# Computing Borel's Regulator

Zacky Choo, Wajid Mannan, Rub\'en J. S\'anchez-Garc\'ia, Victor P.
Snaith

Arxiv ID: 0908.3765•Last 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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