Abstract
We present an infinite series formula based on the Karoubi–Hamida integral, for the universal Borel class evaluated on H2n+1(GL(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(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.
Funding source: EPSRC
Award Identifier / Grant number: EP/C549074/1
We would like to warmly thank Herbert Gangl for his helpful advice and detailed comments on a previous version of our paper. We would also like to thank MathOverflow user `Ralph' for answering a question about the Hurewicz homomorphism, Rob Snocken and Christopher Voll for explaining various aspects of Dedekind zeta functions to us, Bernhard Koeck for useful discussions on K-theory, Matthias Flach for his comment on the orders of some K-theory groups and the Algebra and Number Theory Group at University College Dublin, in particular Robert Osburn, for their invitation to discuss this work. Finally we would like to thank EPSRC for funding this research, Sajeda Mannan for facilitating a research visit, and the universities of Sheffield and Southampton for supporting this work.
© 2015 by De Gruyter
