Elementary Differential Topology | Date: 15 April 2011, 11:56
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Elementary Differential Topology: Lectures Given at Massachusetts Institute of Technology Fall, 1961
Summary: few topics, of very limited value
Munkres' "Elementary Differential Topology" was intended as a supplement to Milnor's Differential topology notes (which were similar to his Topology from the Differentiable Viewpoint but at a higher level), so it doesn't cover most of the material that standard introductory differential topology books do. Rather, the author's purpose was to (1) give the student a feel for the techniques of differential topology and practice in using them, and (2) prove a couple of basic and important results that at the time (1961) had not appeared in book form. Thus this book could not serve as a textbook for a course in the subject, but could be useful perhaps as a workbook for a student who wanted to practice solving problems. The word "elementary" in the title merely indicates that no algebraic topology is used in the proofs (with one minor exception, to show that a disk cannot be mapped homeomorphically onto an annulus) - its use was not intended as an indication of the level of the book, although it is pretty elementary anyway.
That this is not suitable as a text for learning differential topology is apparent from what material has been omitted: Sard's theorem, Whitney's imbedding theorem, Morse theory, transversality (except for a brief mention in the last couple of pages), the degree of a map, intersection theory, differential forms, vector bundles (except for the tangent and normal bundles), etc., to say nothing of more advanced topics such as cobordism or surgery.
So what is covered? Aside from basic definitions of C^r manifolds (i.e., manifolds with charts that have transition functions that are r times continuously differentiable), submanifolds, immersions, diffeomorphisms, bump functions, partitions of unity, and the inverse and implicit function theorems (proved only for Euclidean spaces), the results are divided into 2 sets: Those having to do with approximating a map with certain features by other maps (generally, showing that the set of maps with certain properties, such as imbeddings, immersions, diffeomorphisms, etc., is open in a certain function space). From this follows the well-known result that all C^r (r>=1) manifolds are smooth, the highlight of the first 2/3 of the book. Along the way, a few results are demonstrated that are needed in the proof, such as the existence of tubular neighborhoods and an imbedding theorem that is much weaker than Whitney's, but not much time is spent on them. This part ends with a proof of the uniqueness of the double of a manifold. Virtually all of these results can be found in Hirsch's Differential Topology in the first 2 chapters, proved much simpler and with modern notation. However, by keeping his presentation more geometric and with a minimum of formalism, it may be easier to follow Munkres' proofs (not that Hirsch is hard). As an example, Munkres uses for the topology of his function spaces the strong C^1 topology, rather than the compact-open topology that Hirsch uses.
The second part of the book, the final 40 pages or so, is devoted to proving that smooth manifolds are actually PL manifolds, and that the triangulation of a smooth manifold with a given smooth structure is essentially unique (a kind of smooth Hauptvermutung - this is not true for PL manifolds in general). This classic result is not usually included in differential topology (or PL topology) books - in fact, I can't think of another book which does contain this proof, making this the best (only?) reason to own this book. The proof itself is not that interesting, consisting of the standard manipulations of simplices that one usually sees in PL topology or older homology theory.
There are many "exercises" through the book, which generally ask the reader to fill in the details of proofs or extend the results of them. These tend to be pretty easy, whereas the many "problems" are harder. For these, hints are often given, so they usually aren't that difficult either (although one problem is labeled as "unsolved"). Aside from the proof that smooth => PL, the only other benefit of reading this book is to practice doing these exercises. But overall, this is far inferior to the aforementioned works of Milnor, Hirsch, Wallace (Differential Topology: First Steps), or Guillemin and Pollack (Differential Topology).
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