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.
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.