3-Manifolds
John Hempel
Princeton University Press
It seems odd that nobody has reviewed this book until now, but undoubtedly this is due to the fame of the book and the fact that the majority of those looking for it have no need for a review; however, a minority may find a review of what this book is and isn't useful in their decision to buy or not.<p>What this book isn't: 1) An introduction to topology, or even to low-dimensional topology.  Someone who has heard of 3-manifolds and gotten excited would do better to get a taste of the subject elsewhere first, e.g. in Rolfsen's _Knots and Links_. 2) A research monograph designed to bring the reader up to speed on current research on 3-manifolds.  This book is about 30 years old and doesn't even mention the Geometrization Conjecture of Thurston.   3) A book on the role of knot theory in 3-manifolds.  Knots play an important role in the theory, not only theoretically, but as a rich source of examples to sharpen the intuition and test conjectures (through Dehn surgeries on knots and links).  This role is not discussed in this book.<p>What this book is: 1) A primer for topologists seeking to become specialists in 3-manifolds.  The basic theorems regarding prime decomposition, loop and sphere theorems, Haken hierarchy, and Waldhausen's theorems on Haken manifolds are explained in detail.  These can be considered some of the highlights although much relevant material is necessarily also explained.  As perhaps befitting a primer, the JSJ decomposition and characteristic submanifold theory is not included.  Jaco's book complements Hempel by covering this material. 2) A reference for those already familiar with the material.  The writing style is very concise and to the point.  This makes it simple to look up a theorem to refresh one's memory on a sticky detail in a proof.  As an introduction to the material, some passages may be terse, but inevitably after some effort, they can be "decoded" completely, unlike some texts that may be more verbose but can never be entirely deciphered.  I think there could be a lot more pictures; there aren't very many, to say the least.  But if the reader draws his/her own pictures, this shouldn't be too much of a problem.<p>Some final remarks: This book serves its dual role as a primer and reference admirably, but the reader may get lost in the details and lose the forest for the trees.  Unfortunately, the only way to rectify this seems to be to read various papers on the subject to get a good feel of the various threads that motivate current research.  But with Hempel's _3-manifolds_ in hand, this task is much easier and enjoyable.