Introduction to Homotopy Theory

Author: Paul Selick

Publisher: American Mathematical Soc.

ISBN: 9780821844366

Category: Mathematics

Page: 188

View: 2633

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.

Modern Classical Homotopy Theory

Author: Jeffrey Strom

Publisher: American Mathematical Soc.

ISBN: 0821852868

Category: Mathematics

Page: 835

View: 5766

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.

Toric Topology

International Conference, May 29-June 3, 2006, Osaka City University, Osaka, Japan

Author: Megumi Harada

Publisher: American Mathematical Soc.

ISBN: 0821844865

Category: Mathematics

Page: 401

View: 2587

Toric topology is the study of algebraic, differential, symplectic-geometric, combinatorial, and homotopy-theoretic aspects of a particular class of torus actions whose quotients are highly structured. The combinatorial properties of this quotient and the equivariant topology of the original manifold interact in a rich variety of ways, thus illuminating subtle aspects of both the combinatorics and the equivariant topology. Many of the motivations and guiding principles of the field are provided by (though not limited to) the theory of toric varieties in algebraic geometry as well as that of symplectic toric manifolds in symplectic geometry. This volume is the proceedings of the International Conference on Toric Topology held in Osaka in May-June 2006. It contains about 25 research and survey articles written by conference speakers, covering many different aspects of, and approaches to, torus actions, such as those mentioned above.Some of the manuscripts are survey articles, intended to give a broad overview of an aspect of the subject; all manuscripts consciously aim to be accessible to a broad reading audience of students and researchers interested in the interaction of the subjects involved. We hope that this volume serves as an enticing invitation to this emerging field.

An SO(3)-Monopole Cobordism Formula Relating Donaldson and Seiberg-Witten Invariants

Author: Paul Feehan,Thomas G. Leness

Publisher: American Mathematical Soc.

ISBN: 147041421X

Category: Cobordism theory

Page: 228

View: 6339

The authors prove an analogue of the Kotschick–Morgan Conjecture in the context of monopoles, obtaining a formula relating the Donaldson and Seiberg–Witten invariants of smooth four-manifolds using the -monopole cobordism. The main technical difficulty in the -monopole program relating the Seiberg–Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible monopoles, namely the moduli spaces of Seiberg–Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of monopoles. In this monograph, the authors prove—modulo a gluing theorem which is an extension of their earlier work—that these intersection pairings can be expressed in terms of topological data and Seiberg–Witten invariants of the four-manifold. Their proofs that the -monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze and Witten's Conjecture in full generality for all closed, oriented, smooth four-manifolds with and odd appear in earlier works.

Algebraic Topology

Author: Tammo tom Dieck

Publisher: European Mathematical Society

ISBN: 9783037190487

Category: Mathematics

Page: 567

View: 7773

This book is written as a textbook on algebraic topology. Prerequisites are standard point set topology (as recalled in the first chapter), elementary algebraic notions (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (masters) students to basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included.

Homotopy Theory of Function Spaces and Related Topics

Author: Yves Félix,Gregory Lupton,Samuel B. Smith

Publisher: American Mathematical Soc.

ISBN: 0821849298

Category: Mathematics

Page: 230

View: 7338

This volume contains the proceedings of the Workshop on Homotopy Theory of Function Spaces and Related Topics, which was held at the Mathematisches Forschungsinstitut Oberwolfach, in Germany, from April 5-11, 2009. This volume contains fourteen original research articles covering a broad range of topics that include: localization and rational homotopy theory, evaluation subgroups, free loop spaces, Whitehead products, spaces of algebraic maps, gauge groups, loop groups, operads, and string topology. In addition to reporting on various topics in the area, this volume is supposed to facilitate the exchange of ideas within Homotopy Theory of Function Spaces, and promote crossfertilization between Homotopy Theory of Function Spaces and other areas. With these latter aims in mind, this volume includes a survey article which, with its extensive bibliography, should help bring researchers and graduate students up to speed on activity in this field as well as a problems list, which is an expanded and edited version of problems discussed in sessions held at the conference. The problems list is intended to suggest directions for future work.

