Rational Homotopy Theory And Differential Forms

Author: Phillip Griffiths
Publisher: Springer Science & Business Media
ISBN: 1461484685
Size: 69.46 MB
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This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham’s theorem on simplicial complexes. In addition, Sullivan’s results on computing the rational homotopy type from forms is presented. New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma *Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theory With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.

Rational Homotopy Theory

Author: Yves Felix
Publisher: Springer Science & Business Media
ISBN: 146130105X
Size: 65.72 MB
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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.

Rational Homotopy Type

Author: Wen-tsün Wu
Publisher: Springer
ISBN: 3540390251
Size: 22.39 MB
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This comprehensive monograph provides a self-contained treatment of the theory of I*-measure, or Sullivan's rational homotopy theory, from a constructive point of view. It centers on the notion of calculability which is due to the author himself, as are the measure-theoretical and constructive points of view in rational homotopy. The I*-measure is shown to differ from other homology and homotopy measures in that it is calculable with respect to most of the important geometric constructions encountered in algebraic topology. This approach provides a new method of treatment and leads to various new results. In particular, an axiomatic system of I*-measure is formulated, quite different in spirit from the usual Eilenberg-Steenrod axiomatic system for homology, and giving at the same time an algorithmic method of computation of the I*-measure in concrete cases. The book will be of interest to researchers in rational homotopy theory and will provide them with new ideas and lines of research to develop further.

Rational Homotopical Models And Uniqueness

Author: Martin Majewski
Publisher: American Mathematical Soc.
ISBN: 0821819208
Size: 46.69 MB
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Abstract. The main goal of this paper is to prove the following conjecture of Baues and Lemaire: the differential graded Lie algebra associated with the Sullivan model of a space is homotopy equivalent to its Quillen model. In addition we show the same for the cellular Lie algebra model which we build from the simplicial analog of the classical Adams-Hilton model. It turns out that this cellular Lie algebra model is one link in a chain of models connecting the models of Quillen and Sullivan. The key result which makes all this possible is Anick's correspondence between differential graded Lie algebras and Hopf algebras up to homotopy. In addition we show that the Quillen model is a rational homotopical equivalence, and we conclude the same for the other models using our main result. The construction of the three models is given in detail. The background from homotopy theory, differential algebra, and algebra is presented in great generality.

Differential Forms In Algebraic Topology

Author: Raoul Bott
Publisher: Springer Science & Business Media
ISBN: 1475739516
Size: 55.98 MB
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Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.

Geometry Of Characteristic Classes

Author: Shigeyuki Morita
Publisher: American Mathematical Soc.
ISBN: 0821821393
Size: 22.25 MB
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Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined using connections on vector bundles, thus revealing their geometric side. The purpose of this book is to introduce the reader to the three theories of characteristic classes that were developed in the late 1960s. They include characteristic classes of flat bundles, characteristic classes of foliations, and characteristic classes of surface bundles. The book is intended for graduate students and research mathematicians working in various areas of geometry and topology.

Rational Homotopy Theory Ii

Author: Yves Félix
Publisher: World Scientific
ISBN: 9814651451
Size: 63.44 MB
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This research monograph is a detailed account with complete proofs of rational homotopy theory for general non-simply connected spaces, based on the minimal models introduced by Sullivan in his original seminal article. Much of the content consists of new results, including generalizations of known results in the simply connected case. The monograph also includes an expanded version of recently published results about the growth and structure of the rational homotopy groups of finite dimensional CW complexes, and concludes with a number of open questions. This monograph is a sequel to the book Rational Homotopy Theory [RHT], published by Springer in 2001, but is self-contained except only that some results from [RHT] are simply quoted without proof. Contents:Basic Definitions and ConstructionsHomotopy Lie Algebras and Sullivan Lie AlgebrasFibrations and Λ-ExtensionsHolonomyThe Model of the Fibre is the Fibre of the ModelLoop Spaces and Loop Space ActionsSullivan SpacesExamplesLusternik-Schnirelmann CategoryDepth of a Sullivan Algebra and of a Sullivan Lie AlgebraDepth of a Connected Graded Lie Algebra of Finite TypeTrichotomyExponential GrowthStructure of a Graded Lie Algebra of Finite DepthWeight Decompositions of a Sullivan Lie AlgebraProblems Readership: Researchers in algebraic topology and Lie algebra theory.Key Features:Contains the basis for using rational homotopy theory for non-simply connected spacesContains new important information on the rational homotopy Lie algebra of spacesIs at the frontier of the research in rational homotopyKeywords:Rational Homotopy Theory;Algebraic Topology;Malcev Completion;Graded Lie Algebra

Topology And Condensed Matter Physics

Author: Somendra Mohan Bhattacharjee
Publisher: Springer
ISBN: 9811068410
Size: 51.68 MB
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This book introduces aspects of topology and applications to problems in condensed matter physics. Basic topics in mathematics have been introduced in a form accessible to physicists, and the use of topology in quantum, statistical and solid state physics has been developed with an emphasis on pedagogy. The aim is to bridge the language barrier between physics and mathematics, as well as the different specializations in physics. Pitched at the level of a graduate student of physics, this book does not assume any additional knowledge of mathematics or physics. It is therefore suited for advanced postgraduate students as well. A collection of selected problems will help the reader learn the topics on one's own, and the broad range of topics covered will make the text a valuable resource for practising researchers in the field. The book consists of two parts: one corresponds to developing the necessary mathematics and the other discusses applications to physical problems. The section on mathematics is a quick, but more-or-less complete, review of topology. The focus is on explaining fundamental concepts rather than dwelling on details of proofs while retaining the mathematical flavour. There is an overview chapter at the beginning and a recapitulation chapter on group theory. The physics section starts with an introduction and then goes on to topics in quantum mechanics, statistical mechanics of polymers, knots, and vertex models, solid state physics, exotic excitations such as Dirac quasiparticles, Majorana modes, Abelian and non-Abelian anyons. Quantum spin liquids and quantum information-processing are also covered in some detail.