Topology from the Differentiable Viewpoint by Milnor

[micromass]

• Author: John Milnor
• Title: Topology from the Differentiable Viewpoint
• Amazon Link: https://www.amazon.com/dp/0691048339/?tag=pfamazon01-20
• Prerequisities:
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] Smooth manifolds and smooth maps
[*] Tangent spaces and derivatives
[*] Regular values
[*] The fundamental theorem of algebra
[*] The theorem of Sard and Brown
[*] Manifolds with boundary
[*] The Brouwer fixed point theorem
[*] Proof of Sard's theorem
[*] The degree modulo 2 of a mapping
[*] Smooth homotopy and smooth isotopy
[*] Oriented manifolds
[*] The Brouwer degree
[*] Vector fields and the Euler number
[*] Framed cobordism; the Pontryagin construction
[*] The Hopf theorem
[*] Exercises
[*] Appendix: Classifying 1-manifolds
[*] Bibliography
[*] Index
[/LIST]

 

[mathwonk]

It is hard to imagine a book which will teach more in fewer pages than this one. A significant expansion of much of this material is in the book by Guillemin and Pollack.

 

[Greg Bernhardt]

mathwonk, your library must rival the great Alexandria's :)

 

[mathwonk]

Unfortunately i gave some volumes away when i moved out of my office. But my collection's strong representation in this sample is skewed because these books on this list are of such high quality!

(It is also partly because I am always trying to find books aimed at beginners and students from which I myself can learn new topics. Some of my friends have much larger collections of more advanced and specialized books.)

In fact my copy of Milnor's book which I bought in about 1966, was autographed by him in 1997 on the occasion of his delivering the Cantrell lectures at UGA.

http://www.math.uga.edu/seminars_conferences/milnor-lectures.htm

 

[rezeew]

As a fellow scientist, I am excited to see John Milnor's work on topology from the differentiable viewpoint. This book is a valuable resource for undergraduates interested in understanding the fundamental concepts of topology and its connections to differentiable manifolds.

Milnor's approach to topology is unique and insightful, as he explores the subject through the lens of smooth manifolds and smooth maps. His discussion on tangent spaces and derivatives provides a solid foundation for understanding the more complex topics in the book.

The inclusion of topics such as the fundamental theorem of algebra, Sard and Brown's theorem, and the Brouwer fixed point theorem make this book a well-rounded and comprehensive resource for undergraduates. The exercises at the end of each chapter also serve as a helpful tool for students to deepen their understanding of the material.

One of the highlights of this book is Milnor's proof of Sard's theorem, which is clear and concise, making it accessible to undergraduates. His discussion on smooth homotopy and smooth isotopy is also well-presented, providing a deeper understanding of these important concepts.

Moreover, the inclusion of topics such as oriented manifolds, the Brouwer degree, and the Hopf theorem adds a more advanced level of understanding to the subject. The appendix on classifying 1-manifolds is a helpful addition for students needing a refresher on this topic.

Overall, Topology from the Differentiable Viewpoint is an excellent resource for undergraduates interested in topology and its connections to differentiable manifolds. Milnor's clear writing style and comprehensive coverage of the subject make this book a must-have for any student studying topology. I highly recommend this book to anyone looking to deepen their understanding of this fascinating field.