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 7976

C*-algebras by Example

Author: Kenneth R. Davidson

Publisher: American Mathematical Soc.

ISBN: 9780821871898

Category: Mathematics

Page: 309

View: 8399

The subject of C*-algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*-algebras and Elliott's use of K-theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*-algebras. This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*-algebras. The development of operator algebras in the last twenty years has been based on a careful study of these special classes. While there are many books on C*-algebras and operator algebras available, this is the first one to attempt to explain the real examples that researchers use to test their hypotheses. Topic include AF algebras, Bunce-Deddens and Cuntz algebras, the Toeplitz algebra, irrational rotation algebras, group C*-algebras, discrete crossed products, abelian C*-algebras (spectral theory and approximate unitary equivalence) and extensions. It also introduces many modern concepts and results in the subject such as real rank zero algebras, topological stable rank, quasidiagonality, and various new constructions. These notes were compiled during the author's participation in the special year on C*-algebras at the Fields Institute of Mathematics during the 1994-1995 academic year. The field of C*-algebras touches upon many other areas of mathematics such as group representations, dynamical systems, physics, K-theory, and topology. The variety of examples offered in this text expose the student to many of these connections. A graduate student with a solid course in functional analysis should be able to read this book. This should prepare them to read much of the current literature. This book is reasonably self-contained, and the author has provided results from other areas when necessary.

Introduction to Symplectic Topology

Author: Dusa McDuff,Dietmar Salamon

Publisher: Oxford University Press

ISBN: 0198794894

Category: Mathematics

Page: 632

View: 9067

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

Tensor Categories

Author: Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik

Publisher: American Mathematical Soc.

ISBN: 1470420244

Category: Algebraic topology

Page: 344

View: 9525

Is there a vector space whose dimension is the golden ratio? Of course not--the golden ratio is not an integer! But this can happen for generalizations of vector spaces--objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

Subject Guide to Books in Print

An Index to the Publishers' Trade List Annual

Author: N.A

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 2101

Scientific American

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Technology

Page: N.A

View: 6602

Algebra für Einsteiger

Von der Gleichungsauflösung zur Galois-Theorie

Author: Jörg Bewersdorff

Publisher: Springer-Verlag

ISBN: 3658022620

Category: Mathematics

Page: 214

View: 6706

Dieses Buch ist eine leicht verständliche Einführung in die Algebra, die den historischen und konkreten Aspekt in den Vordergrund rückt. Der rote Faden ist eines der klassischen und fundamentalen Probleme der Algebra: Nachdem im 16. Jahrhundert allgemeine Lösungsformeln für Gleichungen dritten und vierten Grades gefunden wurden, schlugen entsprechende Bemühungen für Gleichungen fünften Grades fehl. Nach fast dreihundertjähriger Suche führte dies schließlich zur Begründung der so genannten Galois-Theorie: Mit ihrer Hilfe kann festgestellt werden, ob eine Gleichung mittels geschachtelter Wurzelausdrücke lösbar ist. Das Buch liefert eine gute Motivation für die moderne Galois-Theorie, die den Studierenden oft so abstrakt und schwer erscheint. In dieser Auflage wurde ein Kapitel ergänzt, in dem ein alternativer, auf Emil Artin zurückgehender Beweis des Hauptsatzes der Galois-Theorie wiedergegeben wird. Dieses Kapitel kann fast unabhängig von den anderen Kapiteln gelesen werden.

Lecture Notes on Motivic Cohomology

Author: Carlo Mazza,Vladimir Voevodsky,Charles A. Weibel

Publisher: American Mathematical Soc.

ISBN: 9780821838471

Category: Mathematics

Page: 216

View: 6776

The notion of a motive is an elusive one, like its namesake 'the motif' of Cezanne's impressionist method of painting. This title includes lectures that correspond to one-hour lectures given by the author during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999-2000.