Algebraic Topology

The single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. The purpose of this book is to help the aspiring reader acquire this essential common sense about algebraic topology in a short period of time. To this end, Sato leads the reader through simple but meaningful examples in concrete terms. Moreover, results are not discussed in their greatest possible generality, but in terms of the simplest and most essential cases. In response to suggestions from readers of the original edition of this book, Sato has added an appendix of useful definitions and results on sets, general topology, groups and such. He has also provided references.Topics covered include fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. Objects and examples considered in the text include the torus, the Mobius strip, the Klein bottle, closed surfaces, cell complexes and vector bundles.

Review

… Sato’s book is a gem, and I am happy to recommend it in very enthusiastic terms. –MAA Reviews

This is an uncommon book with an interesting idea behind it, which is given in its title: to give an intuitive approach to algebraic topology. Instead of stating theorems in full generality or proving them rigorously with all technical details (or proving them at all), the author rather tries to make the reader familiar with “the idea” of the central notions of algebraic topology. –Zentralblatt MATH

A nice supplement for a topology course. –American Mathematical Monthly

Table of Contents

1. Homeomorphisms and homotopy equivalences
2. Topological spaces and cell complexes
3. Homotopy
4. Homology
5. Homology groups of cell complexes
6. Cohomology
7. The Universal coefficient theorem
8. Fiber bundles and vector bundles
9. Spectral sequences