Computing Borel's regulator

Abstract

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H2n+1(GL(double-struck C)). For a cyclotomic field F we define a canonical set of elements in K3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H3(GL(double-struck C)) under the Hurewicz map. Applying our formula to these images yields a value V1(F), which coincides with the Borel regulator R1(F) when our set is a basis of K3(F) modulo torsion. For F = ℚ(e2πi/3) a computation of V1(F) has been made based on our techniques.