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Preface

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Topology Illustrated by Peter Saveliev

A first course in topology is usually a semester in point-set topology. Sometimes a chapter on the fundamental group is included at the end, with very little time left. For the student, algebraic topology often never comes.

The main part of the present text grew from the course Topology I and II that I have taught at Marshall University in recent years. This material follows a two-semester first course in topology with emphasis on algebraic topology. Some common topics are missing, though: the fundamental group, classification of surfaces, and knots. Point-set topology is presented only to the degree that seems necessary in order to develop algebraic topology; the rest is likely to appear in a, typically required, real analysis course. The focus is on homology. An introduction to discrete exterior calculus is also included.

The presentation is often more detailed than one normally sees in a textbook on this subject which makes the text useful for self-study.

Exercises are in the hundreds. They appear just as new material is being developed. Some of them are quite straight-forward; their purpose is to slow you down.

Pictures are also in the hundreds. They are used -- only as metaphors -- to illustrate topological ideas and constructions. When a picture is used to illustrate a proof, the proof still remains complete without it.

Applications are present throughout the book. However, they are neither especially realistic nor (with the exception of a few spreadsheets) computational in nature; they are mere illustrations of the topological ideas. Some of the topics are: the shape of the universe, configuration spaces, digital image analysis, data analysis, social choice, exchange economy, and, of course, anything related to calculus. As the core content is independent of the applications, the instructor can pick and choose what to include.

Both pictures and exercises are spread evenly through the sections (average: 1.3 per page, with standard deviation .3).

The way the ideas are developed may be called “historical”, but not in the sense of what actually happened -- it's been 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. Don't be discouraged!

A rigorous course in linear algebra is an absolute necessity. In the early chapters, one may be able to avoid the need for a modern algebra course but not the maturity it requires.

Chapter 1 contains an informal introduction to homology as well as a sample: homology of graphs. Chapter 2 is the starting point of point-set topology, developed as much as is needed for the next chapter. Chapter 3 introduces first cubical complexes, cubical chains, unoriented and then oriented, and cubical homology. Then a thorough introduction to simplicial homology is provided. Chapter 4 continues to build the necessary concepts of point-set topology and uses them to develop further ideas of algebraic topology, such as homotopy and cell complexes. Chapter 5 presents homology theory of polyhedra. Chapter 6 introduces calculus of discrete differential forms (cochains) and cohomology. Chapter 7 develops some applications of differential forms.

By the end of the first semester, one is expected to reach the section on simplicial homology in chapter 3, but maybe not the section on the homology maps yet. For a single semester first course, one might try this sequence: chapter 2, sections 3.4 - 3.6, chapter 4 (except 4.3), section 5.1. For a one-semester course that follows point-set topology (and modern algebra), one can take an accelerated route: chapters 3 - 5 (skipping the applications). For discrete calculus, follow: sections 3.1 - 3.3, chapters 6 and 7.

The book is mostly undergraduate; it takes the student to the point where the tough proofs are about to start to become unavoidable. Where the book leaves off, one usually proceeds to such topics as: the axioms of homology, singular homology, products, homotopy groups, or homological algebra. Geometry and Topology by Bredon is a good choice.

Best of luck!

Peter Saveliev


References

Here follows the list of (some of) the books I previously used for teaching: