Embedding in Some Singular Manifolds
Author | : Lyle Leonard Welch |
Publisher | : |
Total Pages | : 128 |
Release | : 1971 |
Genre | : Manifolds (Mathematics) |
ISBN | : |
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Author | : Lyle Leonard Welch |
Publisher | : |
Total Pages | : 128 |
Release | : 1971 |
Genre | : Manifolds (Mathematics) |
ISBN | : |
Author | : Lyle Leonard Welch |
Publisher | : |
Total Pages | : 158 |
Release | : 1971 |
Genre | : Topology |
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 | : |
Publisher | : Academic Press |
Total Pages | : 331 |
Release | : 1986-12-22 |
Genre | : Mathematics |
ISBN | : 0080874436 |
Decompositions of Manifolds
Author | : Haynes Miller |
Publisher | : CRC Press |
Total Pages | : 982 |
Release | : 2020-01-23 |
Genre | : Mathematics |
ISBN | : 1351251619 |
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 | : 333 |
Release | : 1973-03-30 |
Genre | : Mathematics |
ISBN | : 0080873677 |
Topological Embeddings
Author | : A.T. Fomenko |
Publisher | : Routledge |
Total Pages | : 226 |
Release | : 2019-06-21 |
Genre | : Mathematics |
ISBN | : 1351405683 |
Many of the modern variational problems of topology arise in different but overlapping fields of scientific study: mechanics, physics and mathematics. In this work, Professor Fomenko offers a concise and clear explanation of some of these problems (both solved and unsolved), using current methods of analytical topology. His book falls into three interrelated sections. The first gives an elementary introduction to some of the most important concepts of topology used in modern physics and mechanics: homology and cohomology, and fibration. The second investigates the significant role of Morse theory in modern aspects of the topology of smooth manifolds, particularly those of three and four dimensions. The third discusses minimal surfaces and harmonic mappings, and presents a number of classic physical experiments that lie at the foundations of modern understanding of multidimensional variational calculus. The author's skilful exposition of these topics and his own graphic illustrations give an unusual motivation to the theory expounded, and his work is recommended reading for specialists and non-specialists alike, involved in the fields of physics and mathematics at both undergraduate and graduate levels.
Author | : Abhishek Bhattacharya |
Publisher | : Cambridge University Press |
Total Pages | : 252 |
Release | : 2012-04-05 |
Genre | : Mathematics |
ISBN | : 1107019583 |
Ideal for statisticians, this book will also interest probabilists, mathematicians, computer scientists, and morphometricians with mathematical training. It presents a systematic introduction to a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes. The theory has important applications in medical diagnostics, image analysis and machine vision.
Author | : Dominic D. Joyce |
Publisher | : OUP Oxford |
Total Pages | : 460 |
Release | : 2000 |
Genre | : Mathematics |
ISBN | : 9780198506010 |
This is a combination of a graduate textbook on Reimannian holonomy groups, and a research monograph on compact manifolds with the exceptional holonomy groups G2 and Spin (7). It contains much new research and many new examples.
Author | : John M. Lee |
Publisher | : American Mathematical Society |
Total Pages | : 377 |
Release | : 2024-05-13 |
Genre | : Mathematics |
ISBN | : 1470476959 |
Complex manifolds are smooth manifolds endowed with coordinate charts that overlap holomorphically. They have deep and beautiful applications in many areas of mathematics. This book is an introduction to the concepts, techniques, and main results about complex manifolds (mainly compact ones), and it tells a story. Starting from familiarity with smooth manifolds and Riemannian geometry, it gradually explains what is different about complex manifolds and develops most of the main tools for working with them, using the Kodaira embedding theorem as a motivating project throughout. The approach and style will be familiar to readers of the author's previous graduate texts: new concepts are introduced gently, with as much intuition and motivation as possible, always relating new concepts to familiar old ones, with plenty of examples. The main prerequisite is familiarity with the basic results on topological, smooth, and Riemannian manifolds. The book is intended for graduate students and researchers in differential geometry, but it will also be appreciated by students of algebraic geometry who wish to understand the motivations, analogies, and analytic results that come from the world of differential geometry.