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Science/Engineering Introduction to piecewise-linear topology
Posted on 2010-03-16
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More Colin Patrick Rourke, Brian Joseph Sanderson: "Introduction to piecewise-linear topology" This monograph is now a classic, and written by some of the major innovators in the subject, this book can still be used as a textbook in piecewise linear topology, in spite of the many developments that have taken place since it was first published. Since the time of publication, a sphere recognition algorithm has been discovered (in the piecewise-linear category), shedding light on a possible resolution of the 3-dimensional Poincare conjecture, the latter of which is proved in this book for dimensions greater than or equal to 6. Also, at the time of publication, Freedman's proof of the Poincare conjecture in dimension 4 had not yet appeared.Thus, there are many reasons why one could consider this book to be out-of-date. However, it can still serve as a reference and as an introduction to elementary piecewise-linear topology. The presentation is excellent, and, as desired in any discussion of geometric topology, there are many diagrams to illustrate the main points. The foundations of piecewise-linear topology have not changed of course, and so the book could still be useful for self-study if one can find a copy of it in a used bookstore.Some of more interesting discussions in the book include the proof of the h-cobordism theorem (using excision, contractibility, and Lefschetz duality), the discussion of simple homotopy type, the proof of the weak Schoenflies theorem, the proof of the Whitney lemma, and the excellent overview of handle theory. Certainly a reading of this book will prepare one for more modern developments, and will hopefully entice readers into seeking out research problems in this area. Remember, the 3-dimensional Poincare conjecture is still open..... LINKS
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