Differential Topology by Victor Guillemin (PDF)

Description

Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack’s book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea–transversality–the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincarøƒ-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Br

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is great for someone like me, who has seen bits and pieces of results from differential topology but would like to see a unified presentation of it. The exposition is concise but includes enough discussion to build some intuition. Easily comprehensible to someone who has had earlier courses in analysis, topology, and differential geometry, and even the latter is only helpful and not strictly required. Good for the advanced undergrad/first year grad student for self-study.The printing and binding are somewhat poor though, and some unconventional notation and definition are occasionally used.

⭐This book is my favorite piece of mathematical literature. The structure might not be for everyone since it is not written like Springers Graduate Text series. By this, I mean the book does not follow the theorem, proof, theorem, proof structure people find in other books. It is more like a story where the authors try to explain how they think about manifolds and how they see them.

⭐The approach taken in this book is a little dated, but with G+P’s witty commentary and valuable insights, there is still much to love. Some choice quotes:“Without transversality, XcapZ may be some frowzy, useless conglomeration.”“If our propaganda has not yet made you a true believer in forms, we invite you to try defining the integral of a function.”“Stand the surface vertically on one end, and coat it evenly with hot fudge topping. Let f_t(x) denote the oozing trajectory of the point x of fudge as time t passes.”

⭐I agree with the reviewer who is not a “higher mathematician”. Neither am I; in fact, I repeatedly found that both Milnor and Hirsch became remarkably clearer after reading the same material from this book. So I stuck to this book. Chapter 4 is particularly well-written, with a very incisive discussion of connections among geometry, algebra, and topology. I hope the publishers decide to republish this book. How hard can that be in the modern small-volume printing era?

⭐I don’t normally write reviews for products, and this is my first time writing a 1 star review for a book. Reading this book makes me recall the famous quote by Albert Einstein “the more I learn, the more I realized I don’t know”.Although a good math text book shouldn’t be a “spoon feed” book, the authors for this book not only have been overly terse on many important definitions, they have also saved tons of inks on providing helpful examples. Since clearly the authors don’t lack paper as they have printed every page with large margins, I’m considering donating some inks to the author to help them compose a better book.If you feel my languages are long-winded and confounding, this is exactly what I’m feeling when I’m reading this book. The authors write not as if they are writing a text book for undergraduate students, but as if they are writing a course outline for their professional peers. On exercise 1.1, the author writes “Show that smooth functions on R^k, considered as a subset of R^l, are the same as usual”. What does “as usual” mean? Should this equivocal term ever exist in an introductory text book that supposes to be rigorous???????In conclusion:It’s good ONLY IF you have VERY INTENSIVE backgrounds.

⭐It’s a perfectly serviceable book but it’s certainly an introduction. If you want anything with depth then you’ll need a more advanced book. Perfect for a low-level graduate course (depending on the school, I guess), but nothing more.

⭐This text is extremely well-organized and well-written. It is excellent as a text for a course or as an addition to a library. Well done!

⭐The book is a great example of a horrible math text. There are several major flaws in it.Firstly, the book is written like a novel, and only makes sense if you think like the authors. Definitions, theorems and proofs are often mixed in paragraphs with a large amount of wishy-washy explanations, which makes the book useless if you are using it for reference. The explanations are often difficult to follow and cluttered, requiring one to dedicate a large amount of time to draw diagrams and try and follow the train of thought of the author. I have wasted large amounts of time trying to find some particular definition required for a simple problem because the definition was hidden in a paragraph of useless gibberish.Secondly, the book is full of errors, omitted assumptions and flaky proofs. Problems at the end of chapters will omit important details like what regularity one should assume for a function, if a manifold is compact, and other such things. Some questions will completely ignore some special cases (such as the dimension of a manifold being 1).Finally, the quality of content varies widely. Sometimes statements are proven in a very hand wavy way, and then there are problems where such hand wavy arguments would make a statement trivial. On the other hand, sometimes explanations are very technical (yet not rigorous).The only way to read this in a sensible way is like reading a novel; from cover to cover, and it is difficult to follow most explanations unless you think in exactly the same way as an author. The general consensus with my friends is that the book is horrible. I would hope that no one would have to read this book.

⭐This book has a good amount of problems in each and every section. A bit concrete but I liked it.

⭐El libro trae una página más ancha que las demás y sobresale, además el fabricante quiso ocultar esto doblando la página sobre sí misma, esto se muestra en la foto.I liked the book very much… Whatever others say most books refer( if they are referring to a differential Topology book) this book for proofs..

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