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Rating: 
Summary: best math book ever written
Review:
Despite the lovely subject matter covered in this book, it more importanty gives one a taste of Mathematics as an intellectual discipline. It in outline shows how a mathematical theory - in this case Differential Topology - is constructed and consquently what mathematicians actually do and think about.
Anyone who would like to appreciate Mathematics as a field of study rather than just learn some math should open this book.
Better still, the prerequisite is only multivariate calculus!I have long thought this book should be the third year of calculus rather than differential equations or complex analysis.
Additionally, for the novice it is the only entry I know of into the mysteries of high dimensional geometry, that amazing almost unbelieveable accomplishment of the human mind.
There is a Star Trek episode in which a blind woman wears a dress of sensors which enable her to know more about her environment than a person can know from seeing. She knows exact distances and dimensions, can detect minute movements, can process the complete spectrum of light. In some sense she sees better. Modern topology and geometry are like that sensor dress for seeing higher dimensions. While we can not visualize the sphere in 5 dimensions, we know more about it from these mathematical theories than a five dimensionally sighted being ever could.
Today, mathematics is often considered to be just a practical tool - like a spread sheet - or a toaster oven. We forget its power to widen our imagination, to frame the unimaginable. This book reminds us of this and shows why Mathematics is the Queen of Sciences.
Rating:
Summary: Take full advantage of the clear, encompassing exposition:
Review: Do the exercises. Many were Ph.D. dissertation-level problems in the 1960s; today, they're aptly described as "elementary"- because Milnor MADE them elementary. This book forms part of the toolkit you will need to fully explore the more modern work in dynamics, complexity, and applications (e.g., economics, physics). The clarity of the exposition also forms an ideal example of how to communicate mathematics powerfully and simply.
Rating: 
Summary: Yet another popular (YAP) Math text
Review: In all practicality, for general math students this book is nice for the library, but by no means is it essential. In fact, it's not worth the $25 in most cases. There are so many outstanding Math texts / topics out there, it is doubtful and political that Milnor's Differential Viewpoint deserves its popularity.
I found this book helpful as supplementary reading for Calculus on Manifolds, so I am a minority student. (Majority student = Linear Algebra Done Right.) Spivak's book motivated the need to look carefully at the first few sections of Milnor's book. The definitions in Milnor coincide perfectly with Spivak.
I left Milnor's book with a good intuition about the inverse function theorem, manifolds, and the rank theorem. I also gave a small study to Sard's Theorem, but I had no need to venture into what apparently was the meat of the book...
It is arguable that the Inverse Function Theorem, Manifolds, and the Rank Theorem alone warrant buying this book, despite it only represents the first 2 sections of it, and is far from the total purpose of the book. Nonetheless, that's all I wanted to gain at the time I was reading it.
On the other hand, if you really are a Differential Topology student (small minority), you are the one who wouldn't need the review because you would know what you are trying to get from the book.
On the other hand, students who buy this book who don't know why will probably do nothing more than collect dust with it.
Rating:
Summary: Excellent starting point for differential topology
Review: One of the best points of this little book is its brevity and clear exposition of the basic ideas. It makes a great reference guide because it's so short and well-organized. Written by a distinguished mathematician, it's no wonder that other graduate-level texts such as Guillemin & Pollacks "Differential Topology" highly recommend reading it alongside their book. Milnor's booklet is a classic, whose style and ideas surely pervade other texts.
Rating:
Summary: An excellent introduction to topology of manifolds.
Review: Perfect for a first-year graduate or advanced undergraduate course, Milnor takes us on a brief stroll through elementary differential topology. Elegant and self-contained, this book serves as an excellent first taste of the subject. Milnor is a master expositor, and is at his best in this book.
Rating:
Summary: Take full advantage of the clear, encompassing exposition:
Review: This book is exceptionally well written and easy to read. Milnor proves a major result on almost every page. One learns a lot per unit time spent on this book. Despite being less than 80 pages, the book covers a significant amount of material in a clear concise manner.
Rating:
Summary: A good place to start
Review: This book is exceptionally well written and easy to read. Milnor proves a major result on almost every page. One learns a lot per unit time spent on this book. Despite being less than 80 pages, the book covers a significant amount of material in a clear concise manner.
Rating:
Summary: Compact, readable text on the topology of manifolds
Review: This book is very short, which is quite an asset for a math book to have. It also requires little knowledge of math beyond advanced calculus and point-set topology. I found it extremely readable, and I greatly enjoyed it. I recommend it highly, and especially enjoyed the proof on page 8 of the fundamental theorem of algebra. How far we've come since Gauss' first proof...
Rating: 
Summary: Compact and useful
Review: This book packs a lot of interesting material into a small volume. E.g., I picked up another book recently that started talking about cobordisms right off the bat; despite my having a couple of shelves full of well-known Dover, Springer, Cambridge UP etc. books on topology, differential geometry, mathematical physics, etc., Milnor's tiny book was the only one I found that could help me understand what cobordisms are right away. The book also uses many illustrations to help understanding.
I demote this to 4 stars only because Princeton UP's price is a bit high; many years ago I was lucky enough to find a used copy of the old U. Virginia edition, and paid much less.
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