Computing Borel's Regulator
Abstract
We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL(\mathbb{C})). For a cyclotomic field F we define a canonical set of elements in K_3(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 H_{3}(GL(\mathbb{C})) under the Hurewicz map. Applying our formula to these images yields a value V_1(F), which coincides with the Borel regulator R_1(F) when our set is a basis of K_3(F) modulo torsion. For F= \mathbb{Q}(e^{2\pi i/3}) a computation of V_1(F) has been made based on our techniques.
Cite
@article{arxiv.0908.3765,
title = {Computing Borel's Regulator},
author = {Zacky Choo and Wajid Mannan and Rubén J. Sánchez-García and Victor P. Snaith},
journal= {arXiv preprint arXiv:0908.3765},
year = {2020}
}
Comments
29 pages, section on computational aspects changed (now in Appendix A), section 5 moved to Appendix B, introduction changed