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The Hauptvermutung According To Sullivan
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Book Synopsis The Hauptvermutung Book by : Andrew Ranicki
Download or read book The Hauptvermutung Book written by Andrew Ranicki and published by Springer Science & Business Media. This book was released on 1996-09-30 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hauptvermutung is the conjecture that any two triangulations of a poly hedron are combinatorially equivalent. The conjecture was formulated at the turn of the century, and until its resolution was a central problem of topology. Initially, it was verified for low-dimensional polyhedra, and it might have been expected that furt her development of high-dimensional topology would lead to a verification in all dimensions. However, in 1961 Milnor constructed high-dimensional polyhedra with combinatorially inequivalent triangulations, disproving the Hauptvermutung in general. These polyhedra were not manifolds, leaving open the Hauptvermu tung for manifolds. The development of surgery theory led to the disproof of the high-dimensional manifold Hauptvermutung in the late 1960's. Unfortunately, the published record of the manifold Hauptvermutung has been incomplete, as was forcefully pointed out by Novikov in his lecture at the Browder 60th birthday conference held at Princeton in March 1994. This volume brings together the original 1967 papers of Casson and Sulli van, and the 1968/1972 'Princeton notes on the Hauptvermutung' of Armstrong, Rourke and Cooke, making this work physically accessible. These papers include several other results which have become part of the folklore but of which proofs have never been published. My own contribution is intended to serve as an intro duction to the Hauptvermutung, and also to give an account of some more recent developments in the area. In preparing the original papers for publication, only minimal changes of punctuation etc.
Book Synopsis The Hauptvermutung Book by : A.A. Ranicki
Download or read book The Hauptvermutung Book written by A.A. Ranicki and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hauptvermutung is the conjecture that any two triangulations of a poly hedron are combinatorially equivalent. The conjecture was formulated at the turn of the century, and until its resolution was a central problem of topology. Initially, it was verified for low-dimensional polyhedra, and it might have been expected that furt her development of high-dimensional topology would lead to a verification in all dimensions. However, in 1961 Milnor constructed high-dimensional polyhedra with combinatorially inequivalent triangulations, disproving the Hauptvermutung in general. These polyhedra were not manifolds, leaving open the Hauptvermu tung for manifolds. The development of surgery theory led to the disproof of the high-dimensional manifold Hauptvermutung in the late 1960's. Unfortunately, the published record of the manifold Hauptvermutung has been incomplete, as was forcefully pointed out by Novikov in his lecture at the Browder 60th birthday conference held at Princeton in March 1994. This volume brings together the original 1967 papers of Casson and Sulli van, and the 1968/1972 'Princeton notes on the Hauptvermutung' of Armstrong, Rourke and Cooke, making this work physically accessible. These papers include several other results which have become part of the folklore but of which proofs have never been published. My own contribution is intended to serve as an intro duction to the Hauptvermutung, and also to give an account of some more recent developments in the area. In preparing the original papers for publication, only minimal changes of punctuation etc.
Book Synopsis Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88), Volume 88 by : Robion C. Kirby
Download or read book Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88), Volume 88 written by Robion C. Kirby and published by Princeton University Press. This book was released on 2016-03-02 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since Poincaré's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.
Book Synopsis A Course on Surgery Theory by : Stanley Chang
Download or read book A Course on Surgery Theory written by Stanley Chang and published by Princeton University Press. This book was released on 2021-01-26 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: An advanced treatment of surgery theory for graduate students and researchers Surgery theory, a subfield of geometric topology, is the study of the classifications of manifolds. A Course on Surgery Theory offers a modern look at this important mathematical discipline and some of its applications. In this book, Stanley Chang and Shmuel Weinberger explain some of the triumphs of surgery theory during the past three decades, from both an algebraic and geometric point of view. They also provide an extensive treatment of basic ideas, main theorems, active applications, and recent literature. The authors methodically cover all aspects of surgery theory, connecting it to other relevant areas of mathematics, including geometry, homotopy theory, analysis, and algebra. Later chapters are self-contained, so readers can study them directly based on topic interest. Of significant use to high-dimensional topologists and researchers in noncommutative geometry and algebraic K-theory, A Course on Surgery Theory serves as an important resource for the mathematics community.
