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Title: Studies in Modern Topology by P.J. Hilton ISBN: 0-88385-105-9 Publisher: The Mathematical Association of America Pub. Date: June, 1940 Format: Textbook Binding List Price(USD): $23.00

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Average Customer Rating: 4 (1 review)

Rating: 4
Summary: Very out of date, but still a good source of intuition.
Comment: One of the things that stands out the most with older books in mathematics is that they emphasize the underlying ideas behind the concepts. They do this via pictures and lots of explanation using ordinary language, but they do eventually make their arguments mathematically rigorous. This is to be contrasted with modern texts, which usually approach a subject from a strictly formal point of view. This is fine from the standpoint of mathematical rigor, but students entering the field may have difficulty absorbing the underlying ideas behind the proofs and definitions. This is especially true for the physics community which must quickly master highly abstract ideas in order to use them in applications.

This book was published in 1968, and therefore is out of date considering the developments that have taken place in the areas discussed in the book since then. But the authors of the articles in the book explain things in a way that could be helpful to those who want a more in-depth understanding of various concepts in topology. The editor of the book gives an overview of the status of topology at that time and the articles that appear in the book. Much of the discussion revolves around geometric and algebraic topology, especially work that resulted from attempts to resolve the 3-dimensional Poincare conjecture. This conjecture in 5 dimensions or above, called the generalized Poincare conjecture, was solved by Stephen Smale some years earlier. The 3- and 4-dimensional Poincare conjecture was still open at the time of publication, but he latter was resolved by Michael Freedman just a few years later.

The 1960's was an exciting decade for topology, due not only to Smale's proof but also to many of the other fascinating results that were taking place at the time. Milnors work on exotic spheres stands out in particular, and even more so due to the later work of Simon Donaldson on exotic differentiable structures in 4-dimensional Euclidean space. Differential topology, which started in the 1950's with the cobordism theory of Rene Thom, was a relatively new branch of topology, but is now very important not only in mathematics but in physical applications. Cobordism theory is discussed in the book in the article by Valentin Poenaru, wherein he brings in the notion of a Thom space of a vector bundle, which can be thought of as a generalization of the one-point compactification of a non-compact topological space (and indeed is just that when the base space of the vector bundle is compact).

The 3-dimensional Poincare conjecture, which drove much of the research discussed in this book, is still open as of the date of this review. Three-dimensional topology is still a very active subject, and many fascinating tools, some of them finding inspiration from physics, such as gauge theories, have been applied to studying it. It is this opinion of this reviewer that the resolution of the 3-dimensional Poincare conjecture lies in these tools from physics, the biggest challenge being to make them mathematically rigorous.