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^{2πi/3}) a computation of V_1(F) has been made based on our techniques.

29 pages, section on computational aspects changed (now in Appendix A), section 5 moved to Appendix B, introduction changed

References in corpus (3)

• Computing Borel's Regulator
• On special elements in higher algebraic K-theory and the Lichtenbaum-Gross Conjecture
• $p$-adic Cocycles and their Regulator Maps

Cited by in corpus (5)

• Computing Borel's Regulator
• $p$-adic Cocycles and their Regulator Maps
• Borel's rank theorem for Artin $L$-functions
• Volume and symplectic structure for l-adic local systems
• Secondary characteristic classes for subgroups of automorphism groups of free groups