A short course in Differential Topology

There are several very good books on Differential Topology, the first to mention Milnor's gem that everybody knows and loves. However, it is difficult to use those books as a text for an introductory one term 60 hours course where you want to teach the basics form scratch (as far as possible, taking into account the nature of Maths degrees nowadays) and get in the end to some interesting applications. This being the situation one confronts often in the real classroom today, this text by Professor Dundas is a wonderful proposal to welcome. Indeed, the topics (summarized in the short description below) are presented in a very friendly and appealing way to motivate abstract notions; this includes many nice examples and assorted references to web links worth clicking. Quite useful too, some parts are explicitely marked when they can be omitted to have all the same a good program. There are in addition some pluses to stress: (1) the use of germs to discuss tangencies, (2) a presentation of vector bundles in general, (3) an introduction to Morse functions and (4) a very nice proof of Ehresmann theorem. Another remarkable feature is the care taken to provide a good bunch of meaningful exercises.

Summing up, the book is highly recommendable for all publics. An interested student can very well go through it quite by her/himself and learn a lot. A professor surely can discover useful aspects that may have skipped her/his attention before. But most important, Professor Dundas offers us a very enjoyable reading!

Reviewer: 
Jesus M Ruiz
Book details

This is a text designed for a real course on Differential Topology for postgraduate students. The topics are as expected: smooth manifolds, tangent spaces and derivatives, regular values and transversality, vector bundles, tangent fields and flows and some global basic facts. There are many examples carefully chosen and described for motivation, as well as many exercises with hints to help the student. A very good book on the matter.

Author:  Publisher: 
Published: 
2018
ISBN: 
978-1-108-42579-7
Price: 
50 EUR
Pages: 
263
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