Rational homotopy theory and differential forms Phillip Griffiths, John Morgan

Format:

Book

Author/Creator:

Griffiths, Phillip, 1938- author.

Morgan, John, 1946 March 21- author.

Series:

Progress in mathematics (Boston, Mass.) volume 16

Progress in mathematics volume 16

Language:

English

Subjects (All):

Homotopy theory.

Differential forms.

Algebra.

algebra.

Physical Description:

1 online resource

Edition:

Second edition

Place of Publication:

New York Birkhäuser 2013

Language Note:

English

System Details:

PDF

text file

Summary:

Rational homotopy theory is today one of the major trends in algebraic topology. Despite the great progress made in only a few years, a textbook properly devoted to this subject still was lacking until now The appearance of the text in book form is highly welcome, since it will satisfy the need of many interested people. Moreover, it contains an approach and point of view that do not appear explicitly in the current literature. Zentralblatt MATH (Review of First Edition) The monograph is intended as an introduction to the theory of minimal models. Anyone who wishes to learn about the theory will find this book a very helpful and enlightening one. There are plenty of examples, illustrations, diagrams and exercises. The material is developed with patience and clarity. Efforts are made to avoid generalities and technicalities that may distract the reader or obscure the main theme. The theory and its power are elegantly presented. This is an excellent monograph. Bulletin of the American Mathematical Society (Review of First Edition) 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 Thams theorem on simplical complexes. In addition, Sullivans results on computing the rational homotopy type from forms is presented

Contents:

Basic Concepts CW Homology Theorem The Whitehead Theorem and the Hurewicz Theorem Spectral Sequence of a Fibration Obstruction Theory Eilenberg-MacLane Spaces, Cohomology, and Principal Fibrations Postnikov Towers and Rational Homotopy Theory deRham's Theorem for Simplicial Complexes Differential Graded Algebras Homotopy Theory of DGAs DGAs and Rational Homotopy Theory The Fundamental Group Examples and Computations Functorality The Hirsch Lemma Quillen's Work on Rational Homotopy Theory A [infinity] -Structures and C [infinity] -Structures Exercises

Machine generated contents note: 2.1. CW Complexes

2.2. First Notions from Homotopy Theory

2.3. Homology

2.4. Categories and Functors

3.1. Statement

3.2. Proof

3.3. Examples

4.1. Definitions and Elementary Properties of Homotopy Groups

4.2. Whitehead Theorem

4.3. Completion of the Computation of πn(Sn)

4.4. Hurewicz Theorem

4.5. Corollaries of the Hurewicz Theorem

4.6. Homotopy Theory of a Fibration

4.7. Applications of the Exact Homotopy Sequence

5.1. Introduction

5.2. Fibrations- over a Cell

5.3. Generalities on Spectral Sequences

5.4. Leray-Serre Spectral Sequence of a Fibration

5.5. Examples

6.1. Introduction

6.2. Definition and Properties of the Obstruction Cocycle

6.3. Further Properties

6.4. Obstruction to the Existence of a Section of a Fibration

6.5. Examples

7.1. Relation of Cohomology and Eilenberg-MacLane Spaces

7.2. Principal K(π, n)-Fibrations

8.1. Rational Homotopy Theory for Simply Connected Spaces

8.2. Construction of the Localization of a Space

9.1. Piecewise Linear Forms

9.2. Lemmas About Piecewise Linear Forms

9.3. Naturality Under Subdivision

9.4. Multiplicativity of the deRham Isomorphism

9.5. Connection with the Cinfinity deRham Theorem

9.6. Generalizations of the Construction

10.1. Introduction

10.2. Hirsch Extensions

10.3. Relative Cohomology

10.4. Construction of the Minimal Model

11.1. Homotopies

11.2. Obstruction Theory

11.3. Applications of Obstruction Theory

11.4. Uniqueness of the Minimal Model

12.1. Transgression in the Serre Spectral Sequence and the Duality

12.2. Hirsch Extensions and Principal Fibrations

12.3. Minimal Models and Postnikov Towers

12.4. Minimal Model of the deRham Complex

13.1. 1-Minimal Models

13.2. π1 Q

13.3. Functorality

13.4. Examples

14.1. Spheres and Projective Spaces

14.2. Graded Lie Algebras

14.3. Borromean Rings

14.4. Symmetric Spaces and Formality

14.5. Third Homotopy Group of a Simply Connected Space

14.6. Homotopy Theory of Certain 4-Dimensional Complexes

14.7. Q-Homotopy Type of BUn and Un

14.8. Products

14.9. Massey Products

15.1. Functorial Correspondence

15.2. Bijectivity of Homotopy Classes of Maps

15.3. Equivalence of Categories

16.1. Cubical Complex and Cubical Forms

16.2. Hirsch Extensions and Spectral Sequences

16.3. Polynomial Forms for a Serre Fibration

16.4. Serre Spectral Sequence for Polynomial Forms

16.5. Proof of Theorem 12.1

17.1. Differential Graded Lie Algebras

17.2. Differential Graded Co-algebras

17.3. Bar Construction

17.4. Relationship Between Quillen's Construction and Sullivan's

17.5. Quillen's Construction

18.1. Operads, Rooted Trees, and Stasheff's Associahedron

18.2. Ainfinity-Algebras and Ainfinity-Categories

18.3. Cinfinity-Algebras and DGAs

Notes:

Includes bibliographical references

Print version record

Other Format:

Print version Griffiths, Phillip, 1938- Rational homotopy theory and differential forms.

ISBN:

9781461484684

1461484685

1461484677

9781461484677

OCLC:

861184041

Access Restriction:

Restricted for use by site license