Book Synopsis Manifolds - Amsterdam 1970 by : N. H. Kuiper
Download or read book Manifolds - Amsterdam 1970 written by N. H. Kuiper and published by Springer. This book was released on 2006-11-15 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis New Developments in Topology by : John Frank Adams
Download or read book New Developments in Topology written by John Frank Adams and published by Cambridge University Press. This book was released on 1974-02-28 with total page 137 pages. Available in PDF, EPUB and Kindle. Book excerpt: Eleven of the fourteen invited speakers at a symposium held by the Oxford Mathematical Institute in June 1972 have revised their contributions and submitted them for publication in this volume. The present papers do not necessarily closely correspond with the original talks, as it was the intention of the volume editor to make this book of mathematical rather than historical interest. The contributions will be of value to workers in topology in universities and polytechnics.
Book Synopsis Induction Theorems for Groups of Homotopy Manifold Structures by : Andrew J. Nicas
Download or read book Induction Theorems for Groups of Homotopy Manifold Structures written by Andrew J. Nicas and published by American Mathematical Soc.. This book was released on 1982 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classifying spaces in surgery theory were first used by Sullivan and Casson in their (independent) unpublished work on the Hauptvermutung for PL manifolds. In his 1968 Ph.D. thesis, F. Quinn developed a general theory of surgery classifying spaces, realizing the Wall surgery groups as the homotopy groups [italic]L[subscript]*([italic]G) = [lowercase Greek]Pi[subscript]*([italic]L([italic]G)) of a spectrum of manifold n-ad surgery problems with fundamental group G. This work presents a detailed account of Quinn's theory. Geometric methods are used to view the Sullivan-Wall manifold structure sequence as an exact sequence of abelian groups (as suggested by Quinn). The intersection of the known induction theorems for generalized cohomology groups and [italic]L-groups then gives an induction theorem for the structure sequence with finite [italic]G.
Download or read book Topology I written by S.P. Novikov and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: This up-to-date survey of the whole field of topology is the flagship of the topology subseries of the Encyclopaedia. The book gives an overview of various subfields, beginning with the elements and proceeding right up to the present frontiers of research.
Book Synopsis Geometric Topology: Localization, Periodicity and Galois Symmetry by : Dennis P. Sullivan
Download or read book Geometric Topology: Localization, Periodicity and Galois Symmetry written by Dennis P. Sullivan and published by Springer. This book was released on 2009-09-03 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: The seminal ‘MIT notes’ of Dennis Sullivan were issued in June 1970 and were widely circulated at the time. The notes had a - jor in?uence on the development of both algebraic and geometric topology, pioneering the localization and completion of spaces in homotopy theory, including p-local, pro?nite and rational homotopy theory, le- ing to the solution of the Adams conjecture on the relationship between vector bundles and spherical ?brations, the formulation of the ‘Sullivan conjecture’ on the contractibility of the space of maps from the classifying space of a ?nite group to a ?nite dimensional CW complex, theactionoftheGalois groupoverQofthealgebraicclosureQof Q on smooth manifold structures in pro?nite homotopy theory, the K-theory orientation ofPL manifolds and bundles. Some of this material has been already published by Sullivan him- 1 self: in an article in the Proceedings of the 1970 Nice ICM, and in the 1974 Annals of Mathematics papers Genetics of homotopy theory and the Adams conjecture and The transversality character- 2 istic class and linking cycles in surgery theory . Many of the ideas originating in the notes have been the starting point of subsequent 1 reprinted at the end of this volume 2 joint with John Morgan vii viii 3 developments . However, the text itself retains a unique ?avour of its time, and of the range of Sullivan’s ideas.
