I think there is no conceptual difficulty at here. For his definition of connected sum we have: Two manifolds M 1, M 2 with the same dimension in. Differential Manifolds – 1st Edition – ISBN: , View on ScienceDirect 1st Edition. Write a review. Authors: Antoni Kosinski. “How useful it is,” noted the Bulletin of the American Mathematical Society, “to have a single, short, well-written book on differential topology.” This accessible.
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The concepts of differential topology form the center of many mathematical disciplines such as differential geometry and Lie group theory. Differential Manifolds presents to advanced undergraduates and graduate students the systematic study of the kosinsski structure of smooth manifolds. Kosinski, Professor Emeritus of Mathematics at Rutgers University, offers an accessible approach to both the h-cobordism theorem and the classification of differential structures on spheres. Subsequent chapters explain the technique of joining manifolds along submanifolds, the handle presentation theorem, and the proof of the h-cobordism theorem based on these constructions.
There follows a chapter on the Pontriagin Construction—the principal link between differential topology and homotopy theory. The final chapter introduces the method of surgery and applies it to the classification of smooth structures of spheres. The text is supplemented by numerous manifoldd historical notes and contains a new appendix, ,anifolds Work of Grigory Perelman,” by John W.
Morgan, which discusses the most recent developments in differential topology. Read more Read less. Discover Prime Book Box for Kids. Add both to Cart Add both to List. These items are shipped from and sold by different sellers.
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Customers who bought this item also bought. Page 1 of 1 Start over Page 1 of 1. Differential Geometry of Curves and Surfaces: Category Theory in Context Aurora: Dover Modern Math Originals. Dover Books on Mathematics Paperback: Dover Publications October 19, Language: Don’t have a Kindle? Try the Kindle edition and experience these mankfolds reading features: Share your thoughts with other customers.
Write a customer review. Showing of 5 reviews. Top Reviews Most recent Top Reviews. There was a problem filtering reviews right now. Please try again later. There are many books on such pre-Ricci-flow differential topology, and they cover much the same material, but this book by Kosinski tries to be helpful to the reader, rather than showing off virtuoso techniques in perplexing ways as some books seem to kosjnski.
This appendix is referred to in an interpolated paragraph on page There’s another interpolated paragraph on page rifferential, which you can recognize by the different font.
As I said above, this book is peripheral to my interests because it is really a differential topology book, not a differential geometry book. So it contains all of the topics regarding differentiable manifolds which do not interest me personally. However, if I ever do want to get into differential topology, this book will be first on my study list.
One person found this helpful. In fact, the purpose of this book is to lay out the theory of higher-dimensional, i. Offhand, I can’t think of another book that covers all these topics as thoroughly and concisely, and does so in a way that is readily comprehensible. The first 4 chapters are an overview of the basic background of differential topology – differential manifolds, diffeomorphism, imbeddings and immersions, isotopy, normal bundles, tubular neighborhoods, Morse functions, intersection numbers, transversality – as one would find in, e.
As the author himself states, with some understatement, “The presentation is complete, but it is assumed, implicitly, eifferential the subject is not totally unfamiliar to the reader. The reader should also have a good knowledge of algebraic topology Dold and Spanier are frequently used as referencesas well as the classification of differntial over spheres as found in Steenrod.
Since the purpose of the first 4 chapters about differsntial pp is to develop the machinery of differential topology to the point where the results on handles, cobordism, and surgery can be proved, several topics are briefly touched upon that are generally not encountered in introductory diff top books, such as the group Gamma of differential structures on the m-sphere mod those that extend over the m-disk or the bidegree of a map from a product of spheres to a sphere, in addition to the aforementioned results of Whitney and Haefliger, but just enough is given so that they may be used in later proofs.
Most perplexing is Chapter V, on foliations, which has only a tenuous connection to the preceding material and absolutely none to the following. It seems that the author just included it because he felt that knowledge of the subject was essential for a topologist, not because it was necessary for the purposes of this book; it certainly could be skipped, but is worth reading as a brief introduction to foliations.
The heart of the book is Chapter VI, where the concept of gluing manifolds together is explored. Normally, connected sums are defined by removing imbedded balls in 2 closed manifolds and gluing them along the spherical boundaries, but Kosinski kosjnski constructs, explicitly in local coordinates, an orientation-reversing diffeomorphism of a punctured ball and then uses that to identify punctured balls in each manifold.
Similarly, handle attachment is defined, rather than by just attaching a handle to an dfiferential sphere in the boundary, but instead by again explicitly constructing an orientation reversing diffeomorphism of a in the 0-dim case punctured hemisphere and then identifying it with the normal bundle of a point in the boundary of the manifold. In this way, one automatically constructs smooth manifolds without having to resort to “vigorous hand waving” to smooth corners.
The mwnifolds to this method which is likely to be unfamiliar to modern readers is that much time is spent constructing explicit formulas for handle attachments, e. As you can see, a lot of important results are derived, whose proofs are complete except for a few technical lemmas that are cited.
Most chapters conclude with a section titled “Historical Remarks” or just “Remarks,” that explains the history of the development differenntial the subject, including many references.
The author himself, now almost 80, had in hand in some of these developments and was personally well-familiar with the giants of c. The text is also interlaced with exercises, most of which are relatively straightforward. The book concludes with a new appendix, written last year by John Morgan my former thesis adviseron Perelman’s proof of the Poincare conjecture.
It’s just an overview of the proof and feels really out of place, the only connections being that it concerns the Poincare conjecture in dim 3, whose proof for dimensions higher koosinski 4 is one of the highlights of this book, and also that Perelman’s proof involves a kind of surgery.
This appendix does little to enhance the value of the book. The book is not without it faults, however. In addition to the above observations manifo,ds it being too advanced for an introductory text and the incongruity of Chapter V, there are the usual batch of typos: There are also errors of exposition, such as reversing the order of the i and j terms in the definition of M1 and M2 on pg. A more serious omission is Theorem X,5. Probably the worst mistake is when the term “framed manifold” is introduced and defined to mean exactly the same thing as “pi-manifold,” without ever acknowledging this fact, and then the terms are used interchangeably afterward, with theorems about framed manifolds being proved by reference to results about pi-manifolds, and even with the redundant expression “framed pi-manifold” being used in a few places.
Moreover, “framed cobordant” is then defined in Chapter X kozinski mean something different than it meant in Chapter IX. Another group of complaints that I have is with the system of references. First of all, the chapter numbers do not appear in either the running heads or the theorem numbers, so when a result is cited in a previous chapter, the reader must flip back and forth through the book to find it, remembering the chapter numbers for each chapter, or must go back to the table of contents to locate it.
Moreover, many theorems from earlier chapters are used kosnski comment, or a reference is made to a theorem when in fact a corollary is being used or vice versa! Sometimes a theorem from another source is cited as the justification for a statement, when in fact the author is directly applying a theorem from his own book that just happened to use that other kosiinski result in its proof – citing his own theorem, by number, would save the reader a lot of effort. And then there’s the important imbedding theorem of Haefliger that he frequently cites, even though he never actually states what the theorem says!
I had to read Haefliger’s paper to verify that it actually could be used to produce the results that Kosinski wanted. This book contains a lot of information about manifolds, particularly those with differentiable structures. It used to only be available with a boring green cover and it was expensive. Now, its cover is colorful and has a wacky picture on it. I thought that this would surely make the price go up but it got cheaper! The picture on the front cover concerns operations on manofolds, particularly differentiable manifolds.
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