Product Description
Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically.

It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike. ... Read more

Customer Reviews (3)

This is where it started
For those individuals who want an in-depth, insightful, and solid understanding of fiber bundles this book must be read. In spite of its date of publication, it still is of considerable value in this regard. Modern treatments of fiber bundles are very formal and the underlying motivation gets swept away in the thirst for rigor. Fiber bundles are now ubiquitous in differential topology, algebraic topology, differential geometry, and algebraic geometry, and have also found a place in theoretical physics, thanks to the success of gauge field theories. Therefore a mastery of fiber bundles is essential for entering any of these fields. But fiber bundles are fascinating objects in and of themselves, and studying them for their own sake needs no apology.

The author does use some antiquated notation, but that is not really a hindrance to the study of the book. The reader will no doubt have some background in differential geometry and topology before attempting this book, so the appropriate translation to more modern notation should be straightforward. Once started, and with a little thought adjustment to the idiosyncracies of the author's writing style, the reader will find a plethora of neat examples and insights into the subject. In particular, part 3 on the cohomology theory of bundles is exceptionally valuable in that it gives the reader a detailed overview of the origin of what are not called Stiefel-Whitney classes. The theory of characteristic classes has of course advanced and matured extensively since this book first appeared, but all of the modern treatments are lacking in that they do not give the reader an appreciation of the fundamentals of the subject. Indeed, the construction of the obstruction to the construction of a cross-section to a bundle is the starting point for many of the ideas in obstruction theory that one finds in differential topology. And yes, the procedures the author uses can be "cleaned-up" and made more concise, but the price one pays in such an endeavor is the loss of an appreciation of the concepts behind the scene.

Since the book is a monograph, there are no exercises, and this is probably the only minus to the book. Also, some knowledge of the German language would be useful to a reader who has it, since the author makes references to papers written in German and much of the terminology in the book shows its roots in the German language. One good example of this is the Reidemeister theory of cohomology groups based on a bundle of coefficients, called Uberdeckung by Reidemeister.

There is no question as to why this book remains in print, and it will no doubt continue to be well into the 21st century. IT is a good example of the idea that something new may not be something better. After finishing it, the reader will be amply prepared to enter into the continually-evolving theory of fiber bundles and their applications, all of which are interesting and important.

Excellent introduction to fiber bundles
This book supplies a lot of intuition and background that more modern texts seem to assume of the reader.Steenrod's writing is meticulous and extremely clear.My opinion is that one can learn just as much out of this seemingly outdated text and probably even more than from the modern texts.

...True, more slick machinery has been developed since Steenrod's time, but those big machines are hardly transparent.Steenrod assumes very little of the reader; he even has a quick course in homotopy groups, although he assumes the reader knows the basics of homology/cohomology.Perhaps most importantly, since many of the ideas in the book were new at the time, he doesn't assume that the reader is already comfortable with those ideas.All together this makes a very accessible book indeed.

Still attractive.
A nostalgic but still attractive book on (homotoy theory of) fiber bundles. This book is not very accessible as it predates the development of modern machinery of algebraic topology, but is worth reading. ... Read more

Product Description
This book contains four essays on expository writing ofbooks and papers at the research level and at the level of graduatetexts. The authors were the four members of the AMS Committee onExpository Writing ... Read more

Customer Reviews (3)

A must read for mathematicians and scientists
This short booklet contains four essays about how to write mathematics papers and books. The essays by Steenrod and Halmos, two prominent figures of 20th century mathematics, stand out for their common sense, depth and lucidity. They bring forth essential strategic issues, such as the need to maintain a clear separation between the formal and informal parts of mathematical papers, as well as useful tactical issues such aschoosing notation. In my opinion, the essays transcend the field of mathematics, and the principles that they delineate are applicable to all areas of scientific writing.

multi-author approach works
The various authors of "How to Write Mathematics" set out to describe their own personal writing styles, beginning with an admission that there is a lack of consensus on this topic. Indeed, the book closes with a couple of pages by the last author critiquing the advice of the first. On including detailed proofs in your publications, for example, you will find one author prescribing and another proscribing the practice.

The essays were written independently and this shows in the overlap and the contradictions. Whereas I have been highly critical of multi-author books in the past, the approach seems to work in this collection. Recalling my own efforts in writing a book on a topic within another field, I found myself agreeing or disagreeing with the authors on the various points. I was relieved to note that as early as at least three decades ago, authors before me shared identical conflicts with editors and copyreaders.

Based on my limited experience in writing, I had developed a number of half-formed rules of thumb. Given the recommendations of writers more experienced than I as so boldly presented, I am likely to be more aggressive in applying these rules. I would recommend this book to those authors with at least a few writing efforts under their belts. Given the specific nature of the examples in the essays, however, it is likely that only mathematicians will fully appreciate this book.

The paper by Halmos is worth the price of the book
Paul Halmos gives wonderful advice on how to write in mathematics.Get it. ... Read more