Classical Topology and Combinatorial Group Theory

Author: N.A

Publisher: Springer Science & Business Media

ISBN: 1468401106

Category: Mathematics

Page: 301

View: 6172

In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does not understand the simplest topological facts, such as the reason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical develop ment where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recrea. ions like the seven bridges; rather, it resulted from the visualization of problems from other parts of mathematics complex analysis (Riemann), mechanics (poincare), and group theory (Oehn). It is these connections to other parts of mathematics which make topology an important as well as a beautiful subject.

Combinatorial Group Theory

A Topological Approach

Author: Daniel E. Cohen

Publisher: CUP Archive

ISBN: 9780521349369

Category: Mathematics

Page: 310

View: 2002

In this book the author aims to show the value of using topological methods in combinatorial group theory.

Combinatorial Group Theory and Applications to Geometry

Author: D.J. Collins,R.I. Grigorchuk,P.F. Kurchanov,H. Zieschang

Publisher: Springer Science & Business Media

ISBN: 9783540637042

Category: Mathematics

Page: 240

View: 3287

This book consists of two parts. The first part provides a comprehensive description of that part of group theory which has its roots in topology. The second more advanced part deals with recent work on groups relating to topological manifolds. It is a valuable guide to research in this field. The text contains numerous examples, sketches of proofs and open problems.

Topology and Combinatorial Group Theory

Proceedings of the Fall Foliage Topology Seminars Held in New Hampshire, 1986-1988

Author: Paul Latiolais

Publisher: Springer Verlag


Category: Mathematics

Page: 207

View: 4813


Author: Tammo Tom Dieck

Publisher: Walter de Gruyter

ISBN: 9783110162363

Category: Mathematics

Page: 454

View: 5813

Algorithmische Geometrie

Grundlagen, Methoden, Anwendungen

Author: Rolf Klein

Publisher: Springer-Verlag

ISBN: 354027619X

Category: Computers

Page: 392

View: 1106

Lehrbuch der Topologie

Author: Herbert Seifert,William Threlfall

Publisher: University of Pennsylvania Press

ISBN: 9780821835951

Category: Mathematics

Page: 353

View: 8729

The 1930s were important years in the development of modern topology, pushed forward by the appearance of a few pivotal books, of which this is one. The focus is on combinatorial and algebraic topology, with as much point-set topology as needed for the main topics. One sees from the modern point of view that the authors are working in a category of spaces that includes locally finite simplicial complexes. (Their definition of manifold is more properly known today as a ""triangulizable homology manifold"".)Amazingly, they manage to accomplish a lot without the convenient tools of homological algebra, such as exact sequences and commutative diagrams, that were developed later. The main topics covered are: simplicial homology (coefficients in $\mathbb{Z}$ or $\mathbb{Z}_2$), local homology, surface topology, the fundamental group and covering spaces, three-manifolds, Poincare duality, and the Lefschetz fixed point theorem. Few prerequisites are necessary. A final section reviews the lemmas and theorems from group theory that are needed in the text. As stated in the introduction to the important book by Alexandroff and Hopf (which appeared a year after ""Seifert and Threlfall""): 'Its lively and instructive presentation makes this book particularly suitable as an introduction or as a textbook.'

Algebraic Topology

A First Course

Author: William Fulton

Publisher: Springer Science & Business Media

ISBN: 1461241804

Category: Mathematics

Page: 430

View: 1353

To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups

Elements of Functional Analysis

Author: Francis Hirsch,Gilles Lacombe

Publisher: Springer Science & Business Media

ISBN: 9780387985244

Category: Mathematics

Page: 396

View: 6756

This book presents the fundamental function spaces and their duals, explores operator theory and finally develops the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems. Includes an assortment of well formulated exercises, with answers and hints collected at the end of the book.

Representations of Compact Lie Groups

Author: T. Bröcker,T.tom Dieck

Publisher: Springer Science & Business Media

ISBN: 9783540136781

Category: Mathematics

Page: 316

View: 1134

This introduction to the representation theory of compact Lie groups follows Herman Weyl’s original approach. It discusses all aspects of finite-dimensional Lie theory, consistently emphasizing the groups themselves. Thus, the presentation is more geometric and analytic than algebraic. It is a useful reference and a source of explicit computations. Each section contains a range of exercises, and 24 figures help illustrate geometric concepts.

Complexity: Knots, Colourings and Countings

Author: Dominic Welsh,David J. Welsh

Publisher: Cambridge University Press

ISBN: 9780521457408

Category: Mathematics

Page: 163

View: 9282

These notes are based on a series of lectures given at the Advanced Research Institute of Discrete Applied Mathematics, Rutgers University.

