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Computing Borel's Regulator

Zacky Choo, Wajid Mannan, Rub\'en J. S\'anchez-Garc\'ia, Victor P. Snaith
Arxiv ID: 0908.3765Last updated: 8/5/2020
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.

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