The regulators of Beilinson and Borel / Jose I. Burgos Gil.

Format:

Book

Author/Creator:

Burgos Gil, José I. (José Ignacio), 1962- author.

Series:

CRM monograph series ; Volume 15.

CRM Monograph Series ; Volume 15

Language:

English

Subjects (All):

Regulators (Mathematics).

Physical Description:

1 online resource (105 pages) : illustrations.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2002.

Summary:

This book contains a complete proof of the fact that Borel's regulator map is twice Beilinson's regulator map. The strategy of the proof follows the argument sketched in Beilinson's original paper and relies on very similar descriptions of the Chern-Weil morphisms and the van Est isomorphism. The book has two different parts. The first one reviews the material from algebraic topology and Lie group theory needed for the comparison theorem. Topics such as simplicial objects, Hopfalgebras, characteristic classes, the Weil algebra, Bott's Periodicity theorem, Lie algebra cohomology, continuous group cohomology and the van Est Theorem are discussed. The second part contains the comparison theorem and the specific material needed in its proof, such as explicit descriptions of theChern-Weil morphism and the van Est isomorphisms, a discussion about small cosimplicial algebras, and a comparison of different definitions of Borel's regulator.

Contents:

Machine generated contents note: Chapter 1. Introduction

Chapter 2. Simplicial and Cosimplicial Objects

2.1. Basic Definitions and Examples

2.2. Simplicial Abelian Groups

2.3. The Geometric Realization

2.4. Sheaves on Simplicial Topological Spaces

2.5. Principal Bundles on Simplicial Manifolds

2.6. The de Rham Algebra of a Simplicial Manifold

Chapter 3. H-Spaces and Hopf Algebras

3.1. Definitions

3.2. Some Examples

3.3. The Structure of Hopf Algebras

3.4. Rational Homotopy of H-Spaces

Chapter 4. The Cohomology of the General Linear Group

4.1. The General Linear Group and the Stiefel Manifolds

4.2. Classifying Spaces and Characteristic Classes

4.3. The Suspension

4.4. The Stability of Homology and Cohomology

4.5. The Stable Homotopy of the General Linear Group

4.6. Other Consequences of Bott's Periodicity Theorem

Chapter 5. Lie Algebra Cohomology and the Weil Algebra

5.1. de Rham Cohomology of a Lie Group

5.2. Reductive Lie Algebras

5.3. Characteristic Classes in de Rham Cohomology

5.4. The Suspension in the Weil Algebra

5.5. Relative Lie Algebra Cohomology

Chapter 6. Group Cohomology and the van Est Isomorphism

6.1. Group Homology and Cohomology

6.2. Continuous Group Cohomology

6.3. Computation of Continuous Cohomology

Chapter 7. Small Cosimplicial Algebras

7.1. Cosimplicial Algebras

7.2. Small Algebras

Chapter 8. Higher Diagonals and Differential Forms

8.1. The Sheaf of Differential Forms

8.2. The Weil Algebra Revisited

8.3. A Description of the van Est Isomorphism

Chapter 9. Borel's Regulator

9.1. Algebraic K-Theory of Rings

9.2. Definition of Borel's Regulator

9.3. The Rank of the Groups Km(o)

9.4. The Values of the Zeta Functions

9.5. A Renormalization of Borel's Regulator

9.6. Borel Elements

9.7. Explicit Representatives of the Borel Element

Chapter 10. Beilinson's Regulator

10.1. Deligne-Beilinson Cohomology

10.2. Deligne-Beilinson Cohomology of B.GLn(C)

10.3. The Definition of Beilinson's Regulator

10.4. The Comparison Between the Regulators.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-3860-7