This publication incorporates a whole evidence of the truth that Borel's regulator map is two times Beilinson's regulator map. the tactic of the evidence follows the argument sketched in Beilinson's unique paper and depends upon very comparable descriptions of the Chern-Weil morphisms and the van Est isomorphism. The booklet has diversified components. the 1st one studies the fabric from algebraic topology and Lie staff concept wanted for the comparability theorem. themes reminiscent of simplicial items, Hopf algebras, attribute periods, the Weil algebra, Bott's Periodicity theorem, Lie algebra cohomology, non-stop team cohomology and the van Est Theorem are mentioned. the second one half comprises the comparability theorem and the categorical fabric wanted in its facts, akin to particular descriptions of the Chern-Weil morphism and the van Est isomorphisms, a dialogue approximately small cosimplicial algebras, and a comparability of other definitions of Borel's regulator.
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