Elements of Homotopy Theory

Author: George W. Whitehead

Publisher: Springer Science & Business Media

ISBN: 1461263182

Category: Mathematics

Page: 746

View: 8012

As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory. It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject. I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical. Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.

Nilpotente Gruppen und nilpotente Räume

Nachdiplomvorlesung gehalten am Mathematik-Departement ETH Zürich 1981/82

Author: P.J. Hilton

Publisher: Springer-Verlag

ISBN: 3540387919

Category: Mathematics

Page: 221

View: 8918

Advances in Homotopy Theory

Papers in Honour of I M James, Cortona 1988

Author: S. Salamon,B. Steer

Publisher: Cambridge University Press

ISBN: 9780521379076

Category: Mathematics

Page: 177

View: 6662

This is a collection of the recorded lectures given at a conference to celebrate the 60th birthday of Professor Ioan James, who has made significant contributions to homotopy theory, highlighting problems and initiating new methods. The present volume contains papers from internationally distinguished research workers that reflect recent exciting breakthroughs in this field. It will be important for all homotopy theorists as both a statement of current research and as a signpost for new directions. This should prove interesting to research workers in algebraic topology.

Rational Homotopy Theory

Author: Yves Felix,Stephen Halperin,J.-C. Thomas

Publisher: Springer Science & Business Media

ISBN: 146130105X

Category: Mathematics

Page: 539

View: 1196

Rational homotopy theory is a subfield of algebraic topology. Written by three authorities in the field, this book contains all the main theorems of the field with complete proofs. As both notation and techniques of rational homotopy theory have been considerably simplified, the book presents modern elementary proofs for many results that were proven ten or fifteen years ago.

Modern Classical Homotopy Theory

Author: Jeffrey Strom

Publisher: American Mathematical Soc.

ISBN: 0821852868

Category: Mathematics

Page: 835

View: 6431

The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

Minimal Resolutions Via Algebraic Discrete Morse Theory

Author: Michael Jöllenbeck,Volkmar Welker

Publisher: American Mathematical Soc.

ISBN: 0821842579

Category: Mathematics

Page: 74

View: 9389

Forman's discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoldberg, the authors show that this theory can be aplied to chain complexes of free modules over a ring and provide four applications of this theory.

Algebraic K-Theory and Its Applications

Author: Jonathan Rosenberg

Publisher: Springer Science & Business Media

ISBN: 1461243149

Category: Mathematics

Page: 394

View: 2378

Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.

Documenta Mathematica

Journal Der Deutschen Mathematiker-Vereinigung

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 8841

Introduction to Homotopy Theory

Author: Paul Selick

Publisher: American Mathematical Soc.

ISBN: 9780821844366

Category: Mathematics

Page: 188

View: 6802

This text is based on a one-semester graduate course taught by the author at The Fields Institute in fall 1995 as part of the homotopy theory program which constituted the Institute's major program that year. The intent of the course was to bring graduate students who had completed a first course in algebraic topology to the point where they could understand research lectures in homotopy theory and to prepare them for the other, more specialized graduate courses being held in conjunction with the program. The notes are divided into two parts: prerequisites and the course proper. Part I, the prerequisites, contains a review of material often taught in a first course in algebraic topology. It should provide a useful summary for students and non-specialists who are interested in learning the basics of algebraic topology. Included are some basic category theory, point set topology, the fundamental group, homological algebra, singular and cellular homology, and Poincare duality. Part II covers fibrations and cofibrations, Hurewicz and cellular approximation theorems, topics in classical homotopy theory, simplicial sets, fiber bundles, Hopf algebras, spectral sequences, localization, generalized homology and cohomology operations. This book collects in one place the material that a researcher in algebraic topology must know. The author has attempted to make this text a self-contained exposition. Precise statements and proofs are given of ``folk'' theorems which are difficult to find or do not exist in the literature.

Algebraic Topology

A First Course

Author: William Fulton

Publisher: Springer Science & Business Media

ISBN: 1461241804

Category: Mathematics

Page: 430

View: 8736

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

A Basic Course in Algebraic Topology

Author: William S. Massey

Publisher: Springer Science & Business Media

ISBN: 9780387974309

Category: Mathematics

Page: 428

View: 7440

This textbook is intended for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. The text consists of material from the first five chapters of the author's earlier book, Algebraic Topology; an Introduction (GTM 56) together with almost all of his book, Singular Homology Theory (GTM 70). The material from the two earlier books has been substantially revised, corrected, and brought up to date.

Field Theory

Author: Steven Roman

Publisher: Springer Science & Business Media

ISBN: 0387276785

Category: Mathematics

Page: 335

View: 5954

"Springer has just released the second edition of Steven Roman’s Field Theory, and it continues to be one of the best graduate-level introductions to the subject out there....Every section of the book has a number of good exercises that would make this book excellent to use either as a textbook or to learn the material on your own. All in all...a well-written expository account of a very exciting area in mathematics." --THE MAA MATHEMATICAL SCIENCES DIGITAL LIBRARY

Representations of Compact Lie Groups

Author: T. Bröcker,T.tom Dieck

Publisher: Springer Science & Business Media

ISBN: 9783540136781

Category: Mathematics

Page: 316

View: 7689

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.