4-manifolds and Kirby Calculus

Author: Robert E. Gompf,András Stipsicz

Publisher: American Mathematical Soc.

ISBN: 0821809946

Category: Mathematics

Page: 558

View: 1925

Since the early 1980s, there has been an explosive growth in 4-manifold theory, particularly due to the influx of interest and ideas from gauge theory and algebraic geometry. This book offers an exposition of the subject from the topological point of view. It bridges the gap to other disciplines and presents classical but important topological techniques that have not previously appeared in the literature. Part I of the text presents the basics of the theory at the second-year graduate level and offers an overview of current research. Part II is devoted to an exposition of Kirby calculus, or handlebody theory on 4-manifolds. It is both elementary and comprehensive. Part III offers in-depth treatments of a broad range of topics from current 4-manifold research. Topics include branched coverings and the geography of complex surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. The authors present many important applications. The text is supplemented with over 300 illustrations and numerous exercises, with solutions given in the book. I greatly recommend this wonderful book to any researcher in 4-manifold topology for the novel ideas, techniques, constructions, and computations on the topic, presented in a very fascinating way. I think really that every student, mathematician, and researcher interested in 4-manifold topology, should own a copy of this beautiful book. --Zentralblatt MATH This book gives an excellent introduction into the theory of 4-manifolds and can be strongly recommended to beginners in this field ... carefully and clearly written; the authors have evidently paid great attention to the presentation of the material ... contains many really pretty and interesting examples and a great number of exercises; the final chapter is then devoted to solutions of some of these ... this type of presentation makes the subject more attractive and its study easier. --European Mathematical Society Newsletter

Perspectives in Analysis, Geometry, and Topology

On the Occasion of the 60th Birthday of Oleg Viro

Author: Ilia Itenberg,Burglind Jöricke,Mikael Passare

Publisher: Springer Science & Business Media

ISBN: 0817682775

Category: Mathematics

Page: 464

View: 9974

The articles in this volume are invited papers from the Marcus Wallenberg symposium and focus on research topics that bridge the gap between analysis, geometry, and topology. The encounters between these three fields are widespread and often provide impetus for major breakthroughs in applications. Topics include new developments in low dimensional topology related to invariants of links and three and four manifolds; Perelman's spectacular proof of the Poincare conjecture; and the recent advances made in algebraic, complex, symplectic, and tropical geometry.

4-Manifolds

Author: Selman Akbulut

Publisher: Oxford University Press

ISBN: 0198784864

Category:

Page: 280

View: 9694

This book presents the topology of smooth 4-manifolds in an intuitive self-contained way, developed over a number of years by Professor Akbulut. The text is aimed at graduate students and focuses on the teaching and learning of the subject, giving a direct approach to constructions and theorems which are supplemented by exercises to help the reader work through the details not covered in the proofs. The book contains a hundred colour illustrations to demonstrate the ideas rather than providing long-winded and potentially unclear explanations. Key results have been selected that relate to the material discussed and the author has provided examples of how to analyse them with the techniques developed in earlier chapters.

Symplectic 4-Manifolds and Algebraic Surfaces

Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 2-10, 2003

Author: Fabrizio Catanese,Denis Auroux,Gang Tian,Marco Manetti,Paul Seidel,Bernd Siebert,Ivan Smith

Publisher: Springer

ISBN: 3540782796

Category: Mathematics

Page: 354

View: 9643

Modern approaches to the study of symplectic 4-manifolds and algebraic surfaces combine a wide range of techniques and sources of inspiration. Gauge theory, symplectic geometry, pseudoholomorphic curves, singularity theory, moduli spaces, braid groups, monodromy, in addition to classical topology and algebraic geometry, combine to make this one of the most vibrant and active areas of research in mathematics. It is our hope that the five lectures of the present volume given at the C.I.M.E. Summer School held in Cetraro, Italy, September 2-10, 2003 will be useful to people working in related areas of mathematics and will become standard references on these topics. The volume is a coherent exposition of an active field of current research focusing on the introduction of new methods for the study of moduli spaces of complex structures on algebraic surfaces, and for the investigation of symplectic topology in dimension 4 and higher.

Deformations of Surface Singularities

Author: Andras Némethi,Agnes Szilárd

Publisher: Springer Science & Business Media

ISBN: 3642391311

Category: Mathematics

Page: 280

View: 5823

The present publication contains a special collection of research and review articles on deformations of surface singularities, that put together serve as an introductory survey of results and methods of the theory, as well as open problems and examples. The aim is to collect material that will help mathematicians already working or wishing to work in this area to deepen their insight and eliminate the technical barriers in this learning process. Additionally, we introduce some material which emphasizes the newly found relationship with the theory of Stein fillings and symplectic geometry. This links two main theories of mathematics: low dimensional topology and algebraic geometry.​ The theory of normal surface singularities is a distinguished part of analytic or algebraic geometry with several important results, its own technical machinery, and several open problems. Recently several connections were established with low dimensional topology, symplectic geometry and theory of Stein fillings. This created an intense mathematical activity with spectacular bridges between the two areas. The theory of deformation of singularities is the key object in these connections.

Geometry and Physics

XVI International Fall Workshop

Author: Rui Loja Fernandes,Roger Picken

Publisher: American Inst. of Physics

ISBN: 9780735405462

Category: Science

Page: 228

View: 4524

All papers have been peer-reviewed. The XVI International Fall Workshop on Geometry and Physics brought together geometers and physicists from within and outside the Iberian peninsula, to exchange ideas on how to describe and understand a variety of phenomena in areas such as mechanics or gravity.

Geometry & Topology

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Geometry

Page: N.A

View: 1074

Fully refereed international journal dealing with all aspects of geometry and topology and their applications.

Einführung in die Geometrie und Topologie

Author: Werner Ballmann

Publisher: Springer-Verlag

ISBN: 3034809018

Category: Mathematics

Page: 162

View: 387

Das Buch bietet eine Einführung in die Topologie, Differentialtopologie und Differentialgeometrie. Es basiert auf Manuskripten, die in verschiedenen Vorlesungszyklen erprobt wurden. Im ersten Kapitel werden grundlegende Begriffe und Resultate aus der mengentheoretischen Topologie bereitgestellt. Eine Ausnahme hiervon bildet der Jordansche Kurvensatz, der für Polygonzüge bewiesen wird und eine erste Idee davon vermitteln soll, welcher Art tiefere topologische Probleme sind. Im zweiten Kapitel werden Mannigfaltigkeiten und Liesche Gruppen eingeführt und an einer Reihe von Beispielen veranschaulicht. Diskutiert werden auch Tangential- und Vektorraumbündel, Differentiale, Vektorfelder und Liesche Klammern von Vektorfeldern. Weiter vertieft wird diese Diskussion im dritten Kapitel, in dem die de Rhamsche Kohomologie und das orientierte Integral eingeführt und der Brouwersche Fixpunktsatz, der Jordan-Brouwersche Zerlegungssatz und die Integralformel von Stokes bewiesen werden. Das abschließende vierte Kapitel ist den Grundlagen der Differentialgeometrie gewidmet. Entlang der Entwicklungslinien, die die Geometrie der Kurven und Untermannigfaltigkeiten in Euklidischen Räumen durchlaufen hat, werden Zusammenhänge und Krümmung, die zentralen Konzepte der Differentialgeometrie, diskutiert. Den Höhepunkt bilden die Gaussgleichungen, die Version des theorema egregium von Gauss für Untermannigfaltigkeiten beliebiger Dimension und Kodimension. Das Buch richtet sich in erster Linie an Mathematik- und Physikstudenten im zweiten und dritten Studienjahr und ist als Vorlage für ein- oder zweisemestrige Vorlesungen geeignet.

Dirac-Operatoren in der Riemannschen Geometrie

Mit einem Ausblick auf die Seiberg-Witten-Theorie

Author: Thomas Friedrich

Publisher: Springer-Verlag

ISBN: 3322803023

Category: Mathematics

Page: 207

View: 8053

Dieses Buch entstand nach einer einsemestrigen Vorlesung an der Humboldt-Universität Berlin im Studienjahr 1996/ 97 und ist eine Einführung in die Theorie der Spinoren und Dirac-Operatoren über Riemannschen Mannigfaltigkeiten. Vom Leser werden nur die grundlegenden Kenntnisse der Algebra und Geometrie im Umfang von zwei bis drei Jahren eines Mathematik- oder Physikstudiums erwartet. Ein Anhang gibt eine Einführung in das aktuelle Gebiet der Seiberg-Witten-Theorie.