Book Synopsis Algebraic Topology, Aarhus 1978 by : J. L. Dupont
Download or read book Algebraic Topology, Aarhus 1978 written by J. L. Dupont and published by Springer. This book was released on 2006-11-15 with total page 705 pages. Available in PDF, EPUB and Kindle. Book excerpt: University of Aarhus, 50. Anniversary, 11 September 1978
Book Synopsis Surgery with Coefficients by : Gerald A. Anderson
Download or read book Surgery with Coefficients written by Gerald A. Anderson and published by Springer. This book was released on 2006-11-14 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Stratified Polyhedra by : David A. Stone
Download or read book Stratified Polyhedra written by David A. Stone and published by Springer. This book was released on 2006-11-15 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: The original block bundle theory of Rourke and Sanderson was concerned with normal block bundles of submanifolds of a manifold. This treatise produces and examines, in some depth, a block bundle theory for a polyhedron in a polyhedron, in which the fibre is permitted to be an arbitrary cone rather than just a disc. The text begins with "Definitions'', followed by chapters establishing the existence and uniqueness of the new bundle structures. This is developed along lines now somewhat traditional amongst bundle-theorists, first pioneered by Thom in the smooth category. A block-transversality theorem is proved, pull-backs and classifying spaces are introduced, and there is a "Thom theorem'' linking cobordism theory with homotopy groups. Here the cobordism theory concerns all polyhedra with links of points in a certain fixed class. The text is a scholarly essay in piecewise linear topology of great generality.
Book Synopsis Surgery on Compact Manifolds by : Charles Terence Clegg Wall
Download or read book Surgery on Compact Manifolds written by Charles Terence Clegg Wall and published by American Mathematical Soc.. This book was released on 1999 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: The publication of this book in 1970 marked the culmination of a period in the history of the topology of manifolds. This edition, based on the original text, is supplemented by notes on subsequent developments and updated references and commentaries.
Book Synopsis Piecewise Linear Structures On Topological Manifolds by : Yuli Rudyak
Download or read book Piecewise Linear Structures On Topological Manifolds written by Yuli Rudyak and published by World Scientific. This book was released on 2015-12-28 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of triangulations of topological spaces has always been at the root of geometric topology. Among the most studied triangulations are piecewise linear triangulations of high-dimensional topological manifolds. Their study culminated in the late 1960s-early 1970s in a complete classification in the work of Kirby and Siebenmann. It is this classification that we discuss in this book, including the celebrated Hauptvermutung and Triangulation Conjecture.The goal of this book is to provide a readable and well-organized exposition of the subject, which would be suitable for advanced graduate students in topology. An exposition like this is currently lacking.
Book Synopsis Current Trends in Transformation Groups by : Anthony Bak
Download or read book Current Trends in Transformation Groups written by Anthony Bak and published by Springer Science & Business Media. This book was released on 2002-07-31 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an overview of some of the most active topics in the theory of transformation groups over the past decades and stresses advances obtained in the last dozen years. The emphasis is on actions of Lie groups on manifolds and CW complexes. Manifolds and actions of Lie groups on them are studied in the linear, semialgebraic, definable, analytic, smooth, and topological categories. Equivalent vector bundles play an important role. The work is divided into fifteen articles and will be of interest to anyone researching or studying transformations groups. The references make it easy to find details and original accounts of the topics surveyed, including tools and theories used in these accounts.
Book Synopsis Proceedings of the Second Conference on Compact Transformation Groups. University of Massachusetts, Amherst, 1971 by : H. T Ku
Download or read book Proceedings of the Second Conference on Compact Transformation Groups. University of Massachusetts, Amherst, 1971 written by H. T Ku and published by Springer. This book was released on 2006-11-15 with total page 465 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis High-dimensional Knot Theory by : Andrew Ranicki
Download or read book High-dimensional Knot Theory written by Andrew Ranicki and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 669 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bringing together many results previously scattered throughout the research literature into a single framework, this work concentrates on the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory.
