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 ChernWeil 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 ChernWeil 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 copublished with the Centre de Recherches Mathématiques. Readership Graduate students and research mathematicians interested in number theory. Reviews "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  Introduction
 Simplicial and cosimplicial objects
 \(H\)spaces and Hopf algebras
 The cohomology of the general linear group
 Lie algebra cohomology and the Weil algebra
 Group cohomology and the van Est isomorphism
 Small cosimplicial algebras
 Higher diagonals and differential forms
 Borel's regulator
 Beilinson's regulator
 Bibliography
 Index
