean topology
  7. Simplicial complexes
  8. Simplicial maps and chain maps

Chapter 4. Spaces

  1. Compact spaces
  2. Quotients
  3. Cell complexes
  4. Triangulations
  5. Manifolds
  6. Products

Chapter 5. Maps

  1. Homotopy
  2. Cell maps
  3. Homology theory
  4. Euler and Lefschetz numbers
  5. Homology of parametric complexes


Appendix: Student's guide to proof writing


Notation


References

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