Book Review: Introduction to differentiable manifolds
Robert B. Hermann · Bulletin of the American Mathematical Society · 1964
Historical notes are found at the ends of some sections and chapters; there are helpful exercises at the end of each section, and the bibliography is fairly extensive.It should be pointed out that the role of resolutions in this book is secondary.They are introduced after the definitions of Ext and of Tor as computational devices.In MacLane's approach they are not (and should not be) basic to the definitions.There is a great temptation to ask that this book include many more applications in many more areas than it does.However, if we keep in mind the fact that the title of the book is Homology, and that the author assumes that we all know by now why we should study homology, we can appreciate the selectivity which the author has displayed.In the light of recent developments, Chapters IX and XII may well be a little out of date, but MacLane points out in his introduction that the subject is still in a state of flux, and it's anybody's guess as to what the final word may be (if any).In any event, this is a book which can be given to a student with the assurance that if he absorbs the material in it he will have learned a great deal.Moreover, the students who have been reading this book have found it to be extremely helpful and clear.