Manifolds can be studied in the topological, differentiable, or piecewise linear categories. Many statements (such as the Poincaré Conjecture) have versions for manifolds in each of these categories, but the techniques for answering the questions (and indeed the answers themselves) often depend both on the chosen category and on the dimensions of the manifolds involved. The Poincaré Conjecture’s history provides a nice example of the dependence on category and dimension. The differentiable version for manifolds of dimension 5 and higher is due to Smale; the piecewise linear version for manifolds of dimension 5 and higher is true by work of Stallings and Zeeman; the topological version for manifolds of dimension 4 was proved by Freedman; and the differentiable 4-dimensional Poincaré Conjecture is still open. In dimension 3, the topological, differentiable, and piecewise linear categories are essentially equivalent. Of course, the 3-dimensional Poincaré Conjecture was proved by Perelman.
Embeddings in Manifolds is concerned exclusively with the topological (TOP) and piecewise linear (PL) categories. It aims to delineate the boundary between the categories and to provide (usually algebraic) criteria for promoting TOP information to PL information.
As befits a book in the series “Graduate Studies in Mathematics”, Embeddings in Manifolds is aimed at graduate students and researchers. Although several sections are accessible to anyone with a course in topology, the vast majority of the book will be accessible only to those with significant topological background and the desire and energy to work carefully through a thorough compendium of foundational results in the theory of TOP and PL manifolds.
The book is very well written: it includes many examples, details, and motivational comments. The proofs are thorough and many references to the literature are provided. The sections which describe wildly embedded knots and surfaces in R3 will be particularly useful in that they provide complete proofs of well-known results which are difficult to find elsewhere. Most of the book is devoted to high dimensional PL topology. Here also many interesting examples are given and theorems proven. For example, in Chapter 3 the authors provide Stallings’ proof of a (slightly weakened) PL Poincaré Conjecture (for manifolds of dimension at least 5). Later chapters give an overview of Siebenmann’s theorem on adding boundaries to non-compact manifolds and Kirby’s solution to the Annulus Conjecture. Every manifold topologist should find something of interest in this book. Those beginning the study of PL methods may well find it indispensible.
A reader’s enjoyment of Embeddings in Manifolds may be hampered by two factors. First, Embeddings in Manifolds relies heavily on the (apparently out-of-print) classic Introduction to Piecewise-Linear Topology by Colin P. Rourke and B. J. Sanderson (Springer-Verlag, 1972). Indeed, the serious reader is advised to have a copy of Rourke and Sanderson’s book at hand; Embeddings on Manifolds makes frequent reference to it. Second, Embeddings in Manifolds could be greatly improved by an expanded index; the current one is sadly lacking. For example, the notion of 1-LCC is important throughout the book but does not appear in the index. It can, however, be found by going to the pages referenced by either LCCk or “locally co-connected” in the index.
Scott Taylor is a 3-manifold topologist and knot theorist. He is only mildly wild.