ABSTRACT
I. Introductory Books 238A. General introductions 238B. Point-set topology 239II. Algebraic Topology 240A. Introductory 240B. Homotopy theory 241C. Cohomology of groups 244D. Homological algebra 245III. Manifold Theory 245A. Differential topology 245B. Piecewise linear topology 247C. Topological manifolds 247D. Surgery theory 247IV. Low-Dimensional Topology 248A. Surfaces 248B. 3-Manifolds 249C. Knot theory 249D. 4-Manifolds 250V. Historical 250A. History of topology 250B. Books of historical interest 251VI. Other Resources 251A. Handbooks 251 237
B. JournalsC. Online resources 252252
Topology is a relatively new branch of mathematics. Before 1900 there were a few hints of what was to come, but the subject is really a creation of the twentieth century. Laying the foundations took much of the first half of the century, as basic point-set topology crystalized into its current form and the proper formulations of homology and cohomology were gradually worked out. After 1950 the pace accelerated rapidly, and the next 20 or 30 years were a period of tremendous growth, with one major theorem after another coming in quick succession. One can get some idea of this by seeing how many Fields Medals were awarded for work in topology. Since around 1980 the major growth in topology seems to have shifted more to the peripheries, particularly in expanding its ties with other areas of mathematics.The late appearance of topology in the mathematical landscape may explain why there has been a shortage of good readable books in many parts of the subject. The situation has finally begun to improve in the last decade, especially in the core topics that have now reached a fairly stable form, but there remain a number of topics that are still not covered as well as they should be. It will be interesting to see how the list of books given below compares with such a list made 10 or 20 years from now.Information about current pricing is also included in the listings. This data was gathered from various online sources in early 2003 and only covers books that are still in print. Out-of-print books are indicated by the abbreviation OP. Discounts that are sometimes available are not taken into account. Complete accuracy for the prices and in-print status is not guaranteed, and in any case this information will soon become outdated. It is impossible not to notice how unreasonably expensive mathematics books from certain publishers have become. One’s local library is obviously the place to go for these high-priced books, and of course for the out-of-print books as well. A few of the books are available online for free download, and one can only hope this becomes the norm in the future. I. INTRODUCTORY BOOKS
To begin, here are two books having the aim of conveying to a general audience some idea of what topology is about, without the usual formal
apparatus of mathematics textbooks. Their prerequisites are minimal, so they could be read by an interested high school student. VV Prasolov. Intuitive Topology. Mathematical World, Vol. 4. Providence, RI: American Mathematical Society, 1995. [Translation from the Russian by A Sossinsky.] [$20] JR Weeks. The Shape of Space. 2nd ed. New York: Marcel Dekker,2002. [$35] B. Point-Set Topology
At the next level are introductory textbooks in point-set topology, or general topology as it is sometimes called. A working knowledge of basic point-set topology is needed not just for more advanced parts of topology but also for quite a few other areas of mathematics, so undergraduate courses in point-set topology are standard fare at most institutions, and many textbooks have been written for such courses. Often they include a little algebraic topology as well. Here are a few that stand out in one way or another. K Janich. Topology. New York: Springer, 1984. [Translation by S Levy of Topologie. Berlin: Springer, 1980.] [$30]Written in a very engaging style, fun to read. Unfortunately there are no exercises, so the book is not suitable as a text for a course unless supplemented by separate exercises. MA Armstrong. Basic Topology. New York: Springer, 1983. [$48]The first third of this book is an attactive exposition of the most basic point-set topology. The rest is an introduction to algebraic topology via the fundamental group and simplicial homology. J Dugundji. Topology. Boston: Allyn and Bacon, 1966. [OP]A good general book on point-set topology, unfortunately out of print. JR Munkres. Topology. 2nd ed. Upper Saddle River, NJ: Prentice Hall,2000. [$98]Not particularly inspiring, but a methodical presentation that many students seem to like. Covers all the basic point-set topology one might need, and then begins algebraic topology via fundamental group and covering spaces, still in the same slow-paced expositional style. If one is looking for less expensive alternatives, there are several introductory topology books published by Dover. One that is particularly well written is: TW Gamelin and RE Greene. Introduction to Topology. 2nd ed. Mineola, NY: Dover, 1999. [$11]
Even less expensive is this free online book: O Viro, O Ivanov, V Kharlamov, and N Netsvetaev. Elementary Topology. https://www.math.uu.se/~oleg/educ-texts.htmlThis is essentially an outline embellished with insightful comments. All proofs are left as exercises, so this is a learn-by-doing textbook. After an introduction to point-set topology, the book continues with introductions to algebraic topology and manifold theory.
