Decompositions of Manifolds by Robert J. Daverman (PDF)

Description

Decomposition theory studies decompositions, or partitions, of manifolds into simple pieces, usually cell-like sets. Since its inception in 1929, the subject has become an important tool in geometric topology. The main goal of the book is to help students interested in geometric topology to bridge the gap between entry-level graduate courses and research at the frontier as well as to demonstrate interrelations of decomposition theory with other parts of geometric topology. With numerous exercises and problems, many of them quite challenging, the book continues to be strongly recommended to everyone who is interested in this subject. The book also contains an extensive bibliography and a useful index of key words, so it can also serve as a reference to a specialist.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Whenever physicists or geometers get interested in manifolds, the manifolds they study usually come equipped with, say, a differentiable structure. Right away, they are able to start doing calculus on the manifold.To gain solid insight into even the most basic properties of smooth manifolds, one can be forced to generalize and consider purely topological manifolds, the country cousins of the smooth analogues. In one of the most celebrated examples of this, M.H. Freedman established the 4-dimensional Poincare Conjecture by starting with a smooth homotopy 4-sphere and deducing that it was, in fact a topological 4-sphere. (Freedman’s technique works if the input to his theory is merely a topological homotopy 4-sphere).So from time-to-time, we need to work with topological manifolds, and now the question is how do we work them? Daverman’s book provides a excellent suite of techniques, known as ‘Decomposition Theory’ to address this question. The book represents the only comprehensive, consolidated text treatment of the state-of-the-art of the theory known to this reviewer.The text begins with the classical theory developed by R.H. Bing and develops Bing’s shrinkability criterion. There is a nice exposition of Bing’s famous proof that the sum of two solid Alexander horned 2-spheres is the 3-sphere. (This work really put the then back-water decomposition theory into the mainstream of topology.)The next theme is the study of cellular and cell-like decompositions in context of AR’s/ANR’s. This is technical, but crucial material.The real highlight of the text is a complete treatment of Edwards’ Disjoint Disks Theorem for manifolds of dimension 6 and up. Edwards originally claimed the theorem for dimensions 5 and up and gave an outline of his proof at an address to the Internationl Congress of Mathematics. However no formal proof was ever written up by Edwards. Daverman’s treatment is a genuine service to the field. The standard application of Edwards’ theorem to the solution of the Double Suspension problem is given as a nice application, by-passing the extremely technical machinery originally developed by Cannon to solve the problem.Note that this text represents a first (and probably only) edition of the book. There are several errors in this first printing, so the reader should be careful going through the material to check statements of theorems, formulas, assumptions, etc.

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