Computing Borels Regulator

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.

💡 Research Summary

The paper “Computing Borel’s Regulator” presents two major advances that turn the abstract theory of Borel regulators and higher algebraic K‑theory into concrete, computable objects. The first advance is an infinite‑series expression for the universal Borel class bₙ∈H^{2n+1}(GL(ℂ),ℝ) derived from the Karoubi‑Hamida integral. Traditionally, the Borel class is given by a complicated integral involving matrix logarithms; the authors rewrite this integral as a convergent series whose terms are explicit traces of powers of the Maurer–Cartan form A⁻¹dA. After a suitable normalization of a matrix A∈GL(ℂ), the series takes the form
  Σ_{k≥0} c_k Tr