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Bing's radial engulfing theorem is proved in codimension three and improved in codimensions greater than three. Infinite engulfing and radial engulfing theorems are proved in codimensions three and greater. The report includes an exposition of basic engulfing results and applications. (Modified author abstract).
The Collected Papers of R.h. Bing by R. H. Bing Pdf
A powerful mathematician and a great problem solver, R. H. Bing laid the foundation for a number of areas of topology. Many of his papers have continued to serve as a source of major theoretical developments and concrete applications in recent years. One outstanding example was Michael H. Freedman's use of Bing's Shrinking Criterion to solve the four-dimensional Poincaré Conjecture. This two-volume set brings together over one hundred of Bing's research, expository, andmiscellaneous papers. These works range over a great variety of topics in topology, including the topology of manifolds, decomposition spaces, continua, metrization, general topology, and geometric topology. In addition, there are a number of papers in the areas of convex functions, linearity, and conformalvarieties. The introductory section in the first volume provides historical background on Bing's life and achievements. This collection will appeal to mathematicians in all areas, and especially those in topology, as well as students, historians, and educators in the mathematical sciences, for it provides a complete historical summary of the mathematical events in the life of the man and the mathematician, R. H. Bing.
Handbook of the History of General Topology by C.E. Aull,R. Lowen Pdf
This book is the first one of a work in several volumes, treating the history of the development of topology. The work contains papers which can be classified into 4 main areas. Thus there are contributions dealing with the life and work of individual topologists, with specific schools of topology, with research in topology in various countries, and with the development of topology in different periods. The work is not restricted to topology in the strictest sense but also deals with applications and generalisations in a broad sense. Thus it also treats, e.g., categorical topology, interactions with functional analysis, convergence spaces, and uniform spaces. Written by specialists in the field, it contains a wealth of information which is not available anywhere else.
Embeddings in Manifolds by Robert J. Daverman,Gerard Venema Pdf
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.
Topics in Low-Dimensional Topology by A Banyaga,H Movahedi-Lankarani,R Wells Pdf
Recent success with the four-dimensional Poincaré conjecture has revived interest in low-dimensional topology, especially the three-dimensional Poincaré conjecture and other aspects of the problems of classifying three-dimensional manifolds. These problems have a driving force, and have generated a great body of research, as well as insight. The main topics treated in this book include a paper by V Poenaru on the Poincaré conjecture and its ramifications, giving an insight into the herculean work of the author on the subject. Steve Armentrout's paper on “Bing's dogbone space” belongs to the topics in three-dimensional topology motivated by the Poincaré conjecture. S Singh gives a nice synthesis of Armentrout's work. Also included in the volume are shorter original papers, dealing with somewhat different aspects of geometry, and dedicated to Armentrout by his colleagues — Augustin Banyaga (and Jean-Pierre Ezin), David Hurtubise, Hossein Movahedi-Lankarani and Robert Wells. Contents: Mathematics of Steve Armentrout: A Review (S Singh)Bing's Dogbone Space Is Not Strongly Locally Simply Connected (S Armentrout)A Program for the Poincaré Conjecture and Some of Its Ramifications (V Poénaru)On the Foundation of Geometry, Analysis, and the Differentiable Structure for Manifolds (D Sullivan)A Conformal Invariant Characterizing the Sphere (A Banyaga & J-P Ezin)Spaces of Holomorphic Maps from ∀P1 to Complex Grassmann Manifolds (D E Hurtubise)Sets with Lie Isometry Groups (H Movahedi-Lankarani & R Wells) Readership: Researchers in mathematics and physics. Keywords:Poincare Conjecture;Topology;Holomorphic Maps;Complex Grassmann Manifolds;Lie Isometry Groups
The Collected Papers of Stephen Smale by F Cucker,R Wong Pdf
'0Keywords:Differential Topology;Dynamical Systems;Economic Theory;Theory of Computation;Global Analysis;Stephen Smale“The three-volume collected works of S Smale are a very welcome addition to every mathematician''s book shelf and a must for a mathematics department library.”Mathematical Reviews'
The Collected Papers of Stephen Smale by Stephen Smale,Felipe Cucker,Roderick Wong Pdf
This invaluable book contains the collected papers of Stephen Smale. These are divided into eight groups: topology; calculus of variations; dynamics; mechanics; economics; biology, electric circuits and mathematical programming; theory of computation; miscellaneous. In addition, each group contains one or two articles by world leaders on its subject which comment on the influence of Smale's work, and another article by Smale with his own retrospective views.