Embeddings in Manifolds (Graduate Studies in Mathematics)

Robert J. Daverman and Gerard A. Venema

ISBN 10: 0821836978 / ISBN 13: 9780821836972

Published by American Mathematical Society, 2009

About this Item

Analyzes how and when objects like polyhedra or manifolds embed in a given higher-dimensional manifold. This book offers description of the various classic examples of wild embeddings. It provides details of the codimension-three proofs, including the requisite piecewise linear tools. Series: Graduate Studies in Mathematics. Num Pages: 474 pages, Illustrations. BIC Classification: PBM; PBP. Category: (P) Professional & Vocational; (UP) Postgraduate, Research & Scholarly. Dimension: 257 x 183 x 31. Weight in Grams: 1034. . 2009. Hardcover. . . . .

Seller Inventory # V9780821836972

Bibliographic Details

• Title: Embeddings in Manifolds (Graduate Studies in Mathematics)
• Publisher: American Mathematical Society
• Publication Date: 2009
• Language: English
• ISBN 10: 0821836978
• ISBN 13: 9780821836972
• Binding: Hardcover
• Condition: New

About this title

Synopsis

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 higher-dimensional 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 codimension-three proofs, including the requisite piecewise linear tools, are provided. The treatment of codimension-two embeddings includes a self-contained, elementary exposition of the algebraic invariants needed to construct counterexamples to the approximation and existence of embeddings. The treatment of codimension-one embeddings includes the locally flat approximation theorem for manifolds as well as the characterization of local flatness in terms of local homotopy properties.

Review

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