Algorithms -- ESA 2010, Part II

18th Annual European Symposium, Liverpool, UK, September 6-8, 2010, Proceedings

Author: Mark de Berg,Ulrich Meyer

Publisher: Springer Science & Business Media

ISBN: 3642157807

Category: Computers

Page: 245

View: 8610

This volume contains the 69 papers presented at the 16th Annual European Symposium on Algorithms (ESA 2010), held in Liverpool during September 6 8, 2010, including three papers by the distinguished invited speakers Artur Czumaj, Herbert Edelsbrunner, and Paolo Ferragina. ESA 2010 was organized as a part of ALGO 2010, which also included the 10th Workshop on Algorithms in Bioinformatics (WABI), the 8th Workshop on Approximation and Online Algorithms (WAOA), and the 10th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS). The European Symposium on Algorithms covers research in the design, use, andanalysisofe?cientalgorithmsanddata structures.As inpreviousyears, the symposium had two tracks: the Design and Analysis Track and the Engineering and Applications Track, each with its own Program Committee. In total 245 papers adhering to the submission guidelines were submitted. Each paper was reviewed by three or four referees. Based on the reviews and the often extensive electronicdiscussionsfollowingthem, thecommittees selected 66papersintotal: 56 (out of 206) to the Design and Analysis Track and 10 (out of 39) to the Engineering andApplicationstrack.We believethat thesepaperstogethermade up a strong and varied program, showing the depth and breadth of current algorithms research."

Wahrheit, Beweis, Unendlichkeit

Eine mathematische Reise zu den vielseitigen Auswirkungen der Unendlichkeit

Author: John Stillwell

Publisher: Springer-Verlag

ISBN: 3642378447

Category: Mathematics

Page: 236

View: 5559

In dem Buch erkundet der preisgekrönte Autor John Stillwell die Konsequenzen, die sich ergeben, wenn man die Unendlichkeit akzeptiert, und diese Konsequenzen sind vielseitig und überraschend. Der Leser benötigt nur wenig über die Schulmathematik hinausgehendes Hintergrundwissen; es reicht die Bereitschaft, sich mit ungewohnten Ideen auseinanderzusetzen. Stillwell führt den Leser sanft in die technischen Details von Mengenlehre und Logik ein, indem jedes Kapitel einem einzigen Gedankengang folgt, der mit einer natürlichen mathematischen Frage beginnt und dann anhand einer Abfolge von historischen Antworten nachvollzogen wird. Auf diese Weise zeigt der Autor, wie jede Antwort ihrerseits zu neuen Fragen führt, aus denen wiederum neue Begriffe und Sätze entstehen. Jedes Kapitel endet mit einem Abschnitt „Historischer Hintergrund“, der das Thema in den größeren Zusammenhang der Mathematik und ihrer Geschichte einordnet.

Combinatorial Methods in Topology and Algebraic Geometry

Author: John R. Harper,Richard Mandelbaum

Publisher: American Mathematical Soc.

ISBN: 9780821850398

Category: Mathematics

Page: 349

View: 617

This collection marks the recent resurgence of interest in combinatorial methods, resulting from their deep and diverse applications both in topology and algebraic geometry. Nearly thirty mathematicians met at the University of Rochester in 1982 to survey several of the areas where combinatorial methods are proving especially fruitful: topology and combinatorial group theory, knot theory, 3-manifolds, homotopy theory and infinite dimensional topology, and four manifolds and algebraic surfaces. This material is accessible to advanced graduate students with a general course in algebraic topology along with some work in combinatorial group theory and geometric topology, as well as to established mathematicians with interests in these areas.For both student and professional mathematicians, the book provides practical suggestions for research directions still to be explored, as well as the aesthetic pleasures of seeing the interplay between algebra and topology which is characteristic of this field. In several areas the book contains the first general exposition published on the subject. In topology, for example, the editors have included M. Cohen, W. Metzler and K. Sauerman's article on 'Collapses of $K\times I$ and group presentations' and Metzler's 'On the Andrews-Curtis-Conjecture and related problems'. In addition, J. M. Montesino has provided summary articles on both 3 and 4-manifolds.

Real and Abstract Analysis

A modern treatment of the theory of functions of a real variable

Author: Edwin Hewitt,Karl Stromberg

Publisher: Springer-Verlag

ISBN: 3662297949

Category: Mathematics

Page: 476

View: 7307