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Books > Science & Mathematics > Mathematics > Topology
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Embeddings in Manifolds (Hardcover, New ed.)
Loot Price: R3,395
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Embeddings in Manifolds (Hardcover, New ed.)
Series: Graduate Studies in Mathematics
Expected to ship within 12 - 19 working days
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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 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.
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