**Graduate Studies in Mathematics**

Volume: 106;
2009;
468 pp;
Hardcover

MSC: Primary 57;

Print ISBN: 978-0-8218-3697-2

Product Code: GSM/106

List Price: $79.00

Individual Member Price: $63.20

**Electronic ISBN: 978-1-4704-1591-4
Product Code: GSM/106.E**

List Price: $79.00

Individual Member Price: $63.20

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#### Supplemental Materials

# Embeddings in Manifolds

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*Robert J. Daverman; Gerard A. Venema*

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.

#### Readership

Graduate students and research mathematicians interested in geometric topology.

#### Reviews & Endorsements

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

-- MAA Reviews

#### Table of Contents

# Table of Contents

## Embeddings in Manifolds

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Acknowledgments xi12 free
- Introduction: The main problem xiii14 free
- Prequel 120 free
- Tame and knotted embeddings 2342
- Wild and flat embeddings 4160
- Engulfing, cellularity, and embedding dimension 97122
- Trivial-range embeddings 145170
- Codimension-three embeddings 181206
- Codimension-two embeddings 273298
- Codimension-one embeddings 349374
- Codimension-zero embeddings 425450
- Bibliography 445470
- Selected symbols and abbreviations 461486
- Index 463488 free
- Back Cover Back Cover1496