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Rating: 
Summary: An important book
Review: Morse theory is about finding geometric information about manifolds using analysis of functions(smooth functions) having these manifolds as their domain. The important fact is that Morse theory can be generalized to infinite dimensional manifolds (i.e. understanding the "geometric" structure of these manifolds using analysis of functions defined on the manifolds). Milnor's book motivate the "infinite" Morse theory in the first chapter explaining finite Morse theory very clearly and in a beautifull manner. The second chapter is preliminaries on Riemannian geometry which leads to the second part of the book about infinite Morse theory. To be more clear:the spaces/manifolds are path spaces on lie groups (compact lie groups mostly ) and the functions on these spaces are "energy" functions of these paths.
Once the reader understand the first chapter, the infinite dimentional Morse theory is much more easy to understand.
And Milnor does a very good job explaining the similarity.
The final chapter shows some implications of Morse theoy in particular Bott periodicity theorem.
One remark is in order for someone who doesnt know(Riemannian) geometry the book can be very hard to read, so I recommend as a starting point the beautiful book of Matsumoto "Morse theory" this is avery good book in a basic level which can serve as a very good introduction to milnor's book.
Rating:
Summary: A real landmark in topology.
Review: Perhaps everyone who has had a bite in topology feels that this book is too famous to be given any kind of reference. The above having been said, this book is really a gem, elegantly explaining the Bott periodicity in the spirit of the original article by Bott. Of course, a simpler proof using K-theory has been available since the sixties, but that does not deteriorate the value of this book.
Rating:
Summary: A Gem of Exposition
Review: This book needs no review, as it is so well-known. But reading the first critique about the need for a more intuitive development of the subject, perhaps one viewpoint is worth restating. This book is a model of concise exposition. All of Milnor's works are written in a fashion that "makes it clear", except that sometimes years of thought are really needed to see the essence of the arguments. The writing sets up the framework for understanding, and the reader must work to fill it in. This is what makes it great writing. The reader accepts the truth of the statements, but Milnor does not bludgeon the reader with details which can be filled in by a professional mathematician. Or as many of us understand it, when you can fill in all the details, you have developed geometric intuition and are on the way to a deeper understanding of the subject. Each chapter of the book is a classic. Chapter 2 on Riemannian geometry gives an overview of the subject which can be used as a basis for teaching a course on the same. When the students can fill in the details, they understand the core of the subject.
Rating:
Summary: The best
Review: When I was just becoming a mathematician, my teacher gave me this book, saying "You're not ready for this yet, but you should have it --- it's the best piece of mathematical exposition there is." Maybe that claim's exaggerated, but I've yet to find one I prefer. Along with Milnor's Lectures on the h-Cobordism Theorem, and his Characteristic Classes, this book is a lesson not only in topology (and wonderful topology, too!), but in clear writing as well.
Rating:
Summary: The best
Review: When I was just becoming a mathematician, my teacher gave me this book, saying "You're not ready for this yet, but you should have it --- it's the best piece of mathematical exposition there is." Maybe that claim's exaggerated, but I've yet to find one I prefer. Along with Milnor's Lectures on the h-Cobordism Theorem, and his Characteristic Classes, this book is a lesson not only in topology (and wonderful topology, too!), but in clear writing as well.
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