Hardcover ISBN:  9780821836972 
Product Code:  GSM/106 
List Price:  $84.00 
MAA Member Price:  $75.60 
AMS Member Price:  $67.20 
Electronic ISBN:  9781470415914 
Product Code:  GSM/106.E 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $63.20 

Book DetailsGraduate Studies in MathematicsVolume: 106; 2009; 468 ppMSC: Primary 57;
A topological embedding is a homeomorphism of one space onto a subspace of another. The book analyzes how and when objects like polyhedra or manifolds embed in a given higherdimensional manifold. The main problem is to determine when two topological embeddings of the same object are equivalent in the sense of differing only by a homeomorphism of the ambient manifold. Knot theory is the special case of spheres smoothly embedded in spheres; in this book, much more general spaces and much more general embeddings are considered. A key aspect of the main problem is taming: when is a topological embedding of a polyhedron equivalent to a piecewise linear embedding? A central theme of the book is the fundamental role played by local homotopy properties of the complement in answering this taming question.
The book begins with a fresh description of the various classic examples of wild embeddings (i.e., embeddings inequivalent to piecewise linear embeddings). Engulfing, the fundamental tool of the subject, is developed next. After that, the study of embeddings is organized by codimension (the difference between the ambient dimension and the dimension of the embedded space). In all codimensions greater than two, topological embeddings of compacta are approximated by nicer embeddings, nice embeddings of polyhedra are tamed, topological embeddings of polyhedra are approximated by piecewise linear embeddings, and piecewise linear embeddings are locally unknotted. Complete details of the codimensionthree proofs, including the requisite piecewise linear tools, are provided. The treatment of codimensiontwo embeddings includes a selfcontained, elementary exposition of the algebraic invariants needed to construct counterexamples to the approximation and existence of embeddings. The treatment of codimensionone embeddings includes the locally flat approximation theorem for manifolds as well as the characterization of local flatness in terms of local homotopy properties.ReadershipGraduate students and research mathematicians interested in geometric topology.

Table of Contents

Chapters

Chapter 0. Prequel

Chapter 1. Tame and knotted embeddings

Chapter 2. Wild and flat embeddings

Chapter 3. Engulfing, cellularity, and embedding dimension

Chapter 4. Trivialrange embeddings

Chapter 5. Codimensionthree embeddings

Chapter 6. Codimensiontwo embeddings

Chapter 7. Codimensionone embeddings

Chapter 8. Codimensionzero embeddings


Additional Material

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The book is very wellwritten: it includes many examples, details, and motivational comments.
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A topological embedding is a homeomorphism of one space onto a subspace of another. The book analyzes how and when objects like polyhedra or manifolds embed in a given higherdimensional manifold. The main problem is to determine when two topological embeddings of the same object are equivalent in the sense of differing only by a homeomorphism of the ambient manifold. Knot theory is the special case of spheres smoothly embedded in spheres; in this book, much more general spaces and much more general embeddings are considered. A key aspect of the main problem is taming: when is a topological embedding of a polyhedron equivalent to a piecewise linear embedding? A central theme of the book is the fundamental role played by local homotopy properties of the complement in answering this taming question.
The book begins with a fresh description of the various classic examples of wild embeddings (i.e., embeddings inequivalent to piecewise linear embeddings). Engulfing, the fundamental tool of the subject, is developed next. After that, the study of embeddings is organized by codimension (the difference between the ambient dimension and the dimension of the embedded space). In all codimensions greater than two, topological embeddings of compacta are approximated by nicer embeddings, nice embeddings of polyhedra are tamed, topological embeddings of polyhedra are approximated by piecewise linear embeddings, and piecewise linear embeddings are locally unknotted. Complete details of the codimensionthree proofs, including the requisite piecewise linear tools, are provided. The treatment of codimensiontwo embeddings includes a selfcontained, elementary exposition of the algebraic invariants needed to construct counterexamples to the approximation and existence of embeddings. The treatment of codimensionone embeddings includes the locally flat approximation theorem for manifolds as well as the characterization of local flatness in terms of local homotopy properties.
Graduate students and research mathematicians interested in geometric topology.

Chapters

Chapter 0. Prequel

Chapter 1. Tame and knotted embeddings

Chapter 2. Wild and flat embeddings

Chapter 3. Engulfing, cellularity, and embedding dimension

Chapter 4. Trivialrange embeddings

Chapter 5. Codimensionthree embeddings

Chapter 6. Codimensiontwo embeddings

Chapter 7. Codimensionone embeddings

Chapter 8. Codimensionzero embeddings

The book is very wellwritten: it includes many examples, details, and motivational comments.
MAA Reviews