Embeddings in Some Singular Manifolds
Author | : Lyle Leonard Welch |
Publisher | : |
Total Pages | : 158 |
Release | : 1971 |
Genre | : Topology |
ISBN | : |
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Author | : Lyle Leonard Welch |
Publisher | : |
Total Pages | : 158 |
Release | : 1971 |
Genre | : Topology |
ISBN | : |
Author | : Lyle Leonard Welch |
Publisher | : |
Total Pages | : 128 |
Release | : 1971 |
Genre | : Manifolds (Mathematics) |
ISBN | : |
Author | : Robert J. Daverman |
Publisher | : American Mathematical Soc. |
Total Pages | : 496 |
Release | : 2009-10-14 |
Genre | : Mathematics |
ISBN | : 0821836978 |
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.
Author | : Robert J. Daverman |
Publisher | : |
Total Pages | : 496 |
Release | : 2009 |
Genre | : Embeddings (Mathematics) |
ISBN | : 9781470415914 |
Author | : Robert Everist Greene |
Publisher | : American Mathematical Soc. |
Total Pages | : 69 |
Release | : 1970 |
Genre | : Embeddings (Mathematics) |
ISBN | : 0821812971 |
Author | : Qing Han |
Publisher | : American Mathematical Soc. |
Total Pages | : 278 |
Release | : 2006 |
Genre | : Mathematics |
ISBN | : 0821840711 |
The question of the existence of isometric embeddings of Riemannian manifolds in Euclidean space is already more than a century old. This book presents, in a systematic way, results both local and global and in arbitrary dimension but with a focus on the isometric embedding of surfaces in ${\mathbb R}^3$. The emphasis is on those PDE techniques which are essential to the most important results of the last century. The classic results in this book include the Janet-Cartan Theorem, Nirenberg's solution of the Weyl problem, and Nash's Embedding Theorem, with a simplified proof by Gunther. The book also includes the main results from the past twenty years, both local and global, on the isometric embedding of surfaces in Euclidean 3-space. The work will be indispensable to researchers in the area. Moreover, the authors integrate the results and techniques into a unified whole, providing a good entry point into the area for advanced graduate students or anyone interested in this subject. The authors avoid what is technically complicated. Background knowledge is kept to an essential minimum: a one-semester course in differential geometry and a one-year course in partial differential equations.
Author | : Haynes Miller |
Publisher | : CRC Press |
Total Pages | : 1043 |
Release | : 2020-01-23 |
Genre | : Mathematics |
ISBN | : 1351251600 |
The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.
Author | : |
Publisher | : Academic Press |
Total Pages | : 331 |
Release | : 1986-12-22 |
Genre | : Mathematics |
ISBN | : 0080874436 |
Decompositions of Manifolds
Author | : Valentin Poenaru |
Publisher | : American Mathematical Soc. |
Total Pages | : 104 |
Release | : 2004 |
Genre | : Mathematics |
ISBN | : 0821834606 |
Shows that at the cost of replacing $V DEGREES3$ by $V_h DEGREES3 = \{V DEGREES3$ with very many holes $\}$, we can always find representations $X DEGREES2 \stackrel {f} {\rightarrow} V DEGREES3$ with $X DEGREES2$ locally finite and almost-arborescent, with $\Psi (f)=\Phi (f)$, and with the ope
Author | : Nikolai Saveliev |
Publisher | : Walter de Gruyter |
Total Pages | : 220 |
Release | : 2011-12-23 |
Genre | : Mathematics |
ISBN | : 3110250365 |
Progress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant. The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments. The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds.