The Hauptvermutung Book

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The Hauptvermutung Book

A.A. Ranicki,A.J. Casson,D.P. Sullivan,M.A. Armstrong,C.P. Rourke,G.E. Cooke
The Hauptvermutung Book Book PDF

Author : A.A. Ranicki,A.J. Casson,D.P. Sullivan,M.A. Armstrong,C.P. Rourke,G.E. Cooke
Publisher : Springer Science & Business Media
Page : 192 pages
File Size : 50,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401733434

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The Hauptvermutung Book by A.A. Ranicki,A.J. Casson,D.P. Sullivan,M.A. Armstrong,C.P. Rourke,G.E. Cooke Pdf

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.

Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)

Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo
Proceedings Of The International Congress Of Mathematicians 2018 Icm 2018 In 4 Volumes Book PDF

Author : Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo
Publisher : World Scientific
Page : 5396 pages
File Size : 53,7 Mb
Release : 2019-02-27
Category : Mathematics
ISBN : 9789813272897

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Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) by Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo Pdf

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

Piecewise Linear Structures on Topological Manifolds

Yuli Rudyak
Piecewise Linear Structures on Topological Manifolds Book PDF

Author : Yuli Rudyak
Publisher : World Scientific
Page : 128 pages
File Size : 51,5 Mb
Release : 2015-12-28
Category : Mathematics
ISBN : 9789814733809

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Piecewise Linear Structures on Topological Manifolds by Yuli Rudyak Pdf

' 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. Contents: PrefaceIntroductionGraphArchitecture of the ProofNormal InvariantApplications and Consequences of the Main TheoremAppendix: Quinn''s Proof of Product Structure Theorem Readership: Researchers working in manifolds, algebraic topology, and K-theory. Key Features:First systematic treatment of the subjectNew treatment of certain topicsKeywords:Hauptvermutung;Triangulation Conjecture'

Foundational Essays on Topological Manifolds, Smoothings, and Triangulations

Robion C. Kirby,Laurence C. Siebenmann
Foundational Essays on Topological Manifolds Smoothings and Triangulations Book PDF

Author : Robion C. Kirby,Laurence C. Siebenmann
Publisher : Princeton University Press
Page : 376 pages
File Size : 55,7 Mb
Release : 1977-05-21
Category : Mathematics
ISBN : 0691081913

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Foundational Essays on Topological Manifolds, Smoothings, and Triangulations by Robion C. Kirby,Laurence C. Siebenmann Pdf

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.

The Wild World of 4-Manifolds

Alexandru Scorpan
The Wild World of 4 Manifolds Book PDF

Author : Alexandru Scorpan
Publisher : American Mathematical Society
Page : 614 pages
File Size : 51,9 Mb
Release : 2022-01-26
Category : Mathematics
ISBN : 9781470468613

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The Wild World of 4-Manifolds by Alexandru Scorpan Pdf

What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds. —MAA Reviews The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it. — Robion C. Kirby, University of California, Berkeley This book offers a panorama of the topology of simply connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. To put things in context, the book starts with a survey of higher dimensions and of topological 4-manifolds. In the second part, the main invariant of a 4-manifold—the intersection form—and its interaction with the topology of the manifold are investigated. In the third part, as an important source of examples, complex surfaces are reviewed. In the final fourth part of the book, gauge theory is presented; this differential-geometric method has brought to light how unwieldy smooth 4-manifolds truly are, and while bringing new insights, has raised more questions than answers. The structure of the book is modular, organized into a main track of about two hundred pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index.

Surveys on Surgery Theory (AM-149), Volume 2

Sylvain Cappell,Andrew Ranicki,Jonathan Rosenberg
Surveys on Surgery Theory AM 149 Volume 2 Book PDF

Author : Sylvain Cappell,Andrew Ranicki,Jonathan Rosenberg
Publisher : Princeton University Press
Page : 446 pages
File Size : 45,5 Mb
Release : 2014-09-08
Category : Mathematics
ISBN : 9781400865215

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Surveys on Surgery Theory (AM-149), Volume 2 by Sylvain Cappell,Andrew Ranicki,Jonathan Rosenberg Pdf

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject's founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as well as its current enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source surveying surgery theory and its applications. Because no one person could write such a survey, the editors asked a variety of experts to report on the areas of current interest. This is the second of two volumes resulting from that collective effort. It will be useful to topologists, to other interested researchers, and to advanced students. The topics covered include current applications of surgery, Wall's finiteness obstruction, algebraic surgery, automorphisms and embeddings of manifolds, surgery theoretic methods for the study of group actions and stratified spaces, metrics of positive scalar curvature, and surgery in dimension four. In addition to the editors, the contributors are S. Ferry, M. Weiss, B. Williams, T. Goodwillie, J. Klein, S. Weinberger, B. Hughes, S. Stolz, R. Kirby, L. Taylor, and F. Quinn.

Introduction to Topological Manifolds

John M. Lee
Introduction to Topological Manifolds Book PDF

Author : John M. Lee
Publisher : Springer Science & Business Media
Page : 395 pages
File Size : 47,9 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780387227276

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Introduction to Topological Manifolds by John M. Lee Pdf

Manifolds play an important role in topology, geometry, complex analysis, algebra, and classical mechanics. Learning manifolds differs from most other introductory mathematics in that the subject matter is often completely unfamiliar. This introduction guides readers by explaining the roles manifolds play in diverse branches of mathematics and physics. The book begins with the basics of general topology and gently moves to manifolds, the fundamental group, and covering spaces.

A Course on Surgery Theory

Stanley Chang,Shmuel Weinberger
A Course on Surgery Theory Book PDF

Author : Stanley Chang,Shmuel Weinberger
Publisher : Princeton University Press
Page : 472 pages
File Size : 44,9 Mb
Release : 2021-01-26
Category : Mathematics
ISBN : 9780691200354

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A Course on Surgery Theory by Stanley Chang,Shmuel Weinberger Pdf

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.

Surgery on Compact Manifolds

Charles Terence Clegg Wall,Andrew Ranicki
Surgery on Compact Manifolds Book PDF

Author : Charles Terence Clegg Wall,Andrew Ranicki
Publisher : American Mathematical Soc.
Page : 321 pages
File Size : 51,9 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821809426

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Surgery on Compact Manifolds by Charles Terence Clegg Wall,Andrew Ranicki Pdf

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.

Mod Two Homology and Cohomology

Jean-Claude Hausmann
Mod Two Homology and Cohomology Book PDF

Author : Jean-Claude Hausmann
Publisher : Springer
Page : 539 pages
File Size : 42,5 Mb
Release : 2015-01-08
Category : Mathematics
ISBN : 9783319093543

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Mod Two Homology and Cohomology by Jean-Claude Hausmann Pdf

Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool in algebraic topology. Compared to a more general approach to (co)homology this refreshing approach has many pedagogical advantages: 1. It leads more quickly to the essentials of the subject, 2. An absence of signs and orientation considerations simplifies the theory, 3. Computations and advanced applications can be presented at an earlier stage, 4. Simple geometrical interpretations of (co)chains. Mod 2 (co)homology was developed in the first quarter of the twentieth century as an alternative to integral homology, before both became particular cases of (co)homology with arbitrary coefficients. The first chapters of this book may serve as a basis for a graduate-level introductory course to (co)homology. Simplicial and singular mod 2 (co)homology are introduced, with their products and Steenrod squares, as well as equivariant cohomology. Classical applications include Brouwer's fixed point theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith theory, Kervaire invariant, etc. The cohomology of flag manifolds is treated in detail (without spectral sequences), including the relationship between Stiefel-Whitney classes and Schubert calculus. More recent developments are also covered, including topological complexity, face spaces, equivariant Morse theory, conjugation spaces, polygon spaces, amongst others. Each chapter ends with exercises, with some hints and answers at the end of the book.

Toric Topology

Victor M. Buchstaber,Taras E. Pano
Toric Topology Book PDF

Author : Victor M. Buchstaber,Taras E. Pano
Publisher : American Mathematical Soc.
Page : 518 pages
File Size : 49,7 Mb
Release : 2015-07-15
Category : Algebraic topology
ISBN : 9781470422141

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Toric Topology by Victor M. Buchstaber,Taras E. Pano Pdf

This book is about toric topology, a new area of mathematics that emerged at the end of the 1990s on the border of equivariant topology, algebraic and symplectic geometry, combinatorics, and commutative algebra. It has quickly grown into a very active area with many links to other areas of mathematics, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a class of manifolds with torus actions defined in combinatorial terms. Construction of moment-angle manifolds relates to combinatorial geometry and algebraic geometry of toric varieties via the notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to important connections with classical and modern areas of symplectic, Lagrangian, and non-Kaehler complex geometry. A related categorical construction of moment-angle complexes and polyhedral products provides for a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate subject of homotopy theory. A new perspective on torus actions has also contributed to the development of classical areas of algebraic topology, such as complex cobordism. This book includes many open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter this beautiful new area.

L2-Invariants: Theory and Applications to Geometry and K-Theory

Wolfgang Lück
L2 Invariants Theory and Applications to Geometry and K Theory Book PDF

Author : Wolfgang Lück
Publisher : Springer Science & Business Media
Page : 604 pages
File Size : 54,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662046876

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L2-Invariants: Theory and Applications to Geometry and K-Theory by Wolfgang Lück Pdf

In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.

Current Trends in Transformation Groups

Anthony Bak,Masaharu Morimoto,Fumihiro Ushitaki
Current Trends in Transformation Groups Book PDF

Author : Anthony Bak,Masaharu Morimoto,Fumihiro Ushitaki
Publisher : Springer Science & Business Media
Page : 272 pages
File Size : 41,6 Mb
Release : 2002-07-31
Category : Mathematics
ISBN : 1402007833

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Current Trends in Transformation Groups by Anthony Bak,Masaharu Morimoto,Fumihiro Ushitaki Pdf

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.

Surveys on Surgery Theory (AM-145), Volume 1

Sylvain Cappell,Andrew Ranicki,Jonathan Rosenberg
Surveys on Surgery Theory AM 145 Volume 1 Book PDF

Author : Sylvain Cappell,Andrew Ranicki,Jonathan Rosenberg
Publisher : Princeton University Press
Page : 448 pages
File Size : 48,9 Mb
Release : 2014-09-08
Category : Mathematics
ISBN : 9781400865192

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Surveys on Surgery Theory (AM-145), Volume 1 by Sylvain Cappell,Andrew Ranicki,Jonathan Rosenberg Pdf

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. There have been some extraordinary accomplishments in that time, which have led to enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source that surveys surgery theory and its applications. Indeed, no one person could write such a survey. The sixtieth birthday of C. T. C. Wall, one of the leaders of the founding generation of surgery theory, provided an opportunity to rectify the situation and produce a comprehensive book on the subject. Experts have written state-of-the-art reports that will be of broad interest to all those interested in topology, not only graduate students and mathematicians, but mathematical physicists as well. Contributors include J. Milnor, S. Novikov, W. Browder, T. Lance, E. Brown, M. Kreck, J. Klein, M. Davis, J. Davis, I. Hambleton, L. Taylor, C. Stark, E. Pedersen, W. Mio, J. Levine, K. Orr, J. Roe, J. Milgram, and C. Thomas.

Topology of Infinite-Dimensional Manifolds

Katsuro Sakai
Topology of Infinite Dimensional Manifolds Book PDF

Author : Katsuro Sakai
Publisher : Springer Nature
Page : 619 pages
File Size : 50,5 Mb
Release : 2020-11-21
Category : Mathematics
ISBN : 9789811575754

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Topology of Infinite-Dimensional Manifolds by Katsuro Sakai Pdf

An infinite-dimensional manifold is a topological manifold modeled on some infinite-dimensional homogeneous space called a model space. In this book, the following spaces are considered model spaces: Hilbert space (or non-separable Hilbert spaces), the Hilbert cube, dense subspaces of Hilbert spaces being universal spaces for absolute Borel spaces, the direct limit of Euclidean spaces, and the direct limit of Hilbert cubes (which is homeomorphic to the dual of a separable infinite-dimensional Banach space with bounded weak-star topology). This book is designed for graduate students to acquire knowledge of fundamental results on infinite-dimensional manifolds and their characterizations. To read and understand this book, some background is required even for senior graduate students in topology, but that background knowledge is minimized and is listed in the first chapter so that references can easily be found. Almost all necessary background information is found in Geometric Aspects of General Topology, the author's first book. Many kinds of hyperspaces and function spaces are investigated in various branches of mathematics, which are mostly infinite-dimensional. Among them, many examples of infinite-dimensional manifolds have been found. For researchers studying such objects, this book will be very helpful. As outstanding applications of Hilbert cube manifolds, the book contains proofs of the topological invariance of Whitehead torsion and Borsuk’s conjecture on the homotopy type of compact ANRs. This is also the first book that presents combinatorial ∞-manifolds, the infinite-dimensional version of combinatorial n-manifolds, and proofs of two remarkable results, that is, any triangulation of each manifold modeled on the direct limit of Euclidean spaces is a combinatorial ∞-manifold and the Hauptvermutung for them is true.