**CRM Monograph Series**

Volume: 15;
2002;
104 pp;
Hardcover

MSC: Primary 19;
Secondary 14

**Print ISBN: 978-0-8218-2630-0
Product Code: CRMM/15**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-3860-9
Product Code: CRMM/15.E**

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

# The Regulators of Beilinson and Borel

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*José I. Burgos Gil*

A co-publication of the AMS and Centre de Recherches Mathématiques

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, Hopf algebras, 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 the Chern-Weil morphism
and the van Est isomorphisms, a discussion about small cosimplicial algebras,
and a comparison of different definitions of Borel's regulator.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students and research mathematicians interested in number theory.

#### Reviews & Endorsements

Contains a lot of expository material, … The monograph is extremely valuable, not only in settling the question but in doing so in a readable way.

-- Mathematical Reviews

… an excellent background source for graduate students.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## The Regulators of Beilinson and Borel

- Cover Cover11
- Dedication v6
- Title page vii8
- Contents ix10
- Acknowledgments xi12
- Introduction 114
- Simplicial and cosimplicial objects 518
- 𝐻-spaces and Hopf algebras 1528
- The cohomology of the general linear group 2336
- Lie algebra cohomology and the Weil algebra 3548
- Group cohomology and the van Est isomorphism 5164
- Small cosimplicial algebras 5770
- Higher diagonals and differential forms 6578
- Borel’s regulator 7588
- Beilinson’s regulator 89102
- Bibliography 99112
- Index 103116
- Back Cover Back Cover1118